Rubik’s cube has a natural extension to fourdimensional space. This article constructs the basic concepts of the puzzle and implements it in a program. The wellknown threedimensional Rubik’s cube consists of 27 unit subcubes. Each face of determines a set of nine subcubes that have a face in the same plane as . The set can be rotated around the normal through the center of . Rubik’s 4cube (or 4D hypercube) consists of 81 unit 4subcubes, each containing eight 3D subcubes. Each 3face of determines a set of 27 4subcubes that have a cube in the same hyperplane as . The set can be rotated around the normal (a plane) through the center of . Projecting the whole 4D configuration to 3D exhibits Rubik’s 4cube as a fourdimensional extension of Rubik’s cube. Starting from a random coloring of the 4cube, the goal of the puzzle is to return to the initial coloring of the 3faces.
To understand the 4D hypercube, it helps to first see how its lowerdimensional analogs relate to each other. The zerodimensional hypercube (or 0cube) is a point, with one vertex. The 1D hypercube (or 1cube) is a segment, with two vertices and one edge. The 2D hypercube (or 2cube) is a square, with four vertices, four edges and one face (the square including its interior). The 3D hypercube is a cube (or 3cube), with eight vertices, edges, six square faces and one volume. Going up a dimension doubles the number of vertices. More generally, the number of cubes (points, segments, squares, …) in an cube, , is .
The 3D cube can be represented in a 2D plane using central projection, defined by taking the intersection of the plane with the line joining the two points and . This projection maps the point to . Choose to obtain the projection shown on the right in
Figure 1. Five of the faces overlap with a sixth face, the price to pay for the loss of one dimension.
Figure 1. A cube and its image under a central projection.
Overall, the 4D Rubik puzzle is a 4cube [1] (or 4D hypercube or tesseract), with vertices, edges, squares, eight cubes and one 4cube. The eight cubes are called cells, which are like the six square faces of a 3D cube. The proper faces of the 4cube are its vertices, edges, squares and cells.
Each point of a proper face is on the 3D hypersurface of the 4cube. No point of a proper face is strictly in the interior of the 4cube; that is, a hypersphere at such a point contains points inside and points outside the 4cube. In particular, no interior point of a cell as a 3D object is in the interior of the hypercube; all the points of a cell are on the boundary of the 4cube.
The 16 vertices of a 4cube can be defined as lists of length four of all possible combinations of
and .
The 24 squares of the 4cube are described in terms of their vertex indices.
Besides the 4cube, there are five other regular polytopes in four dimensions. The .csv and .m files containing information for these polytopes are provided at [2]: the positions of the vertices, vertex indices for the proper faces and which faces are neighbors.
To display the 4cube in 3D, central projection from 4D to 3D is analogous to central projection from 3D to 2D; the function is the natural extension of ; see Figure 2.
Figure 2. Projected image of a 4cube by means of center projection. The larger outer cube is one of the cells of the 4cube.
An axis of rotation in 3D is a fixed line. In 4D, an axis of rotation in four dimensions is a fixed plane [3]. For example, the rotation matrix about the  plane is defined by:
(1) 
There are six planes of rotation spanned by pairs of coordinate axes, namely , , , , , .
Here is the first one, for example, which leaves points in the  plane fixed.
This animation shows two successive rotations of the 4cube projected to 3D.
Consider a 4cube with center at the origin , side length 3, and with all proper faces of positive dimension parallel to the coordinate axes. Then its 16 vertices are:
(2) 
The eight cells of the initial 4cube are colored differently. The word “initial” means that no rotations have been applied. The coloring touches every point of a cell, including its 3D interior points.
Just as the faces of Rubik’s cube are divided into nine squares by dividing each edge into three, the edges of Rubik’s 4cube are also divided into three. Then the initial 4cube is divided into small 4cubes, each with edge length 1. The boundary (a hypersurface) of a small 4cube contains eight small cubes, its cells.
The Rubik 3cube has 27 subcubes in ; no square of the center cube is colored and some squares of other cubes are colored. These 26 subcubes are classified into three types according to whether they are at a corner, at an edge or at the center of a face of the larger cube. Figure 3 shows one of each type.
Figure 3. Three types of small cubes: in the center of a square face, at an edge and at a vertex, with one, two or three colored squares.
Analogously, the 81 small 4cubes of include the uncolored one at the center and 80 partially colored small 4cubes. These are classified into four types according to the dimension of their intersection with . The type of a small 4cube does not change after rotation. Table 1 summarizes the numbers for each type for and .
The number of colored small squares for Rubik’s cube is calculated using the data in Table 1:
Another way is to count the number of faces times the number of squares per face: .
The small 4cubes with nonzero coordinates form the hypersurface of . In particular, a small 4cube with center given by four nonzero coordinates contains a vertex of . Again from Table 1, the number of colored small cells is:
This number can also be obtained as the number of cells of times the number of small cells per cell of : .
We define several global variables to be used here and later. Figure 4 shows the divided 4cube with 216 colored small cells.
Figure 4. Center projection of a hypercube consisting of 216 colored small cells.
Each edge is divided into three parts, so that the length of the 4subcubes is 1. Consider a 4subcube with center at . Then, the vertices of the 4subcube are . The coordinates of the center of each 4subcube are a combination of one of , 0 and . When the value is nonzero ( or ), the 4subcubes face outward in the corresponding directions. In other words, the nonzero values in coordinates denote the outwardfacing 4subcubes.
For the initial state, the colors of the cells are set according to the coordinates of their centers:
For example, in the small 4cube with center , the two small cells with vertices and are colored because both the and coordinate values are nonzero.
The geometry of the 216 small colored cells is used to manage the puzzle. Each element of the datasets consists of four elements: (1) the vertices of six squares; (2) the location of the center of the small 4cube to which the small cell belongs; (3) color; and (4) the location of the center of the small cell. The vertices of the six squares are used for drawing the subcubes, and the locations of the centers of the subcubes are used to judge the completeness of the puzzle. The dataset of the initial state is obtained by the following procedures. First, the vertex numbers of the squares making up each small cell are defined.
Next, the 216 small cells are selected by checking all possible small cells.
This list contains 216 entries and each entry contains four components corresponding to a small cell.
For example, here is entry 123 of . The components for this small cell are its six square faces, center, color and current position.
sets up a 4cube for drawing.
Here is an example.
sets up a cell (with 27 4cubes) for drawing.
Figure 5 shows the initial state of Rubik’s 4cube.
Figure 5. Center projection of initial state of the Rubik 4cube with its eight cells.
In the 3D case of Rubik’s cube , a block is a set of nine small cubes whose centers have one coordinate that is constant, , 0 or 1. There are nine blocks, three per coordinate axis. A natural technique to rotate a middle block, for example, the one cut by the plane by , is to rotate the block above by , the block below by , and then the whole cube by .
In the 4D case, a block is a set of 27 small 4cubes whose centers have one coordinate that is constant. There are 12 blocks, four per axis and three per choice of constant coordinate , 0 or 1. Under a rotation, the small 4cubes in a block change position simultaneously. Each block is a fourdimensional hyperprism with height 1.
Figure 6 shows an example of the block ; the cell opposite the orange cell is not colored.
Figure 6. Example of a block of 27 small 4cubes (orange, ).
A block is rotated by , or around an axis, which is a fixed plane. Therefore, the information needed for an action on the Rubik 4cube is (1) the block to be rotated; (2) the axis of rotation; and (3) the angle. For Rubik’s cube , the axis of rotation is automatically determined by selecting a block. But for , two coordinate axes must be chosen to determine the fixed plane. One is the constant coordinate axis used to select the block, and the other must be chosen from the remaining three coordinate axes. There are 108 possible actions on : 12 choices of block, three choices for the second coordinate axis and three choices of angle: . Therefore, 108 buttons are required for the rotations in the Rubik 4cube computer program. Table 2 lists the properties of Rubik’s cube and Rubik’s 4cube.
The program to realize Rubik’s 4cube in 3D relies on central projection of a hypercube and rotation matrices in 4D. The program is shown in the next section.
Implementing an interface consists of three parts: constructing the buttons for the rotations, displaying the current state and judging whether the puzzle is complete.
The buttons for the rotations are placed in grids. The player can rotate a block by clicking one of the buttons. The rows correspond to the selection of the axis of the coordinates of the block and the columns correspond to the coordinate values for that axis. The player can select a block by choosing one of the rows and one of the columns. For example, clicking the button where row crosses column chooses the block on . For each block, the other three axes are listed. Then, the selection of the second axis is required to verify the rotational plane. Finally, one of the three buttons (up, diagonal and down) must be chosen to determine the rotation angle of (▲), (■) and (▼). (The 0 rows can be ignored—the player can perform an equivalent pair of actions instead in the parallel blocks.)
When the colors of the 27 subcubes on a cell are all the same, that cell is complete. The puzzle is solved when all the cells are complete.
Although we succeeded in implementing Rubik’s 4cube, some problems remain to be addressed. We aimed for ease of implementation rather than efficiency. Therefore, in the future, we should consider enhancing the application to get a more effective visualization method and an intuitive interface.
The program redraws 1,296 squares after each rotation, so that efficient coding is important. There is a great deal of redundancy in calculating the vertices of the 4subcubes for each rotation. The most effective method for handling the vertices using subcubes or 4subcubes remains to be clarified. Note that we must transfer the vertices of the 4subcubes to those of the subcubes when we handle the vertices of 4subcubes as a dataset rather than handling them as subcubes.
Effective visualization is a common problem for fourdimensional geometry. In this article, we used central projection to represent 4cubes. However, the proposed projection does not completely represent the features of the puzzle. Although there are other projections for representing a 4cube, the most suitable method is not yet clear.
Another possible improvement would be to animate the rotation. The animation of the rotation of the colored small 4cubes would help the player intuitively understand their rearrangement.
An intuitive interface is important for playing this puzzle. The interface and visualization issues are related, and their development may provide a new method for understanding fourdimensional space.
[1]  H. S. M. Coxeter, Introduction to Geometry, 2nd ed., Hoboken: Wiley, 1989. 
[2]  T. Yoshino. “Activities of Dr. Takashi Yoshino.” (Dec 11, 2017) takashiyoshino.randomwalk.org/basicdataof4dregularpolytopes. 
[3]  K. Miyazaki, M. Ishii and S. Yamaguchi, Science of HigherDimensional Shape and Symmetry, Kyoto: Kyoto University Press, 2005 (In Japanese). 
T. Yoshino, “Rubik’s 4Cube,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.198. 
Profession: Science of Form. Fields of interest: skeletal structure of plankton, nonEuclidean geometry, hyperspace, pattern formation.
Takashi Yoshino
Toyo University,
Kujirai 2100, Kawagoe, 3508585, JAPAN
tyoshino@toyo.jp
Given a finite vertex set, one can construct every connected spanning hypergraph by first choosing a spanning hypertree, then choosing a blob on each of its edges.
If is a finite vertex set and is a collection of finite subsets (called edges), none of which is a subset of another, we recursively define the swell of , , to be the collection of all sets that either:
For example, if
then
If we also have , then the set system is called a clutter. This condition means that (except in the case ) each edge contains at least two vertices, and the hypergraph spanned by the edge set is connected. Here the hypergraph spanned by a set of edges is defined to have vertex set and edge set . (There is no agreedupon definition of “hypergraph.” For some authors it is any set system; for others it is a simplicial complex; for others it is an antichain of sets.)
Here is a larger clutter.
The number of clutters spanning vertices is given by A048143 (oeis.org/A048143),
This sequence varies as , so the number of digits required roughly doubles with each consecutive term. Our main example is just one of some 56 sextillion members of .
This normalizing function is a universal invariant for the species of labeled clutters, meaning two clutters are isomorphic iff they have the same image.
Here is a list of nonisomorphic representatives for all clutters with up to four vertices, corresponding to “unlabeled” clutters. This bruteforce enumeration may not work for .
A kernel of is a clutter (the restriction of to edges that are subsets of ) for some .
Define .
A set partition is a set of disjoint sets with .
Suppose is a set partition of such that each block is a kernel of (i.e. and ). Since would imply , it follows that the set of unions is itself a clutter, which we call a cap of .
Equivalently, a cap of is a clutter satisfying both:
To see that this does not establish a partial order of clutters with a vertex set, observe that
is a nontransitive chain of caps. The following is the set of all set partitions of the edge set indices corresponding to each cap of the clutter.
In these plots of clutter partitions, the filled squares correspond to all pairs of a vertex and an edge such that the vertex belongs to the edge; these squares are then shaded according to which block of the partition the edge belongs to.
The density of a clutter is
where the sum is over all edges .
A clutter with two or more edges is a tree iff . This is equivalent to the usual definition of a spanning hypertree [1].
A clutter is a blob iff no cap of is a tree.
The trees and blobs among the caps and kernels (respectively) of our running example are as follows.
Suppose a clutter decomposes into a cap and corresponding set of kernels . Then
where the sum is over all . In particular, , and iff every is a tree. Using this simple identity, one easily proves the following.
Lemma
The following is also straightforward.
Proposition
We now come to the main result.
For our running example, the theorem corresponds to the following decomposition into a tree of blobs.
Theorem
Proof
First we show that any blob (kernel) is contained within a single branch of any tree (cap). Suppose that is a kernel of and is a blob, and that is a cap of that is a tree. Let be the subtree of contributing to the set partition of nonempty intersections for each branch . The set of unions forms a clutter that is obtained from by deleting in turn all vertices not in , a process that weakly decreases density. Let be the set partition comprised of maximal kernels (i.e. connected components) contained in blocks of . Then is a cap of and . Since is a connected clutter, we have , and therefore . But since is a blob, cannot be a tree, hence it must be a maximal cap (viz. , ).
Next we show that . If any two blobs overlap, both blobs must be contained entirely in whatever branch (of any given tree) contains their intersection. This implies that there is another blob containing their union, and hence that the maximal blobs are disjoint. Since every singleton is also a blob, we conclude that is a cap of .
Finally, if any kernel of were a blob, so would be the restriction of to its union, contradicting maximality of . This proves that the set of unions of is a tree. ■
The following are the decompositions for each nonisomorphic clutter with four vertices.
Let be the set of all kernels of . If is itself a (connected) clutter with vertex set , then there exists a unique subsetminimal upper bound satisfying both
In general, we can only define uniquely for a connected set of kernels, so is not strictly a join operation for the poset of subsets . But if is not connected as a clutter, then letting be its (maximal) connected components, we say that is a connected set of kernels iff is connected as a set of kernels, in which case the join is given by
In practice, the verification of connectedness and the computation of may require several iterations constructing joins of connected components. For example, consider the connected set of kernels.
It has the following sequence of joins of connected components.
One Problem
If is a connected set of kernels, we define its compression to be the number of iterations in the computation of by constructing consecutive joins of connected components. For the previous example, we have . Although it seems unlikely that is a bounded invariant, we do not know how to construct an example with compression greater than .
Define an invariant by
for all , where the sum is over all clutter partitions . Here denotes the indicator function for a proposition , equal to 1 or 0 depending on whether is true or false, respectively.
Theorem
Proof
Let be the set of all clutter partitions . What we have essentially shown above is that , regarded as a subposet of the lattice of set partitions ordered by refinement, is a lattice. We have the simple enumerative identity
where the product is over all nonsingleton kernels , here regarded as elements of whose only nonsingleton block is . Expanding the righthand side gives
where the sum is over all sets of nonsingleton kernels , again regarded as lattice elements. Here is algorithmically the same operation as the connectedjoin operation on . Expanding and factoring accordingly, this becomes
where the outer sum is over all , the product is over all , and the inner sum is over all connected sets of kernels spanning . For any kernel , define
where the outer sum is over all clutter partitions , and where the product and inner sum are as before. Letting be the set partition of whose only nonsingleton block is , we have shown that
Hence our theorized expansion does indeed satisfy the defining identity of . ■
Note that it is sufficient in the preceding theorem and proof to consider only connected sets of subsetminimal nonsingleton kernels, and it is often practical to do so. The hypergraph whose edges are minimal nonsingleton kernels is also of some interest. The wellknown Möbius function of a hypergraph is defined on the lattice of connected set partitions, and in this context an element of may be called a pseudokernel. In comparison, however, our invariant , which is defined on essentially all clutters, seems to be more interesting; we do not know if it has been studied before.
A semiclutter is any antichain of subsets . For each finite set , let be the set of semiclutters spanning . A species [2] is an endofunctor on the category of finite sets and bijections, so here we have defined a species of semiclutters. The compound semiclutter of a decomposition , as defined by Billera [3], is obtained as a disjoint “sum” of Cartesian “products.” Interpreted in the language of species theory, this is a certain natural transformation
where denotes the composition operation on species, a generalization of composition of exponential formal power series. Let be the set of (connected) clutters spanning , let be the set containing only the maximal clutter on , and let be the set of clutters having no expression as a compound of a proper decomposition (i.e. is the species of “prime” clutters). Billera’s main theorem (attributed to Shapley) establishes a unique reduced compound representation, which is itself a species of decompositions
From this it is evident that can ultimately be reduced to a nested compound expression using only trivial and prime clutters. Hence the problem of enumerating semiclutters on a vertex set is reduced to the problem of constructing, for any connected clutter, its “maximal proper committees,” which is the nontrivial solution of [2] for the enumeration of prime clutters considered. This is a particularly interesting application of formal species.
[1]  R. Bacher, “On the Enumeration of Labelled Hypertrees and of Labelled Bipartite Trees.” arxiv.org/abs/1102.2708. 
[2]  A. Joyal, “Une théorie combinatoire des séries formelles,” Advances in Mathematics, 42(1), 1981 pp. 1–82. doi:10.1016/00018708(81)900529. 
[3]  L. J. Billera, “On the Composition and Decomposition of Clutters,” Journal of Combinatorial Theory, Series B, 11(3), 1971 pp. 234–245. doi:10.1016/00958956(71)900335. 
G. Wiseman, “Every Clutter Is a Tree of Blobs,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.197. 
The author is a former graduate student of pure mathematics at the University of California, Davis. He is interested in categorical technology applied to discrete mathematics.
Gus Wiseman
gus@nafindix.com
We present a Mathematica implementation of an algorithm for computing new closedform evaluations for classes of triglogarithmic and hyperboliclogarithmic definite integrals based on the substitution of logarithmic functions into the Maclaurin series expansions of trigonometric and hyperbolic functions. Using this algorithm, we offer new closedform evaluations for a variety of triglogarithmic integrals that stateoftheart computer algebra systems cannot evaluate directly. We also show how this algorithm may be used to evaluate interesting infinite series and products.
Although there are many wellknown techniques for the symbolic evaluation of definite integrals, such as Slater’s convolution method, there are many open issues concerning symbolic definite integration using computer algebra systems [1–4]. Computing a closedform expression for a definite integral often easily reduces to the evaluation of the corresponding indefinite integral, but there are many natural definite integrals of elementary functions that cannot be directly evaluated following that procedure [5]. This article considers the general problem of the symbolic computation of triglogarithmic and hyperboliclogarithmic definite integrals. Integrals of this form are interesting in part because they often have surprising, simple and elegant closedform evaluations involving special functions such as the gamma function and the generalized Riemann zeta function. They may also be used to evaluate infinite series:
In this article, we are primarily concerned with the evaluation of integrals involving an expression of the form or , where is a complex number and is a variable. Integrals of this form emerge in a natural context within physics and engineering, since the Cauchy–Euler differential equation yields the basic solutions and . For example, there are applications based on integrals involving these basic solutions related to the Einstein–Barber field equations [6, 7].
Integrals with an integrand involving a trigonometric function composed with a logarithmic function as a factor are also interesting because there are many integrals of this form that cannot be directly evaluated by stateoftheart computer algebra systems. For example, consider the definite integral . Mathematica 11.2 is not able to directly evaluate this definite integral nor the underlying indefinite integral.
Here is a numerical approximation.
A natural way to evaluate the symbolic definite integral would be to expand the integrand by substituting into the Maclaurin series for the sine function and then use Mathematica to evaluate the corresponding infinite series.
Evaluate the expression using Mathematica. This integral would occur in the termbyterm expansion.
The Maclaurin series substitution technique applied to this integral gives the series . Here is a numerical approximation.
Mathematica is able to evaluate the series involving the Riemann zeta function, leading to the elegant closedform evaluation for the definite integral that we have previously noted [8].
In this article, we generalize the strategy illustrated using a Mathematica program for a function that extends the builtin Wolfram Language function with respect to definite integrals so as to be able to find new evaluations for a large variety of triglogarithmic and hyperboliclogarithmic integrals that Mathematica 11.2 is not able to directly evaluate otherwise. The underlying algorithm of the function is based on the substitution of logarithmic functions into the Maclaurin series expansions of certain trigonometric/hyperbolic expressions within numerators of integrands, as illustrated in the preceding example. Since Mathematica is able to evaluate a large variety of integrals involving powers of logarithmic functions, the function is able to evaluate a large variety of interesting and natural definite integrals. Also, the function may be used to prove new evaluations for interesting infinite series:
We begin by offering a variety of illustrations of applications of this algorithm. The integration results given in this article are new in the sense that Mathematica 11.2 is not able to evaluate the definite integrals. The function is documented in the Mathematica package corresponding to this article.
In Section 4, we discuss some nuances concerning . In Section 5, we summarize the main capabilities of this function. In Section 6, we discuss some avenues for future research related to the function.
In this article, Mathematica 11.2 was used to generate results; earlier versions may be too slow.
Given an integral involving an expression of the form , where is a complex number, the function generalizes the strategy outlined in Section 1 to attempt to evaluate the corresponding integral. The function uses the Maclaurin series for the sine function, substitutes a logarithmic function into this power series and integrates it term by term. The resulting infinite series often involves the generalized Riemann zeta function. For example, consider the definite integral .
The function is also able to compute new evaluations of integrals involving expressions such as .
is also able to compute new evaluations of integrals involving a factor such as .
Integration results such as those in Section 2 may be used to prove interesting infinite series formulas using this identity:
(1) 
This elegant infinite series formula is easily verified using equation (1) together with the function:
(2) 
To prove it, use (1) with the Maclaurin series expansion of the expression within the integrand of . It is known that [9], and several elegant proofs of this formula are given in [9]. Similar formulas, such as the following new infinite series formulas, may be proven similarly.
Proposition 1
Proof
Begin by evaluating .
Expand the factor in the integrand as a Maclaurin series, . The desired result follows by integrating both sides of this equality using the evaluation and simplifying the resultant summand. □
Proposition 2
Proof
First evaluate .
We have that . Integrating both sides of this equality using the evaluation may be used to finish the proof. □
Proposition 3
Proof
Evaluate .
To evaluate the preceding infinite series, expand the factor within the integrand, integrate the resultant summand and use the evaluation of the integral. □
Proposition 4
Proof
Evaluate .
Expand the factor and follow the technique used in the previous proofs.
The function is able to evaluate a variety of definite integrals involving expressions of the form , where is a complex number.
The evaluation of this triglogarithmic integral involves Catalan’s constant.
This one involves the Glaisher–Kinkelin constant.
Here .
We prove the following wellknown infinite product formula following the strategy used in Section 2.1: evaluate , substitute the Maclaurin series for into this integral and integrate term by term.
Similarly, we prove the following new infinite series formulas.
Proposition 5
Proof
First evaluate .
Expanding in the preceding integrand,
Integrating both sides of this equation using the integral evaluation and simplifying yields the desired result. □
Proposition 6
Proof
Evaluate .
To evaluate the series, expand in the integrand, integrate the resultant summand and simplify, and use the integral evaluation. □
Proposition 7
Proof
Evaluate .
The series may then be evaluated like the others. □
As discussed in Section 1, the function is based on the substitution of logarithmic functions into Maclaurin series. Our implementation is in many ways based on the use of string manipulation through Wolfram Language string operations such as . Given an integrand as input, the function converts the numerator of to a string and determines whether the numerator of contains trigonometric, hyperbolic or logarithmic expressions. If contains a suitable combination of functional expressions, such as a sine function and a logarithmic function, uses string manipulation to determine whether Mathematica 11.2 is able to evaluate a definite integral involving logarithmic powers that is needed to evaluate an infinite series, following the technique described in Section 1. If Mathematica 11.2 is able to compute a closedform evaluation for this logpower integral, then the function uses such an evaluation to evaluate the infinite series.
The function is an extension of the function in terms of definite integrals, in the sense that if the function is able to evaluate a definite integral, then returns the same evaluation. However, does not extend with respect to indefinite integrals. Also, there are certain classes of definite integrals that neither the function nor can evaluate (as an infinite series or otherwise).
The function is able to directly compute new evaluations for a variety of integrals of the following forms that cannot be directly computed by stateoftheart computer algebra systems (here , , , etc. denote complex numbers).
where
We have shown how may be used to construct new evaluations of a variety of interesting infinite series, such as series involving the inverse tangent function. is also able to directly compute closedform evaluations of definite integrals involving products of triglogarithmic and hyperboliclogarithmic expressions.
We currently leave it as an open problem to generalize . There are many natural ways to generalize it. Many definite integrals have an integrand with a factor consisting of the composition of two elementary transcendental functions that may be evaluated using Maclaurin series, but that cannot be directly evaluated using stateoftheart computer algebra systems. For example, Mathematica 11.2 is not able to compute the following integral.
Using the Maclaurin series for the square of the inverse sine, it is easily seen that
Implementing a generalization of this strategy to evaluate using Mathematica may be difficult, since Mathematica 11.2 is not able to compute integrals such as .
We currently leave it as an open problem to construct an analog of the function that is able to directly evaluate infinite series such as the summation evaluated in Section 2.
A natural way to generalize the function would be to construct a program for directly evaluating more general classes of logarithmicpower integrals. For example, these logarithmicpower integrals have simple closedform evaluations, but Mathematica 11.2 is not able to directly evaluate general formulas for these integrals.
The function is designed to evaluate definite integrals with an integrand with a factor in the form of a triglogarithmic or hyperboliclogarithmic function. However, it is clear that the general strategy outlined in the Section 1 may be applied more generally. That is, this technique may be applied to certain types of definite integrals with factors different than or . We currently leave it as an open problem to generalize the function so as to be able to apply this Maclaurin series substitution technique to integrands without such factors.
The author would like to thank a reviewer for many useful comments and suggestions.
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[8]  N. J. A. Sloane, “The OnLine Encyclopedia of Integer Sequences,” sequence A265011. oeis.org. 
[9]  Namagiri, “A Closed Form for the Infinite Series .” Mathematics Stack Exchange. (Apr 30, 2014) math.stackexchange.com/questions/776182. 
J. M. Campbell, “An Algorithm for TrigonometricLogarithmic Definite Integrals,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.196. 
Additional electronic files:
Available at: www.mathematicajournal.com/data/uploads/2017/10/TIntegratePackage.m
John M. Campbell received an M. Math. degree from the University of Waterloo and graduated first class with distinction with a Specialized Honours Bachelor of Science degree from York University. He has worked as a research assistant at the Fields Institute for Research in Mathematical Sciences and at York University.
John M. Campbell
York University
4700 Keele Street
Toronto, ON M3J 1P3
jmaxwellcampbell@gmail.com
Rice–Rampsberger–Kassel–Marcus (RRKM) theory calculates an energydependent microcanonical unimolecular rate constant for a chemical reaction from a sum and density of vibrational quantum states. This article demonstrates how to program the Beyer–Swinehart direct count of the sum and density of states for harmonic oscillators, as well as the Stein–Rabinovitch extension for anharmonic oscillators. Microcanonical rate constants are calculated for the decomposition of vinyl cyanide () into , and as an example.
The essential framework for our understanding of unimolecular reaction rates was developed largely in the first half of the twentieth century [1]. The Lindemann–Hinshelwood mechanism, proposed in 1922 by Lindemann and later expanded upon by Hinshelwood [2], posited that the gas phase unimolecular reactions were initiated by bimolecular collisions. This allowed the rate of the reaction to be calculated as a function of internal energy, regardless of the method of activation. Later that decade, Rice and Ramsperger and, independently, Kassel developed an improvement on the model known as the Rice–Ramsperger–Kassel (RRK) model [3, 4]. The RRK model viewed the molecule as a system of identical harmonic oscillators and introduced the idea of activation energy: that sufficient energy must be deposited into specific modes of motion in order for the reaction to occur. In 1952, Rice and Marcus extended this into what is now known as Rice–Rampsberger–Kassel–Marcus theory (RRKM), which incorporates a more complete quantum mechanical description of the molecule and utilizes the concept of the transition state, a specific conformation that the molecule must adopt in order for the reaction to proceed [5]. The RRKM model remains the prevailing scientific description of unimolecular chemical kinetics to this day.
Chemical transformations require bonds to break in the reactants and new bonds to form as products are generated. The potential energy associated with these transformations defines an dimensional potential energy surface (PES), where is the number of atoms [6]. The reaction pathway is a onedimensional cross section of the PES composed of the precursor, intermediates, transition states and products of a particular reaction [6]. Reactants and intermediates of a reaction correspond to minima, whereas transition states are located at maxima along the reaction pathway.
The rate of the reaction depends on the probability of the molecule adopting the conformations along the reaction pathway, which, in turn, stems from the probability that a sufficient amount of energy is partitioned into the necessary modes of motion. The RRKM formalism assumes that: (1) once the molecule adopts the transition state orientation, the reaction proceeds to products; and (2) the internal vibrational energy redistribution (IVR) is fast with respect to the timescale of the reaction. Under these constraints, the RRKM microcanonical rate constant, , is given by
(1) 
where is the internal energy of the system, is the activation energy for the reaction, is the sum of states of the transition state from to , is the density of states of the precursor at energy , is Planck’s constant, and is the degeneracy of the reaction pathway [3, 4, 5]. A state is defined as any unique vibrational configuration that determines the internal energy of the molecule. The sum of states, , is the total number of states within a specific energy range [6]. The density of states is the number of states per energy level. Equivalently, the density of states is the derivative of the sum of states with respect to energy [6]. The density is expressed in units of inverse energy
(), while the sum of states is a dimensionless quantity. Thus, the microcanonical rate constant of equation (1) has units .
There are vibrational frequencies for nonlinear polyatomic molecules and vibrational frequencies for the corresponding transition state. It is the values of these frequencies, which typically range from , that dictate how the internal energy is distributed in the molecule. The calculation for the sum and density of states depends on these frequencies, which can be measured or looked up in standard data tables. However, for transition states, these values must be estimated or, more likely, calculated using theoretical packages.
This article is divided into the following sections: In Section 2, the sum and density of states for vinyl cyanide () are determined through a direct count method analogous to that developed by Beyer and Swinehart [7]. In Section 3, the method is extended to anharmonic oscillators using the Stein–Rabinovitch algorithm [8]. In Section 4, the RRKM microcanonical rate constants are calculated for the unimolecular dissociation of vinyl cyanide and are used to predict various dynamic properties of the chemical reaction.
Beyer and Swinehart (BS) developed a surprisingly simple yet computationally intensive direct count method to calculate the sum and density of states at defined energies of a harmonic oscillator [7]. The advantage of conducting such calculations in Mathematica is that the user can leverage highquality numerical ODE solvers and interactive graphical features to visualize dynamic properties of the chemical reaction. This is demonstrated in Section 4, where the temporal development of nine different species is plotted and automatically updated as the user sets the internal energy of the system with a slider. All of this can be done in real time after some initial computations that are exact with Mathematica.
There are a number of equivalent ways to implement the BS direct count of states. To ease the exposition, we first illustrate the task with a loop. The sum and density of states are determined by the vibrational frequencies of the species of interest, which we define in the list . In more complex situations, such as biomolecules or reactions at very high energies, to lessen the computational burden it may be necessary to divide the value of the total energy into larger packets and modify the vibrational frequencies to be multiples of the new packet size [8]. In this case, the packet size is 1 and the vibrational frequencies for vinyl cyanide are taken from reference [9].
The final result for the sum or density of states is a list with elements. For the sum of states, this list is initialized with all values set to 1. The sum of states can then be calculated using a nested (or doubly indexed) loop.
Because the density of states is the derivative of the sum of states, the density of states can also be determined from the sum of states table by setting the element of the density of states to the difference between the and elements of the sum of states table. Alternatively and equivalently, this can be done to the initial sum of states table to derive the initial density of states table, which is a list with elements with the initial element set to 1 and all other elements set to 0. This can be conveniently done with .
A visual comparison of the sum and density of states, calculated for vinyl cyanide (), shows the behavior of the two quantities over the energy range. As the general trend of the graphic is clear, we subsample the lists of values to avoid plotting millions of points, needlessly increasing file size.
Thus, when vinyl cyanide has an internal energy of , the sum and density of quantum states are approximately and .
Notice that the loops for computing the sum and density of states perform the same basic process—the BS direct count method—on different initial lists. As we proceed, adopting a functional programming approach eases the exposition and vastly improves code clarity. We do this by introducing two new functions, and . The first updates the lists of sums and densities for a single vibrational frequency, and the second uses to iteratively update for each of the vibrational frequencies in the table.
We then use these to create the two functions and , which take a frequency list and total energy as arguments and do just as their names imply. The last argument to these functions, , is used in Section 4.
For completeness, we recompute the sum and density of states using these functions to illustrate how that can be done. It is readily verified that these compute the same quantities.
The harmonic oscillator model assumes that the difference between successive vibrational energy levels remains constant. In real oscillators, this difference decreases as the vibrational excitation increases. These anharmonic effects increase both the sum and density of states and thus often cancel when taking the ratio to determine the RRKM rate constant in equation (1). However, there are some cases where this effect does not completely cancel or where highly resolved measurements necessitate a more accurate description of the vibrational energy.
The Stein–Rabinovitch (SR) algorithm extends the BS method to incorporate the effects of anharmonicity into the RRKM rate constant [8]. Following [10], the anharmonic effects are incorporated into the energy level expression through the equation
(2) 
where is the energy of the vibrational level, is the vibrational frequency, and is the anharmonicity constant. The subscript stands for equilibrium. It is chosen because this would be the vibrational frequency if the oscillator vibrated harmonically about its equilibrium position (the minimum in the well). The SR extension counts each energy level of the anharmonic model to calculate the sum and density of states in a way that is functionally similar to the BS method; however, the input table is modified to include the anharmonicity constant for each vibrational mode. For this example, we have organized the anharmonicity constants, , into the table . Using the vibrational frequencies of vinyl cyanide from [9], and assigning 0.01 as the anharmonicity constant for each vibration, the energy for each level of each mode can be calculated.
With these we can calculate the sum and density of states, initializing the lists in the same manner as the BS method (a list of 1s for the sum of states, and a 1 followed by 0s for the density of states). The process is functionally similar to that used to calculate the sum and density of states for a set of harmonic oscillators. In the BS method, the elements of a single table are modified with each iteration of the inner loop. By contrast, in the SR extension, the elements of the table are updated through separable computations combined after each complete loop for each oscillator.
The implementations can be verified by replicating the calculations in the appendix of [8], which shows the calculation of the density of states up to 10 units of energy for a model system with two oscillators. The energies of the first oscillator are and the energies of the second are .
With these, we can define SR analogs to and .
We can then use them to compute the SR sum and density of states.
Here are two plots comparing the harmonic and anharmonic models for the density and sum of states calculated for vinyl cyanide.
As a more substantive example, we calculate rate constants for the decomposition of vinyl cyanide () into hydrogen cyanide (), hydrogen isocyanide () and acetylene (), compare them to published values, and interactively visualize the temporal dependence of the various species. In [9], Homayoon and colleagues considered the following kinetic model that involved seven different paths, three different intermediates, 13 rate constants and three different products:
Each step of each path passes through a transition state to yield either an intermediate or a final product. The following differential rate equations characterize the temporal dependence of the amounts of each species:
The determination of the sum and density of states requires the vibrational frequencies for each species located at minima and maxima along the reaction pathway. For the decomposition reaction of vinyl cyanide, there are six minima (the geometry of along with each intermediate) and 11 maxima (each unique transition state). These 17 sets of frequencies are taken from reference [9] and are provided in Table 1. In the following definitions, we have chosen the same notation as in reference [9]. TS indicates a transition state, the particular path is indicated by a Roman numeral, and the minimum energy of each species relative to the minimum energy of vinyl cyanide has the prefix re.
We now turn to the calculation of the sum and density of states and then use the values to calculate each rate constant. We begin by defining the vibrational constants.
Because the internal energy is defined relative to the well depth of vinyl cyanide, the sums and densities of states for the other species must be calculated using energy adjusted by the lowest energy of each species relative to the lowest energy of vinyl cyanide (peak height for the transition states and well depth for the intermediates).
Using and from Section 1, we can now compute the sums and densities of states in parallel.
Using equation (1), we now compute the 13 rate constants, taking care to avoid indeterminate values such as 0/0.
The rate constants can now be visualized. Note that as energy increases, all the reaction rates also increase.
The temporal dependence of the relative concentrations of each stable species (precursor, intermediates and products) can be determined using the rate equations and . The reaction dynamics of the decay of the precursor vinyl cyanide, the subsequent build up and decay of the intermediates, and the formation of products, each normalized with respect to , are plotted at a given internal energy determined by a variable slide bar.
A number of qualitative and quantitative results can be gleaned by simple inspection of this dynamic plot. One example is the amount of time it takes for the reaction to run to completion as energy increases. The plot also indicates that there is never a significant amount of any intermediate apart from that in path III. This is not surprising, given the significantly lower energy for the first transition state in path III relative to the first transition state in pathways IV and V, and given that three pathways share .
A final element determined from this analysis is the exit channel ratios (or product distribution ratios) determined at reaction completion and for a specific system internal energy. This is important because many experimental measurements are only capable of measuring product ratios as a function of precursor internal energy. Thus, experimental scientists require such an RRKM analysis to provide a dynamic and atomistic picture of the chemical reaction. The following code determines and plots the final to ratio at internal energies up to .
This determines the energy at which HNC reaches its maximum relative abundance.
The last plot indicates that production is favored at all system internal energies. Isocyanide () only becomes a noticeable product at a system internal energy above . This is due to paths IV–VII becoming competitive with path I at higher energies. Surprisingly, the product ratio reduces at energies greater than (equivalently, there is a maximum in the product distribution curve). This result would be difficult to predict, even qualitatively, with just the calculated rate constants, demonstrating the usefulness of the powerful tools available in Mathematica.
[1]  D. I. Giacomo, “A Short Account of RRKM Theory of Unimolecular Reactions and of Marcus Theory of Electron Transfer in a Historical Perspective,” Journal of Chemical Education, 92(3), 2015 pp. 476–481. doi:10.1021/ed5001312. 
[2]  F. A. Lindemann, S. Arrhenious, I. Langmuir, N. R. Dhar, J. Perrin and W. C. McC. Lewis, “Discussion on ‘The Radiation Theory of Chemical Action’,” Transactions of the Faraday Society, 17, 1922 pp. 598–606. doi: 10.1039/TF9221700598. 
[3]  O. K. Rice and H. C. Ramsperger, “Theories of Unimolecular Gas Reactions at Low Pressures,” Journal of the American Chemical Society, 49(7), 1927 pp. 1617–1629. doi:10.1021/ja01406a001. 
[4]  L. S. Kassel, “Studies in Homogeneous Gas Reactions I,” The Journal of Physical Chemistry, 32(2), 1928 pp. 225–242. doi:10.1021/j150284a007. 
[5]  R. A. Marcus, “Unimolecular Dissociations and Free Radical Recombination Reactions,” The Journal of Chemical Physics, 20(3), 1952 pp. 359–364. doi:10.1063/1.1700424. 
[6]  IUPAC. Compendium of Chemical Terminology, 2nd ed. (the “Gold Book”) (compiled by A. D. McNaught and A. Wilkinson), Blackwell Scientific Publications: Oxford, 1997. goldbook.iupac.org (XML online corrected version, created by M. Nic, J. Jirat and B. Kosata; updates compiled by A. Jenkins). 
[7]  T. Beyer and D. F. Swinehart, “Algorithm 448: Number of MultiplyRestricted Partitions,” Communications of the ACM, 16(6), 1973 pp. 379. doi.org/10.1145/362248.362275. 
[8]  S. E. Stein and B. S. Rabinovitch, “Accurate Evaluation of Internal Energy Level Sums and Densities Including Anharmonic Oscillators and Hindered Rotors,” The Journal of Chemical Physics, 58(6), 1973 pp. 2438–2445. doi.org/10.1063/1.1679522. 
[9]  Z. Homayoon, S. A. Vázquex, R. RodríguezFernández and E. MartínezNúñez, “Ab Initio and RRKM Study of the HCN/HNC Elimination Channels from Vinyl Cyanide,” The Journal of Physical Chemistry A, 115(6), 2011 pp. 979–985. doi.org/10.1021/jp109843a. 
[10]  P. M. Morse, “Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels,” Physical Review, 34(1), 1929 pp. 57–64. doi:10.1103/PhysRev.34.57. 
A. C. Mansell, D. J. Kahle and D. J. Bellert, “Calculating RRKM Rate Constants from Vibrational Frequencies and Their Dynamic Interpretation,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.195. 
Adam C. Mansell is a Ph.D. candidate in the Bellert Group of the Department of Chemistry and Biochemistry at Baylor University. He holds a B.A. in chemistry from Concordia University Irvine.
David J. Kahle is a computational statistician and assistant professor of statistics at Baylor University. He holds a B.A. in mathematics from the University of Richmond and M.A. and Ph.D. degrees in statistics from Rice University.
Darrin J. Bellert is an associate professor of chemistry at Baylor University. He received his B.S. from Wright State University in Ohio, and his Ph.D. from Florida State University.
Adam C. Mansell
Department of Chemistry and Biochemistry
Baylor University
One Bear Place #97348
Waco, TX 76798
adam_mansell@baylor.edu
David J. Kahle
Department of Statistical Science
Baylor University
One Bear Place #97140
Waco, TX 76798
david.kahle@gmail.com
Darrin J. Bellert
Department of Chemistry and Biochemistry
Baylor University
One Bear Place #97348
Waco, TX 76798
darrin_bellert@baylor.edu
This article continues the presentation of a variety of applications around the theme of inversion: quandles, inversion of one circle into another and inverting a pair of circles into congruent or concentric circles.
Recent decades have seen a rebirth of geometry as an important subject both in the curriculum and in mathematical and computational research. Dynamic geometry programs have met the demand for visual, specialized computational tools that help bridge the gap between purely visual and algebraic methods. This development has also extended the understanding of the theoretical and computational foundations of geometry, which in turn has stimulated the proliferation of several new branches of geometry, producing a more mature and modern discipline.
In this spirit, these articles [1, 2] have been written to be useful as additional material in a teaching environment on computational geometry, following the practice of the author in teaching the subject at the beginning university level. This third article includes a section on quandles (algebraic generalizations of inversion) that describes their properties and generates all finite quandles up to order five. Also, we include the construction of a circle inverting one circle into another, followed by a section on the construction of a circle inverting two circles, or a circle and a line, into a pair of concentric circles.
Let mean the inverse of the object in the circle with center and radius , drawn as a red dashed circle.
We repeat the definitions of the functions , , and from the previous article [2].
The function computes the square of the Euclidean distance between two given points. (It is more convenient to use the following definition than the builtin Mathematica function .)
The function tests whether three given points are collinear. When exactly two of them are equal, it gives , and when all three are equal, it gives , because there is no unique line through them.
The function computes the unique circle passing through three given points; if they are collinear, then the function is applied first.
The function computes the circle passing through the points , and . If the points are collinear, it gives the line through them; if all three points are the same, it returns an error message, as there is no meaningful definition of inversion in a circle of zero radius.
The function computes the inverse of in a circle or line . The object can be a point (including the special point that inverts to the center of ), a circle or a line (specified by two points).
The geometric definition of inversion of circles can be formalized algebraically and thus be generalized. Let denote the result of inverting in . Quandles arise mostly in knot theory and group theory and are characterized by the following axioms [3]:
The first two axioms correspond to wellknown properties of inversion.
The following figure illustrates the third axiom. Red arrows go from the center of the circle to be inverted to the center of its inversion.
A set equipped with a binary operation whose elements satisfy these three axioms is called an involutory quandle, or quandle for short. The operation is neither commutative nor associative. Inversive geometry applied to generalized circles is an example of an infinite quandle. There are other sets that also verify the axioms; for example, if , the operation is a quandle.
Finite quandles are somewhat curious; for instance, the following is the operation matrix corresponding to a sixelement quandle (due to Takasaki).
This verifies that under any modulus, this structure generalizes to a quandle.
A matrix corresponding to a finite quandle has different elements appearing in its main diagonal and also different elements in each of its columns (i.e. for all elements , , there exists a unique such that ). Also, for every triple of indices, we must have
Is there an arrangement of generalized circles that forms a finite quandle under mutual inversion, that is, is closed by inversion? Any two orthogonal circles form a twoelement quandle. Also, consider a set of lines equally spaced, passing through the origin. This set is closed under reflection. Taking and labeling the lines from to gives the Takasaki matrix. A circle centered at the origin and lines equally spaced produce a set closed under inversion; if we label the circle as , the matrix in this case has all elements in the last row equal to . Let us now generate all finite quandles of size .
Computing the number of finite quandles by this method is computationally expensive, as the variable is of length . With , using the previous code, it took about an hour reporting 404 instances (time measured on a Mac Pro 3.1 GHz, 16 Gb).
The following is an example of a set of four circles closed under inversion (that is, any circle in the set inverted in any other circle results in a circle in the set), also called the inversive group of three points [3]. The function computes a disk or a line passing through point such that points and form an inverse pair under inversion in . In the following , you can drag a locator to modify one of the four circles; the others are computed accordingly.
The function slightly varies three points that are coincident or collinear.
For more on quandles, see [4].
Throughout this section, let and be circles with , and let the inversion circle be , such that . Call such an the midcircle of and . There are three cases, depending on the relative positions of and .
If , a reflection takes into , so assume from now on.
Theorem 1
Assume without loss of generality that and . Draw two parallel radii from and defining variable points and . Extend the lines and to intersect at point . From similar triangles, , and we easily conclude that .
Extend the lines and to intersect at the point . Construct the circle , which is tangent to and .
The first part of the following output checks that the circle inverts into , with .
The second part checks that , so that and invert to each other in .
The circle separates and , while connects them, so intersects in two points and , each of which is fixed under inversion in . Since , and on invert into , and , also on , inverts to itself (though not pointwise), and and are orthogonal.
Let be orthogonal to and , so that both and are invariant under inversion in . If inverts into in , then and invert into each other in , so , and and are orthogonal.
This notation is followed in the next . (A circle orthogonal to and is not drawn.)
Theorem 2
To verify inverts the circle into the circle , proceed as in the previous section.
The next follows the previous construction of the midcircle of and .
Theorem 3
To verify this, is inverted in the two circles from theorems 1 and 2.
The following shows the construction of both midcircles following the previous notation.
Theorem 4
Proof
Let be a circle centered on the midcircle . Inverting in , we obtain a line (drawn in black in the following ). As separates and , must separate and ; in fact, must invert into . As is a line, inversion in is reflection, hence and are congruent. □
The following shows additionally that the brown line joining the centers of and inverts into a brown circle orthogonal to the blue line and to the congruent circles and , as was expected.
For more on the geometry of circles and inversion, see [5, 6, 7].
The radical axis of two circles is the locus of a point from which tangents to the two circles have the same length. It is always a straight line perpendicular to the line joining the centers of the two circles. If the circles intersect, the radical axis is the line through the points of intersection. If the circles are tangent, it is their common tangent. We will need the following property of the radical axis of two circles [8, 9, 10].
Theorem 5
This checks that any point on has tangents to and of equal length.
Theorem 6
A few words on the assumptions in the following Simplify to verify theorem 6. The first three, , , , hold in general. The fourth, , is for the Solve that computes to find a solution. The next two, and , are for the inversion of to be feasible, and the last two are for the inversion of to be feasible.
Theorem 7
Proof
Let and be the given circles. Choose a circle centered at a point on the radical axis with radius equal to the length of the tangent from to . This circle intersects the line in two points , . Any circle with or as center inverts and into concentric circles. □
The next shows the circles in blue and in yellow, their radical axis, a point on the radical axis, the circle now moving freely on the radical axis, the circle of inversion centered at one of the intersections of and , and two concentric circles obtained by inverting and in . Only one of the inversive circles is shown; the other is a mirror image in the radical axis.
The center of the inversive circle does not depend on the position of . This checks the inverses of the circles and are concentric.
Let be the center of the other inversive circle.
Theorem 8
A line and a circle that do not intersect or a pair of nonintersecting circles can be inverted into two concentric circles. The key is to obtain a common orthogonal circle and then choose the inversion center at its intersection with a particular line [11].
Theorem 9
I am grateful to the Center of Investigation and Advanced Studies in Mexico City for the use of its extensive library and to Wolfram Research for providing an ideal environment to develop this series of articles.
[1]  J. RangelMondragon, “Inversive Geometry: Part 1,” The Mathematica Journal, 15, 2013. doi:10.3888/tmj.15–7. 
[2]  J. RangelMondragon, “Inversive Geometry: Part 2,” The Mathematica Journal, 18, 2014. doi:10.3888/tmj.185. 
[3]  F. Morley and F. V. Morley, Inversive Geometry, New York: Ginn and Company, 1933. 
[4]  B. Ho and S. Nelson, “Matrices and Finite Quandles,” Homology, Homotopy and Applications, 7(1), 2005 pp. 197–208. doi:10.4310/HHA.2005.v7.n1.a11. 
[5]  H. S. M. Coxeter, “Inversive Geometry,” Educational Studies in Mathematics, 3(3), 1971 pp. 310–321. doi:10.1007/BF00302300. 
[6]  D. Pedoe, Geometry, A Comprehensive Course, New York: Dover Publications Inc., 1988. 
[7]  H. S. M. Coxeter and S. L. Geitzer, Geometry Revisited, New York: Random House, 1967. 
[8]  E. W. Weisstein. ”Radical Line” from MathWorld–A Wolfram Web Resource. mathworld.wolfram.com/RadicalLine.html. 
[9]  S. R. Murthy. “Radical Axis and Radical Center” from the Wolfram Demonstrations Project–A Wolfram Web Resource. demonstrations.wolfram.com/RadicalAxisAndRadicalCenter. 
[10]  J. RangelMondragón, “The Arbelos,” The Mathematica Journal, 16, 2014. doi:10.3888/tmj.165. 
[11]  D. E. Blair, Inversion Theory and Conformal Mapping, Providence, RI: American Mathematical Society, 2000. 
J. RangelMondragón, “Inversive Geometry: Part 3,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.194. 
Jaime RangelMondragón received M.Sc. and Ph.D. degrees in Applied Mathematics and Computation from the School of Mathematics and Computer Science at the University College of North Wales in Bangor, UK. He was a visiting scholar at Wolfram Research, Inc. and held positions in the Faculty of Informatics at UCNW, the Center of Literary and Linguistic Studies at the College of Mexico, the Department of Electrical Engineering at the Center of Research and Advanced Studies, the Center of Computational Engineering (of which he was director) at the Monterrey Institute of Technology, the Department of Mechatronics at the Queretaro Institute of Technology and the Autonomous University of Queretaro in Mexico, where he was a member of the Department of Informatics and in charge of the Academic Body of Algorithms, Computation and Networks. His research included combinatorics, the theory of computing, computational geometry and recreational mathematics. Jaime RangelMondragón died in 2015.
]]>Input shaping is an established technique to generate prefilters so that flexible mechanical systems move with minimal residual vibration. Many examples of such systems are found in engineering—for example, space structures, robots, cranes and so on. The problem of vibration control is serious when precise motion is required in the presence of structural flexibility. In a wind turbine blade, untreated flapwise vibrations may reduce the life of the blade and unexpected vibrations can spread to the supporting structure. This article investigates one of the tools available to control vibrations within flexible mechanical systems using the input shaping technique.
Among other choices [1, 2] for reducing vibrations in flexible systems, input shaping control is an openloop control technique that is implemented by convolving a sequence of impulses with a desired command. The amplitudes and time locations of the impulses are determined from the system’s natural frequency and damping ratio by solving a set of constraint equations. Historically, input shaping dates from the late 1950s. Originally named “Posicast Control,” the initial development of input shaping is largely credited to Smith [3, 4], with one notable precursor due to Calvert and Gimpel [5]. All three works proposed a simple technique to generate a nonoscillatory response from a lightly damped system subjected to a step input, which was motivated by a simple wave cancellation concept for the elimination of the oscillatory motion of the underdamped system. The early forms of command generators suffered from poor robustness properties, as they were sensitive to modeling errors of natural frequencies and damping ratios. Since this initial work, there have been many developments in the area of input shaping control, with one of the pacing elements being the progress in microprocessor technology to implement the concept. More recent robust command generators have proven beneficial for real systems with, for example, Swigert [6] proposing techniques for the determination of torque profiles that considered the sensitivity of the terminal states to variations in the model parameters. Other examples of input shapers have been developed that are robust to natural frequency modeling errors, the first of which was called the Zero Vibration and Derivative (ZVD) shaper [7], which improved robustness to modeling errors by adding additional constraints on the derivative of the residual vibration magnitudes. With this robustness present, input shaping has been implemented in a variety of systems, including movement of cranes [8, 9], precise movement of disk drives [10], flexible spacecraft [11, 12], industrial robots [13, 14] and coordinate measuring machines [15]. There have also been developments using hybrid input shaping [16] and threestep input shaping techniques [17].
Many types of solutions are possible for the problem of flexible dynamics—for example, feedback control, command shaping or redesigning the physical geometry [18]. A simple example of this challenging area is that of an overhead traveling crane, as shown in Figure 1, which consists of a point mass of the moveable structure (crab or crane) of a point mass of the payload and of a nonextensible load carrying rope (cable) of length .
Figure 1. Schematic of overhead gantry crane.
The equations of motion for such a system can be set up either directly from Newtonian mechanics or indirectly using Lagrangian methods. Using either results in the nonlinear system of equations for the motion as
(1) 
When the gantry crane is accelerating or retarding, then the hanging cable starts to vibrate. The code for equation (1) is shown in Figure 2, with the force set as positive, and the movement of the cable in particular is shown when the crane is accelerating. A result for retardation can easily be found by simply assigning a negative value for in the code for .
Figure 2. Overhead gantry crane accelerating.
A special case is when the gantry crane is moving at a constant velocity; that is, is set to zero. Then the hanging cable for lifting is not swinging, and the crane and pendulum move as in Figure 3.
Figure 3. Overhead gantry crane moving with constant speed.
The upper end of the cable is attached to a trolley that travels along a rail to position the payload. Cranes are usually controlled by a human operator who moves levers or presses buttons to cause the trolley to move. If the operator presses the control button for a finite time period, then the trolley will move a finite distance and come to rest. However, the payload usually oscillates about some support on the trolley due to the trolley motion, as shown in Figure 3 by the uncontrolled oscillation. The crane driver can smooth this situation by suitably pressing the button multiple times. The payload motion for this scenario could be as shown in Figure 4 labeled as “Operator controlled.”
Figure 4. Payload response for uncontrolled and operator controlled.
Here we start with the simplest commands to move systems without vibration. An impulse applied to a system usually causes it to vibrate, but that can be canceled by a second impulse. This concept is shown in Figure 5, where each input is piecewise constant and the system being considered is purely oscillatory with no damping. As can be seen, the response functions add together to give zero.
Figure 5. Simple cancellation of a vibration.
Next, Figure 6 shows the response of a typical forced damped system to a twoimpulse command.
Figure 6. Typical springdamped system.
For the preceding system, the equations of motion are
(2) 
where and are coefficients due to drag and spring stiffness, respectively.
Again, each input is piecewise constant, but the equation of motion has an additional damping term dependent on the speed of motion.
It is instructive to derive the amplitudes and time locations of the twoimpulse command shown in Figure 7.
Figure 7. Twoimpulse response with damping.
If a reasonable estimate of the system’s natural frequency and damping ratio is known, then the residual vibration that results from a sequence of impulses can be described [10] using the expression
(3) 
where
(4) 
and are the amplitudes and time locations of the impulses, is the number of impulses in the impulse sequence, and .
Equation (3) is actually the percentage of residual vibration, which is a measure of the amount of vibration a sequence of impulses will cause relative to the vibration caused by a single impulse with unit magnitude. On setting equation (3) equal to zero and avoiding a trivial solution, values for the impulse amplitudes and time locations that would lead to a zero residual vibration can be found. To avoid the zerovalued trivial solution and to obtain a normalized result, the impulses are required to sum to one; that is,
(5) 
However, impulses can still satisfy equation (5) by taking very large numbers, both positive and negative. To alleviate this, a bounded solution is imposed that limits the values of to positive values
(6) 
For a twoimpulse sequence, there are four unknowns, , , , . Without loss of generality, we can set the time location of the first impulse equal to zero. For equation (3) to be satisfied, the expressions in equation (4) must both be equal to zero. Therefore, we get
(7) 
The second of these two expressions can be satisfied nontrivially by setting the sine term equal to zero. This occurs when
(8) 
where is the damped period of vibration. This of course means that there are an infinite number of possible values for the location of the second impulse, but to cancel the vibration in the shortest amount of time, the smallest value of is chosen:
(9) 
For this case, the amplitude constraint given in equation (5) reduces to
(10) 
Using the expression for the damped natural frequency and substituting equations (9) and (10) into the first expression of equation (5) gives
(11) 
The sequence of two impulses that leads to zero residual vibration can be summarized as
(12) 
where .
A zerovibration (ZV) input shaper, as just described, is useful in situations where the parameters of the system are known with a high degree of accuracy. Also, if little faith is held in the input shaping approach, the application will never increase vibration beyond the level before shaping [19]. It has been pointed out [20] that previous articles on input shaping have confused the issue of natural frequency, even if the conceptual explanation when using the method is generally acceptable. Kang [20] differentiates between (a variable), which is the actual value of the undamped natural frequency of the system, and (a constant), which is the “modeled” value of the undamped natural frequency . Kang [20] also proves that vibration approaches zero as . The article shows clearly that for a vibratory system, a modeling frequency is chosen such that at the modeling frequency .
The following code generates the sensitivity curve (Figure for a ZV shaper by plotting the amplitude of residual vibration as a function of the system parameters. In this case, the modeling frequency was set as rad/s and the damping ratio as 0.0.
Figure 8. Sensitivity curve for ZV input shaper.
The amplitudes and time locations of the impulses depend on the system parameters and . If there are errors in these values, (and there always are [18]), then the impulse sequence will not result in zero vibration. A Zero Vibration and Derivative (ZVD) shaper is a command generation scheme designed to make the input shaping process more robust to these modeling errors. To increase robustness to modeling error, the ZVD input shaper adds two constraints [20], the derivatives
(13) 
The sequence for the ZVD shaper can be summarized as
(14) 
where .
An alternative to the ZV shaper is the ZVD shaper, which is much more robust than the ZV shaper, as shown in Figure 9. However, the ZVD shaper has a time duration equal to one period of the vibration frequency, as opposed to the onehalf period length of the ZV shaper. This tradeoff is typical of the input shaper design process; that is, increasing insensitivity usually requires increasing the length of the input shaper. An input shaper with even more insensitivity than the ZVD can be obtained by setting the second derivative of equation (3) with respect to equal to zero. This is called the ZVDD shaper. The algorithm can be extended indefinitely with repeated differentiation of equation (3). Closedform solutions of the ZV, ZVD and ZVDD shapers for damped systems exist [7]. An alternative procedure for increasing insensitivity using extrainsensitive constraints has been derived [21]. Instead of forcing the residual vibration to zero at the modeling frequency, the residual vibration is limited to a low level of . The width of the notch in the sensitivity curve is then maximized by forcing the vibration to zero at two frequencies, one lower than the modeling frequency and the other higher. Figure 9 indicates that there are two inner maxima, say at frequencies and , where the vibration must equal as defined in equation (15) and the derivative must equal zero. These two constraints translate to
(15) 
and
(16) 
where and is the difference between and . Note that represents the frequency shift from the modeling frequency to the frequency that corresponds to the first hump in the sensitivity curve; depends on and does not appear in the final formula for the shaper. Other conditions are that the impulse amplitudes must sum to one, and following the hypothesis that the shaper contains four evenly spaced impulses with a duration of one and a half periods to form the sensitivity curve [21], then
(17) 
Using these conditions, it can be shown that
(18) 
Expanding equations (15) and (16), combining terms and using equation (18) gives
(19) 
and
(20) 
Equation (19) can be solved for :
(21) 
Substituting equation (21) into equation (20) yields
(22) 
where
The twohump shaper for an undamped system can now be summarized as
(23) 
The following code generates a twohump shaper based on the above analysis and compares it to the ZVD shaper. When , the insensitivity to modeling errors (i.e. the width of the sensitivity curve) is increased by over 100%. Again, the modeling frequency is set at rad/s.
Figure 9. Sensitivity curves.
Robustness is not restricted to errors in the frequency. Figure 10 shows a threedimensional sensitivity curve for a shaper that was designed to suppress vibration over the range of damping ratios between 0 and 0.1.
Figure 10. Threedimensional curve including variation with damping ratio.
A damped oscillatory dynamic system model has the transfer function
(24) 
where again and are the natural frequency and damping ratio, respectively. Figure 11 gives various responses, depending on the damping factor.
Figure 11. Responses to input for different damping factors.
The equation for the responses shown in Figure 11 is
(25) 
where and are the amplitude and time of the impulse, respectively. Further, the response to a sequence of impulses can be obtained using the superposition principle. Thus for impulses, the impulse response can be expressed as , where
(26) 
(27) 
where and are again the magnitude and times at which the impulses occur.
To demonstrate the effect on the response when the model is not perfect, the following code was written using a robust fourimpulse ZVDD shaper. Figure 12 shows the response when no shaping is imposed, when the model is perfect, and when there is a 20% error in the frequency estimate. The initial peak response is cut to 57% when there is no input shaper applied.
Figure 12. Responses when model is not perfect.
Wind turbine blade vibration is a serious problem because it will reduce the life of the blade and vibrations can also be transferred to the supporting tower, causing the complete structure to vibrate. One source of an increase in vibration amplitudes is the change of pitch angle input. Use is made here of a ZV input shaper to demonstrate a decrease in amplitude when the pitch angle changes from large to small. Although many ways to suppress wind turbine blade vibration have been developed, there has not been much work done on the effect on the vibrations when changing the pitch angle rapidly. A rapidly changing pitch angle input could be considered as a step input, causing some additional vibration (larger amplitudes) to the blade. In this example, the effect of using an input shaper to reduce the blade angle deflection is investigated. We consider the wind turbine blade as a cantilever beam with the hub end clamped and the other end free to move. The effect of the rotation is taken into account by the inclusion of centrifugal stiffening, and the modal shapes were calculated using the Adomian modified decomposition method [22]. To incorporate the effect of changing the pitch angle, the wellknown blade element theory [23] was used to form a generalized normal force consisting of components of lift and drag forces as functions of pitch angle. The expressions for kinetic energy, potential energy and aerodynamic forces were then used to form a Lagrangian of the blade that governs the motion of blade flapwise deflection.
A ZV input shaper is used in a scheme summarized in Figure 13, which is a block diagram of an input shaping control scheme dealing with unexpected wind disturbances.
Figure 13. Schematic of input shaper controller.
The input shaping control is a feedforward control method, and only the shaped input is used to control the system. The idea is to see how the blade flapwise deflection reacts to a pitch angle change. The pitch angle is initially set at a 4° angle of attack. Figure 14 shows the flapwise deflection with the pitch angle is 4° when it has reached its steady state, and Figure 15 shows the deflection at 14°. It can be seen that the deflection of the blade is worse at the smaller angle, and this is due to the wind turbine blade being a pitchtofeather type.
Figure 14. Flapwise deflection (pitch angle 4°).
Figure 15. Flapwise deflection (pitch angle 14°).
To see how the pitch angle affects the flapwise deflection, the pitch angle is changed from 14° to 4° at 30 seconds. Figure 16 shows that some residual vibration is caused (solid blue curve) since the deflection after 30 seconds is different from when the pitch angle was set at 14°. This is because in the model there is no damping at first. Next, the input shaper is added, and clearly, from the dashed orange curve in Figure 16, the residual vibration is reduced.
Figure 16. Pitch angle change effect.
Some of the tools available for input shaping have been investigated here, where the input to a given system has been shaped so as to minimize the residual vibration. Important to future use of these techniques is that they have been shown to be robust and able to tolerate errors within the system parameters; that is, although a residual vibration may not become zero due to the shaper, there is generally a large reduction in vibration.
[1]  J.H. Park and S. Rhim, “Experiments of Optimal Delay Extraction Algorithm Using Adaptive TimeDelay Filter for Improved Vibration Suppression,” Journal of Mechanical Science and Technology, 23(4), 2009 pp. 997–1000. doi:10.1007/s1220600903281. 
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[8]  K. L. Sorensen, W. E. Singhose and S. Dickerson, “A Controller Enabling Precise Positioning and Sway Reduction in Bridge and Gantry Cranes,” Control Engineering Practice, 15(7), 2007 pp. 825–837. doi:10.1016/j.conengprac.2006.03.005. 
[9]  M. A. Ahmad, R. M. T. R. Ismail, M. S. Ramli, R. E. Samin and M. A. Zawawi, “Robust Input Shaping for AntiSway Control of Rotary Crane,” Proceedings of TENCON 2009—IEEE Region 10 Conference, Singapore, Jan. 23–26, 2009 pp. 1039–1043. doi:10.1109/TENCON.2009.5395891. 
[10]  W. E. Singhose, W. Seering and N. C. Singer, “TimeOptimal Negative Input Shapers,” Journal of Dynamic Systems, Measurement, and Control, 119(2), 1997 pp. 198–205. doi:10.1115/1.2801233. 
[11]  D. Gorinevsky and G. Vukovich, “Nonlinear Input Shaping Control of Flexible Spacecraft Reorientation Maneuver,” Journal of Guidance, Control, and Dynamics, 21(2), 1998 pp. 264–270. doi:10.2514/2.4252. 
[12]  L. Y. Pao and W. E. Singhose, “Verifying Robust TimeOptimal Commands for Multimode Flexible Spacecraft,” Journal of Guidance, Control, and Dynamics, 20(4), 1997 pp. 831–833. doi:10.2514/2.4123. 
[13]  J. Park, P. H. Chang, H. S. Park and E. Lee, “Design of Learning Input Shaping Technique for Residual Vibration Suppression in an Industrial Robot,” IEEE/ASME Transactions on Mechatronics, 11(1), 2006 pp. 55–65. doi:10.1109/TMECH.2005.863365. 
[14]  C.G. Kang, K. S. Woo, J. W. Kim, D. J. Lee, K. H. Park and H. C. Kim, “Suppression of Residual Vibrations with Input Shaping for a TwoMode Mechanical System,” Proceedings of International Conference on Service and Interactive Robotics, Taipei, Taiwan, 2009 pp. 1–6. 
[15]  S. D. Jones and A. G. Ulsoy, “An Approach to Control Input Shaping with Application to Coordinate Measuring Machines,” Journal of Dynamic Systems, Measurement, and Control, 121(2), 1999 pp. 242–247. doi:10.1115/1.2802461. 
[16]  S. Kapucu, G. Alici and S. Bayseç, “Residual Swing/Vibration Reduction Using a Hybrid Input Shaping Method,” Mechanism and Machine Theory, 36(3), 2001 pp 311–326. doi:10.1016/S0094114X(00)000483. 
[17]  S. S. Güreyük and S. Cinal, “Robust ThreeImpulse Sequence Input Shaper Design,” Journal of Vibration and Control, 13(12), 2007 pp.1807–1818. doi:10.1177/1077546307080012. 
[18]  T. Singh and W. Singhose, “Tutorial on Input Shaping/Time Delay Control of Maneuvering Flexible Structures,” Proceedings of the 2002 American Control Conference, Vol. 3, Anchorage, AK, May 8–10, 2002 pp. 1717–1731. doi:10.1109/ACC.2002.1023813. 
[19]  I. Arolovich and G. Agranovich, “Control Improvement of UnderDamped Systems and Structures by Input Shaping,” Proceedings of the 8th International Conference on Material Technologies and Modeling (MMT2014), Ariel, Israel, Jul. 28–Aug. 1, 2014 pp. 3.1–3.10. (May 23, 2017) www.semanticscholar.org/paper/ControlImprovementofUnderdampedSystemsandStArolovichAgranovich/5cd5f119710edc81be912aea09a66c64e92d48a2. 
[20]  C.G. Kang, “On the Derivative Constraints of Input Shaping Control,” Journal of Mechanical Science and Technology, 25(2), 2011 pp. 549–554. doi:10.1007/s1220601012057. 
[21]  T. Singh and S. R. Vadali, “Robust TimeOptimal Control: Frequency Domain Approach,” Journal of Guidance, Control, and Dynamics, 17(2), 1994 pp. 346–353. doi:10.2514/3.21204. 
[22]  D. Adair and M. Jaeger, “Simulation of Tapered Rotating Beams with Centrifugal Stiffening Using the Adomian Decomposition Method,” Applied Mathematical Modelling, 40(4), 2016 pp. 3230–3241. doi:10.1016/j.apm.2015.09.097. 
[23]  D. Adair and M. Alimaganbetov, “Propeller Wing Aerodynamic Interference for Small UAVs during VSTOL,” 56th Israel Annual Conference on Aerospace Sciences, Tel Aviv/Haifa, 9–10 Mar., 2016. (May 23, 2017) www.researchgate.net/publication/285356494_Propeller_Wing_Aerodynamic_Interference_ for_Small_UAVs_during_VSTOL. 
D. Adair and M. Jaeger, “Aspects of Input Shaping Control of Flexible Mechanical Systems,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.193. 
Desmond Adair is a professor of Mechanical Engineering in the School of Engineering, Nazarbayev University, Astana, Republic of Kazakhstan. His recent research interests include developing analytical methods for solving vibration problems, and computational fluid dynamics.
Martin Jaeger is an associate professor of Civil Engineering and manager of the Project Based Learning Centre in the School of Engineering, Australian College of Kuwait, Mishref, Kuwait. His recent research interests include construction management and total quality management, as well as developing strategies for engineering education.
Desmond Adair
School of Engineering
Nazarbayev University
53 Kabanbay Batyr Ave.
Astana, 010000, Republic of Kazakhstan
dadair@nu.edu.kz
Martin Jaeger
School of Engineering and ICT
University of Tasmania
Churchill Ave.
Hobart, TAS 7001, Australia
mjaeger@utas.edu.au
This didactic synthesis compares three solution methods for polynomial approximation and systematically presents their common characteristics and their close interrelations:
1. Classical Gram–Schmidt orthonormalization and Fourier approximation in
2. Linear leastsquares solution via QR factorization on an equally spaced grid in
3. Linear leastsquares solution via the normal equations method in and on an equally
spaced grid in
The first two methods are linear leastsquares systems with Vandermonde matrices ; the normal equations contain matrices of Hilbert type . The solutions on equally spaced grids in converge to the solutions in All solution characteristics and their relations are illustrated by symbolic or numeric examples and graphs.
Let . Consider the Hilbert space of realvalued square integrable functions (or , for short), equipped with Lebesgue measure and scalar product
(1) 
and the corresponding norm
scalar products can be approximated by scalar products on discrete grids in based on Riemann sums and similarly for norms.
Partition the finite interval into subintervals by the points
and set
Suppose that is a bounded function on . Let be any point in the subinterval and define the grid
The Riemann sums on the partition and grid are defined by
(2) 
(3) 
(4) 
(5) 
Equation (3) is called the lefthand Riemann sum, (4) the righthand Riemann sum and (5) the (composite) midpoint rule.
For an equally spaced partition, the step size is
The equally spaced partition points are
and the equally spaced grid of length (excluding the endpoint ) is
It is also possible to use the grid points or grid shifted by an amount , where so that , as
Let
(6) 
For equally spaced grids, the Riemann sums simplify to
(7) 
Setting , and gives the lefthand Riemann sum, the composite midpoint rule and the righthand Riemann sum, respectively. The error of the Riemann sums is defined as
The set of continuous realvalued functions forms a dense subspace of , [1], Theorem (13.21). For , the restrictions and to this grid are welldefined. Define the dimensional scalar product on this grid:
(8) 
The dimensional 2norm is
The factor ensures that the norms of constant functions agree:
Denote the linear space of polynomials with real coefficients of degree at most by and define the polynomial by
The polynomial can be written as a scalar product (or dot product) of two tuples, the monomials up to degree and the coefficients:
Introducing the Vandermonde matrix
(9) 
every polynomial of degree can be written as the product of a matrix and a vector as
The product of a matrix and an vector is regarded as a 1vector, not a scalar, as in Mathematica.
Restricting the Vandermonde matrix to the interval gives an operator mapping into :
(10) 
Whereas is an unbounded operator, is a bounded operator with respect to the
2norms on and .
The polynomial approximates in the norm, as measured by
(11) 
In matrixvector notation, this constitutes a linear leastsquares problem for the coefficients :
(12) 
where
Now take a discrete grid
and sample on this discrete grid:
The polynomial of degree approximates in the 2norm on this grid as measured by
(13) 
In matrixvector notation, this constitutes a linear leastsquares problem for the coefficients :
(14) 
where
(15) 
A rectangular or square matrix of this form is called a Vandermonde matrix.
Let , be Hilbert spaces with scalar products , and let be a bounded linear operator defined everywhere on . Then the adjoint operator of is defined by the identity ([2], p. 196, Definition 1)
(16) 
For Hilbert spaces , over the reals, one writes instead of .
All Riemann sums integrate constant functions exactly on any grid, since
If is bounded on and continuous except for finitely many points, it has a Riemann integral. Consult [3], Chapter 2, for proofs and further references on quadrature formulas.
Theorem 1
(17) 
A similar result holds for the lefthand Riemann sums (with ).
If , the error term of the elementary midpoint formula is given in [3], (2.6.5):
(18) 
Therefore the error of the composite midpoint formula can be bounded by
(19) 
By Theorem 1, for functions and , the discrete scalar product converges at least as fast as to the scalar product:
By equation (19), for functions , the discrete scalar product converges at least as fast as to the scalar product:
See [4], sections 2.4 and 2.6.
Theorem 2
(20) 
Define to be the index of the last positive singular value
Then
(21) 
The condition number of a rectangular matrix with full column rank with respect to the 2norm (in short, the 2condition number) is defined as the quotient of its largest to its smallest singular value ([4], equation (2.6.5)):
(22) 
By equation (20), the 2condition number has the properties
(23) 
(24) 
If is an invertible matrix, . The SVD of is obtained from the SVD of as
(25) 
(26) 
If is a real matrix with orthonormal columns , it can be completed to an orthogonal matrix . Therefore the SVD of is
(27) 
The 2condition number of is
(28) 
See [2], sections VII.1 and VII.2.
Theorem 3
.
For and a subset , linear combinations from the subset define a finitedimensional linear operator
(29) 
Obviously, and if and only if is linearly independent.
is bounded and has the matrix representation
Apply the definition of the adjoint operator and notice that the first scalar product is that of , ; then
Therefore the adjoint operator is
(30) 
Substituting into the preceding equation gives the representation of as an matrix (note the scalar products are taken in ):
(31) 
Here is Hermitian positive semidefinite if is over the complex numbers, and symmetric positive semidefinite if is over the reals, and
A polynomial of degree has at most distinct zeros, therefore the set of monomials is a linearly independent subset of and
(32) 
By [5], the determinant of the Vandermonde matrix is the product
Therefore, the rectangular Vandermonde matrix (15) has full rank if and only if the points are pairwise disjoint.
Names are chosen according to previous notation and terminology.
This defines the 2condition number.
This defines symbolic integration with time constrained to 10 seconds.
This defines numerical integration.
The function (which is ) first attempts symbolic integration; if that is unsuccessful or takes too long, it performs numerical integration.
This defines the scalar product in .
Since is listable in and , it also implements the adjoint operator for a set of functions according to equation (32).
This defines the norm in .
This defines functions for discrete grids.
If , or is a machine number, the functions , , and return machine numbers.
This defines the functions , , and .
To avoid potential clashes with predefined values for the variables , and , the script letters , and are used for symbolic results.
These sample functions are used in the commands.
Let . Given a rectangular data matrix and an observation vector ,
(33) 
the linear leastsquares (LS) problem is to find:
(34) 
A comprehensive description of the QR factorization of an matrix via Householder, Givens, fast Givens, classical Gram–Schmidt and modified Gram–Schmidt methods is given in [4], section 5.2. Here only the essential steps are presented.
Let be an orthogonal matrix. Such a matrix preserves lengths of vectors:
Given the real matrix , , the goal is to construct an orthogonal matrix such that
where is an uppertriangular matrix of the form
Obviously
The Mathematica function QRDecomposition deviates from the full QR factorization as follows:
is output as an matrix. The rows of are orthonormal. Only the uppertriangular
submatrix is output.
Then the unique solution of the uppertriangular system is straightforward:
It gives the unique solution of the fullrank linear leastsquares problem (34):
Since multiplication with an orthogonal matrix does not change the singular values, the condition numbers do not change either:
(35) 
This holds in particular for the Vandermonde matrices, both in the discrete and continuous case.
This defines and .
Applying the classical Gram–Schmidt orthonormalization process in a preHilbert space, described in [2], p. 88 ff., to the monomials in gives an orthonormal system of polynomials
(36) 
that satisfy
(37) 
The Fourier coefficients of any function with respect to this orthonormal system are defined according to [2], p. 86, equation (1) (the dot here denotes the placeholder for the integration argument in the scalar product):
The best approximation to is given as the Fourier sum of terms:
The orthonormal system of polynomials is given by the function . The functions and are also defined.
This defines the polynomials .
These polynomials differ from the classical Legendre polynomials LegendreP built into Mathematica only by normalization factors.
This shows the Fourier approximation of a sample set of functions.
Proposition 1
(38) 
(39) 
Proof
Theorem 4
Proof
For this QR factorization, the inverse of the uppertriangular matrix is already calculated by the Gram–Schmidt process:
Equations (26), (28) and (35) give the relations on the 2condition numbers of the operators or matrices:
Corollary 1
The numerical instability of the classical Gram–Schmidt process in machine arithmetic is discussed in [4], section 5.2.8. However, since the Gram–Schmidt orthonormalization of the monomials with respect to the scalar product can be performed by symbolic calculations, numerical algorithm stability is not an issue here, contrary to the dimensional scalar product .
This defines the Gram–Schmidt coefficient matrix .
This defines the lower and uppertriangular matrices of QR decomposition.
This defines the orthogonal matrix .
Here is a set of orthonormal polynomials (, ).
This gives the Gram–Schmidt coefficient matrix , with .
Multiply the matrix of monomials from the right or left.
This reproduces orthogonal matrix .
By construction, the polynomials contained in the matrix columns are orthonormal with respect to the scalar product in .
This verifies the QR decomposition , as in Theorem 4.
Select one of the sample functions and compare the results from the Gram–Schmidt orthonormalization and QR factorization interpretation.
For the continuous case, because , the singular values of equal the singular values of .
Here is the case of a discrete grid.
Obviously, is the lowertriangular Cholesky factor of the .
Consequently, the 2condition number of the Vandermonde matrix of size on is the square root of the 2condition number of .
This gives summary results.
The approach for deriving the normal equations for the leastsquares problem (34) is described in [4], section 5.3, for example. Define
A necessary condition for a minimum is , or equivalently,
(40) 
These are called the normal equations. The minimum is unique if the Hessian matrix has full rank . Then there is a unique solution of the linear leastsquares problem (34) or (40):
(41) 
For an equally spaced , defined as in equation (5), the Vandermonde matrix has full rank, so the Hessian matrix for polynomial approximation has full rank as well.
If is rank deficient (), then there are an infinite number of solutions to the leastsquares problem. There is still a unique solution with minimum 2norm, which can be written with the pseudoinverse matrix ([4], section 5.5):
For full rank, .
Suppose with (see [4] section 5.3.1). Then this algorithm computes the unique solution to the linear leastsquares problem (34):
The approach for deriving the normal equations for the case applies with one modification to the continuous leastsquares approximation equation (12) (see [6]):
The matrix transpose has to be replaced by the adjoint operator :
A necessary condition for a minimum is , or equivalently,
(42) 
These are called the normal equations.
(43) 
is called the Hessian matrix. The minimum is unique if has full rank . The elements can be calculated via equation (31):
Obviously, the Hessian matrix is symmetric and positive semidefinite. Since has full rank for any nonzero interval , then has full rank (as well by equation (32)) and is therefore positive definite. Then there exists a unique solution of equations (42) and (12).
Finally, calculate the elements on the righthand side of via equation (30):
This subsection investigates under which conditions and how fast the polynomial approximations on discrete grids converge to the continuous polynomial approximation in .
For an equally spaced , the normal equations, multiplied by the step size , read
(44) 
Define
(45) 
then by equation (7), the matrix elements of are just the Riemann sums for the matrix elements (integrals) of . Therefore, the Hessian matrices on the discrete grid converge to the continuous Hessian matrix in any matrix norm according to:
Define
(46) 
then by equation (7), the elements of are just the Riemann sums for the elements of , the moments of . Therefore, the righthand side of the normal equations on the discrete grid converge to the righthand side of the continuous normal equations in any vector norm according to:
(47) 
Proposition 2
Proof
From equation (42), the solution of the polynomial approximation in is
From equation (44), the solution for the discrete grid is
(48) 
For the matrix inverses
Expanding the difference of the solution coefficient vectors completes the proof:
Theorem 5
Proof
From Theorem 4 and because ,
By [4], Theorem 4.2.7, the Cholesky decomposition of a symmetric positivedefinite square matrix is unique.
Equations (24) and (26) give the relation for the 2condition numbers of the matrices.
Corollary 2
This defines the continuous and discrete Hessian matrices.
These are the right‐hand sides of the continuous and discrete normal equations.
This gives the solution of the normal equations system.
This gives the approximation polynomials for the continuous and discrete cases.
This gives the Gram–Schmidt coefficient matrix, its inverse and inverse transpose.
The matrix times its transpose equals the Hessian matrix of the normal equations.
Equivalently, is the lowertriangular and is the uppertriangular Cholesky factor of the Hessian matrix .
For , the Hessian matrix is identical to the Hilbert matrix of dimension :
Hilbert matrices are illconditioned already for dimensions (i.e. have condition numbers greater than about 10,000) and soon reach the limits of 64bit IEEE arithmetic.
Here are the summary results. It takes some time for .
This performs Gram–Schmidt orthonormalization for a special case. For other cases, change the 6 in to an integer between 1 and 10.
Here are the results from the normal equations.
The two solutions agree both symbolically and in exact arithmetic.
But there are numeric differences in IEEE 758 machine arithmetic.
These differences come from the lower error amplification expressed in a lower 2condition number,
The numerical solution via the Gram–Schmidt orthonormalization solution is usually more accurate than the normal equations solution.
Here is the QR factorization. Again, for other cases, change the 3 in to an integer between 1 and 10.
Normal equations.
The difference between the numerical solutions is due to the difference in how their rounding errors propagate.
Because of the lower error amplification expressed in a lower 2condition number, the numerical solution via QR factorization is more accurate than the normal equations solution.
This calculates the convergence order for grids of powers of 2.
Choose problem parameters. Yet again, for other cases, change the 7 in to an integer between 1 and 10.
Here are the approximation errors for grids of powers of 2.
Here is the convergence order. To see the result for another function, select another sample function in . Zero errors are suppressed.
For case 3, all but the first two elements of are zero. For case 5, all but the first element of are zero; therefore the logarithmic plots look incomplete.
Sample functions 3, 4, 5, 6 have discontinuities in the zeroth, first or second derivative; 7 and 8 have singularities in the first or second derivative. These sample functions and do not satisfy all the assumptions of Theorem 1. Therefore the convergence order of the righthand side can be lower than 1 (respectively 2) as predicted by equation (47). Sample functions 1, 2, 9, 10 are infinitely often continuously differentiable; therefore they have maximum convergence order 1 (respectively 2) according to equation (47).
These are the approximation errors for grids of powers of 2.
This takes a few minutes.
This plots the convergence order.
Analyzing polynomial approximation, this article has systematically worked out the close relations between the solutions obtained by:
The interrelations are:
This article has analyzed the polynomial approximation of a realvalued function with respect to the leastsquares norm in different settings:
Three different solution methods for this leastsquares problem have been compared:
All definitions and solution methods were implemented in Mathematica 11.1.
All solution characteristics and their relations were illustrated by symbolic or numeric examples and graphs.
The author thanks Alexandra Herzog and Jonas Gienger for critical proofreading and the anonymous referees for valuable improvements of the paper. The paper evolved from a lecture series in numerical analysis, given by the author for ESOC staff and contractors from 2012 to 2016, using Mathematica:
[1]  E. Hewitt and K. Stromberg, Real and Abstract Analysis, New York: SpringerVerlag, 1975. 
[2]  K. Yosida, Functional Analysis, 5th ed., New York: SpringerVerlag, 1978. 
[3]  P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed., London: Academic Press, 1984. 
[4]  G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Baltimore: The John Hopkins University Press, 2013. 
[5]  E. W. Weisstein. “Vandermonde Determinant” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/VandermondeDeterminant.html. 
[6]  J. D. Faires and R. Burden, Numerical Methods, 3rd ed., Pacific Grove, CA: Thomson/Brooks/Cole, 2003. 
G. Gienger, “Polynomial Approximation,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.192. 
2011–2016 ESA/ESOC research and technology management office
2010–2011 ESOC navigation support office
1987–2009 Mathematical analyst in ESOC Flight Dynamics division; supported more than 20
space missions
1988 Received Dr. rer. nat. from Heidelberg University, Faculty of Science and Mathematics
1984–1987 Teaching assistant at University of Giessen
1982–1984 Research scientist at Heidelberg University
1975–1981 Studied mathematics and physics at Heidelberg University
Gottlob Gienger
European Space Operations Centre ESOC
Research and Technology Management Office OPSGX
RobertBoschStrasse 5, 64293 Darmstadt, Germany
Retired staff
Gottlob.Gienger@gmail.com
The derivation of the scattering force and the gradient force on a spherical particle due to an electromagnetic wave often invokes the Clausius–Mossotti factor, based on an ad hoc physical model. In this article, we derive the expressions including the Clausius–Mossotti factor directly from the fundamental equations of classical electromagnetism. Starting from an analytic expression for the force on a spherical particle in a vacuum using the Maxwell stress tensor, as well as the Mie solution for the response of dielectric particles to an electromagnetic plane wave, we derive the scattering and gradient forces. In both cases, the Clausius–Mossotti factor arises rigorously from the derivation without any physical argumentation. The limits agree with expressions in the literature.
Recently, we made a theoretical study of a system to sort submicrometer dielectric spheres in the interference field of a laser in slowly flowing air [1, 2]. In the course of that project, we derived the scattering and gradient forces rigorously from Maxwell’s equations. The derivation was too detailed for that article, so we are presenting it here. The results agree with expressions from Harada and Asakura [3], as we show in the Appendix. In addition, we present the code we used to derive the force on a spherical particle used in [1, 2].
In this article, we develop code to generate the Mie scattering coefficients and the stress tensor formulas, and we combine them to form first the scattering force and then the gradient force. The scattering force comes first because it requires an incident plane wave, whereas the gradient force requires an incident standing wave that is a little more difficult to set up.
The solution for the response of a spherical dielectric particle in a vacuum was given more than 100 years ago [4]. The problem has been studied extensively by Bohren and Huffman [5]. Our formulation follows the textbook of Zangwill [6]. Since the problem is so well studied, we go directly to the solution. We omit an implicit time dependence in the plane wave traveling in the positive direction with the electric field linearly polarized in the direction. The electric field and magnetic field are thereby given by
where and are scalar functions related to the transverse electric and transverse magnetic parts of the solution, respectively, is the speed of light, and is a point in space. The International System of Units (SI) is used; in these units has the same dimensions as . Compared to the equations in Zangwill, the overall sign differs here, and we have inserted a factor of to simplify the implementation. We can obtain from by the substitution ; we will use this relationship later to simplify the calculation. The scalar function is given by
Here, and are the usual spherical coordinates, and the are associated Legendre polynomials. If , where is a spherical Bessel function, the expressions describe the incident fields for a plane wave in the case of an incident plane wave with spatial dependence . The sum of the incident and scattered fields is given by
for , where is a spherical Hankel function. For , we substitute and in the previous two expressions.
Explicit forms for the Mie coefficients and are given here for a particle with index of refraction and radius . We restrict attention to the case of a nonmagnetic dielectric sphere:
where and are Riccati–Bessel functions; a prime denotes differentiation with respect to the argument. The total scattering cross section is given by
written here in dimensionless form by incorporating the geometric cross section of the spherical particle; this form is commonly denoted by .
The function calculates the Mie coefficients and using equations (5) and (6); the table of those coefficients is . The functions and are and , respectively. The variable (with maximum value ) is an index used for Legendre coefficients in physics. The variable is , the product of the wavevector and the particle radius ; ranges from to with step size . The function returns a list pairing with the cross section normalized to the geometry. A simpler threeargument function calls a fiveargument function. The parameter helps to tell how many terms to calculate to achieve convergence; 1.5 seems to work well, but the reader may wish to test this.
Next, we plot the Mie cross section, similar to the one found in Zangwill [6] and Bohren and Huffman [5]. This is slow: it took 442 seconds on a 3.7 GHz personal computer. The code could be written to run significantly faster, but it would become more cryptic. The main point here is to verify the correctness of the code and to clarify the exposition. The red line is the asymptotic value for large , a dimensionless parameter comparing the particle size to the wavelength. The fact that this value is exactly twice the geometric cross section is discussed in [3, 4].
Figure 1. Mie cross section for a sphere of radius and wavevector compared to the geometric cross section and its large limit, in red. For the physical significance of the sharp peaks, see [7].
Here we develop the Taylor expansion of the Mie coefficients for small . This subsection confirms equations (8–11). The results are used in Section 5.
We start with a simplification.
The following definitions are motivated by equations (5) and (6).
Later, we will find we need these for . The index of refraction is .
The Taylor series is taken next. A high order is necessary even for the small limit. In many cases, the spherical Neumann function enters the calculation, which leads to diverergence in this limit. The real and imaginary parts are represented by the suffixes 1 or 2 in the variable names. The series expansions are chosen so as to include terms to the lowest nonvanishing order.
The results are given here as mathematical formulas, which should agree with results in the cell above. Superscripts 1 and 2 refer to real and imaginary parts.
For a harmonic electromagnetic field, the timeaveraged electromagnetic force on a dielectric particle in a vacuum is given by [1, 2, 4]
where the angle brackets mean the time average, is the differential of the surface normal, and the integral is taken over any surface enclosing the particle. Thus the Maxwell stress tensor is given by
where
and where is the electric constant, is a dyadic, and is the identity matrix. The companion matrix is given by with . (The magnetic constant used in the references is related by .)
Since the particle is spherical, it is natural to do the integral on the surface of a sphere at radius . Since we are in a vacuum, the electromagnetic field exhibits no force there. Looking ahead, we will check to see that the result is independent of . The direction of is , so it is sufficient to calculate a mixedbasis tensor component , where refers to the component along and is a Cartesian direction. Rewriting equation (12) in spherical coordinates leads to
We accept the complexity of the mixedbases tensor because the dot product leads to a single component in spherical coordinates, but the integral requires coordinates that are independent of the point of integration. In practice, these are Cartesian. Looking ahead, we will see that only the component will be nonzero, which further motivates the choice. The calculation will proceed by forming the electric field in spherical coordinates and making a row of , where the index implies spherical coordinates. The vector will be transformed to Cartesian coordinates by rightmultiplying by a rotation matrix. (The same process is used for the magnetic field .)
The effect of the time average is the following: given , . The stress tensor will be computed from the second term. The first term is included later by adding the complex conjugate.
For the scattering force, we take the incident electric field to be a plane wave traveling in the positive direction and linearly polarized in the direction. Explicitly,
As discussed, we need to transform a vector from spherical to Cartesian coordinates. We give that function first.
The electric or magnetic stress tensor is computed from the following code. is a factor required for the time average.
gets the component, since the field components have the usual order (, , ); is the complex conjugate, but we need to help the kernel find simplifications. (The magnetic part will be found by symbolic substitution.) The conjugation is performed as follows.
is used for real expansion.
Following are some additional simplifications that we will use. We operate in the dimensionless radial variable called , which is (i.e. times ), where and do not enter the calculation independently. The assumptions are made because of the range of integration.
We introduced extra factors of into equations (1) and (2) because we implement rather than to enable the use of . The operation is named , the curl for a dimensionless radial variable.
Next we define a function to create the electric and magnetic fields (“em”), omitting the prefactor . Before the final answer for the force is obtained, the factor will be reintroduced as the variable .
This gives a simplified interface for a positive plane wave.
It is sufficient to set for both the scattering and gradient forces, since both occur in the limit of small particles. Setting does not capture all contributing terms to lowest order, and setting or higher does not cause the limit to change. For the scattering, we will have , which is for a plane wave going in the positive direction. For the gradient force where we have a standing wave, we decompose that into the sum of a plane wave going toward and one going toward . The backward wave is still polarized in the direction, so we produce it by mapping . Flipping is necessary because the direction of travel is ; if we wish to preserve , it is necessary to change the sign of . We will always set and ; these may be set to zero to suppress the external field or the induced field, respectively.
We next determine the electric and magnetic fields. Although we could increase , the run time increases dramatically. For example, the case of from [1, 2] was run overnight.
For , use 1 for a quick test, 2 to get the main results, and 3 or more to confirm that a higherorder expansion does not change the limit. The run time increases rapidly as is increased. Run times can vary, but typically it is best to start the code and come back to it after a few minutes to a couple of hours. The parameter is used both here and for the gradient force in Section 4.
Next, we make rows from the electric part and magnetic part (multiplied by ) of the stress tensor in Cartesian coordinates.
We do the integrals over next, yielding the three Cartesian components of the electric part of the stress tensor, after azimuthal integration.
The azimuthal integrals of the and components of the electric part of the stress tensor are zero, as they should be. The component is nonzero in general and will be used later.
The magnetic terms are similar.
The integral over is done next. Although these variables contain the word , they lack some constant factors to be forces, hence the prefix . These factors will be included after a few manipulations. The notation ending with is for the component. The successive variables lack the constant, but the final result is the force.
Combining the electric and magnetic terms into a total force at this point leads to some cancellations, so it is convenient to do it next.
Up to this point, we have done the manipulations without telling the kernel that and are, in fact, spherical Bessel functions and . Moreover, we use (a complex function of real arguments) because we will be obtaining real and imaginary parts shortly.
As discussed previously, we have calculated only one of two terms in the force. Next, we add the complex conjugate.
We introduce real and imaginary parts for the Mie coefficients.
The expression is still quite complicated. However, knows (for ) that the Wronskian of the spherical Bessel functions [8]. We will see that the force does not depend on the radius of the integration sphere, as long as . We expect this on physical grounds: there should be no force on the vacuum region surrounding the physical sphere. Mathematically, the Wronskian plays a key role in achieving this condition.
We next include some constant prefactors, namely , and . The factors arise from the definition of the stress tensor. The factor comes from the fact that when we performed the two angular integrals over the sphere, we did not yet include the dimensional constant associated with the area element. This extra factor of is the only exception to the radius appearing in the dimensioness variable . The substitution allows powers of to be canceled.
The real and imaginary parts of the Mie coefficient are and , respectively, and similarly, for they are and .
In the previous section, we used the general form for the electric and magnetic fields for the Mie expansion as input to the Maxwell stress tensor to derive the force on a particle. We do the same in this section, borrowing from the previous section to the extent possible. However, the spherical particle is in a standing wave field instead of a plane wave traveling in the direction. The standing wave is a linear superposition of two plane waves going toward and toward , so we need a solution from the latter field. We obtain this by symmetry from the existing solution. Sending does what is needed: the polarization of is unchanged, but the direction of propagation changes sign. This is implemented with the wrapper to the function described in Section 3.
Although the standing wave is the sum of two plane waves, more precisely, a phased sum is required. Although we do not show it here, the answer is proportional to . This is because the dielectric sphere is located at the coordinate origin, and we need to have a maximum of the electric field there. We do this as follows. (In the variable names, ▽ stands for gradient.)
From here, the manipulation is the same as for the scattering force, so the code is written without further explanation. Certain terms integrate to zero.
First, the electric terms in the stress tensor are calculated. As previously, the and components
are 0.
Second, the magnetic terms in the stress tensor are calculated. Again, the and components are 0 and the component is nonzero in general.
Finally, the electric and magnetic terms are combined and the force is found.
Our next task is to determine the scattering and gradient forces on spheres up to the leading order in . Physically, these are spheres that are small compared to a wavelength. We combine the results of Section 3 for the scattering force and Section 4 for the gradient force with those of Section 2 for the Mie coefficients. In Section 1.2, we showed that , , , , , , and using superscripts and for the real and imaginary parts, respectively. Therefore, the scattering force, to lowest order in , retains only the terms in , and for the gradient force, the lowestorder coefficient is . The terms were given in equation (9). The key point is that the Clausius–Mossotti factor appears with the second and first powers in the two terms. The result falls out of a Taylor expansion without any appeal to physical arguments about the response of dipoles. All of the terms are proportional to , leading to the physically required result that if the sphere in fact contains a vacuum , there is no electromagnetic response and thus no force. However, the denominator is characteristic only of the terms giving the lowestorder response. Higher terms have different dependencies on the index of refraction , such as , as seen previously. The gradient and scattering forces properly exist only in the limit of small .
Having selected the lowestorder terms for the two force expansions, these are the forces to lowest order in .
These formulas agree with those given by Harada and Asakura [3], as shown in the Appendix.
Our goal was to derive the scattering force and the gradient force rigorously from the Mie solution and the Maxwell stress tensor. We began by presenting the Mie solution for a plane wave incident on a dielectric sphere in a vacuum and showing that our implementation matches a figure from a textbook. We then presented the formula for the force in terms of a surface integral of the Maxwell stress tensor, which we take on a sphere of arbitrary radius centered on and including the whole dielectric sphere. We analyzed the scattering force first, giving a formula for the force in terms of the Mie coefficients and then taking the limit as the radius of the sphere tends to zero. This yields agreement with the usual formula for the scattering force in a vacuum. The forces were reformulated for a standing wave, and a similar program was carried out, leading to agreement for the gradient force.
The Claussius–Mossotti term that appears in expressions for the scattering force and the gradient force is seen to be implicit in the rigorous Mie solutions to Maxwell’s equations. By finding the Maxwell stress tensor for a plane wave or a standing wave acting on a dielectric sphere, we are able to show the response in lowest order is in agreement with a widely used formula for the scattering force and the gradient force, respectively. Since part of the derivation includes a tenthorder Taylor expansion of special functions, it is difficult to see how the result could be obtained without computerassisted algebra.
First, we wish to match equation (17) to the scattering force as given in equation (12) of [3], hereafter called equation (HA12). Variables with superscript (HA) are from the reference. We consider only the case of the external medium being a vacuum, so . This implies , the index in the sphere, so that the Clausius–Mossotti factor is present in both our equation (17) and equation (HA12). The same holds for equation (18) and equation (HA16). Next, [3] considers a Gaussian beam profile, so we simply pick the point in the middle, setting . The factor is the intensity at the center of the beam . The beam intensity is related to the electrical field by . Given these expressions, all the factors can be matched by inspection.
Next, we wish to match equation (18) to the gradient force given in equation (HA16). The one additional equation to note is that . The factor of 2 occurs in the numerator because, by our definition, represents the field of one of the two interfering beams. The 2 in the denominator appears in the intensityfield conversion equation. The twos in are due to the physical fact that a standing plane wave interference pattern is periodic with half the wavelength of the electric fields of the plane waves of which the interference pattern is composed. Again, a match can be made by inspection.
Eric L. Shirley provided a key step in the derivation.
J. J. Curry and Z. H. Levine, “ContinuousFeed Optical Sorting of Aerosol Particles,” Optics Express, 24(13), 2016 pp. 14100–14123. doi:10.1364/OE.24.014100.  
Z. H. Levine and J. J. Curry, “OptSortSph: Optical Sorting in a Standing Wave Field Calculated with Effective Velocities and Diffusion Constants,” Journal of Research of the National Institute of Standards and Technology, 121, 2016 pp. 420–421. doi:10.6028/jres.121.020.  
Y. Harada and T. Asakura, “Radiation Forces on a Dielectric Sphere in the Rayleigh Scattering Regime,” Optics Communications, 124(56), 1996 pp. 529–541. doi:10.1016/00304018(95)007539. 

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der physik, 330(3), 1908 pp. 377–445. onlinelibrary.wiley.com/doi/10.1002/andp.19083300302/pdf.  
C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, New York: Wiley, 1998. 

A. Zangwill, Modern Electrodynamics, Cambridge: Cambridge University Press, 2013.  
P. Chylek, J. T. Kiehl and M. K. W. Ko, “Optical Levitation and PartialWave Resonances,” Physical Review A, 18(5), 1978 pp. 2229–2233. doi:10.1103/PhysRevA.18.2229.  
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th printing, New York: Dover, 1972, Equation (10.1.6). 

Z. H. Levine and J. J. Curry, “Scattering and Gradient Forces from the Electromagnetic Stress Tensor Acting on a Dielectric Sphere,” The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.191. 
Zachary H. Levine is a physicist at NIST and a Fellow of the American Physical Society. He has concentrated on various aspects of the interaction of light and matter, including the calculation of dielectric constants, second harmonic coefficients and higher optical tensors.
J. J. Curry is an electrical engineer at the National Institute of Standards and Technology. His current interests include manipulation and measurement of particles using optical forces. In recent years, he has worked on problems of interest to the lighting industry.
Zachary H. Levine
The National Institute of Standards and Technology
Gaithersburg, MD 208998410
zlevine@nist.gov
J. J. Curry
The National Institute of Standards and Technology
Gaithersburg, MD 208998441
john.curry@nist.gov
This article illustrates how Mathematica can be employed to model stochastic processes via stochastic differential equations to compute trajectories and their statistical features. In addition, we discuss parameter estimation of the model via the maximum likelihood method with global optimization. We consider handling modeling error, system noise and measurement error, compare the stochastic and deterministic models and use the likelihoodratio test to verify the possible improvement provided by stochastic modeling. The Mathematica code is simple and can be easily adapted to similar problems.
Stochastic differential equations (SDEs) have received great attention in a large number of fields, including finance, physics, system biology, biochemical processes and pharmacokinetics. SDEs serve as a natural way of introducing uncertainty into a deterministic model represented by ordinary differential equations (ODEs). In contrast to the classical approach where uncertainty only exists in the measurements, SDEs can provide a more flexible framework to account for deviation in states and parameters that describe the underlying system. For an introduction to the theory and numerical solution of SDEs, see [1].
In this article, we illustrate how Mathematica deals with the problem of simulation and parameter estimation of such systems, represented here by the FitzHugh–Nagumo model describing excitable media. The first section sets up the model equations. The second section deals with the simulation of the stochastic system and computes the trajectories, their distribution at a specified time point and the change of the statistical data of these distributions in time, namely the trajectory of the mean value and standard deviation. The third section describes the parameter estimation of the model via the maximum likelihood method (ML). The last section analyzes the goodness of fit between deterministic and stochastic models using the likelihood ratio test (LRT).
The FitzHugh–Nagumo model for excitable media is a nonlinear model describing the reciprocal dependencies of the voltage across an exon membrane and a recovery variable summarizing outward currents. The model was developed in [2]. The model is general and can also model excitable media, for example, heart tissue. The deterministic ODE model (whitebox model) is described by the following system of ordinary differential equations:
(1) 
(2) 
with parameters , and and initial condition and .
Whitebox models are mainly constructed on the basis of knowledge of physics about the system. Solutions to ODEs are deterministic functions of time, and hence these models are built on the assumption that the future value of the state variables can be predicted exactly. An essential part of model validation is the analysis of the residual errors (the deviation between the true observations and the onestep predictions provided by the model). This validation method is based on the fact that a correct model leads to uncorrelated residuals. This is rarely obtainable for whitebox models. Hence, in these situations, it is not possible to validate ODE models using standard statistical tools. However, by using a slightly more advanced type of equation, this problem can be solved by replacing ODEs with SDEs that incorporate the stochastic behavior of the system: modeling error, unknown disturbances, system noise and so on.
The stochastic SDE graybox model can be considered as an extension of the ODE model by introducing system noise:
(3) 
(4) 
where is a Wiener process (also known as Brownian motion), a continuoustime random walk. The next section carries out the numerical simulation of the SDE model using the parameter settings , and .
Equations (3) and (4) represent an Itostochastic process that can be simulated in Mathematica employing a stochastic Runge–Kutta method.
First, a single realization is simulated in the time interval .
Figure 1. The trajectories of the state variables (blue) and (brown) in the case of a single realization of the Ito process.
The values of and can be found at any time point. Here is an example.
This computes the trajectories for 100 realizations.
Figure 2. The trajectories of the state variables (blue) and (brown) in the case of 100 realizations of the Ito process.
Slice distributions of the state variables can be computed at any time point. First, let us simulate the trajectories in a slightly different and faster way, now using 1000 realizations.
At the time , we can compute the mean and standard deviation for both state variables and , as well as their histograms.
Figure 3. The distribution of (left) and (right) in the case of 100 realizations of the Ito process, at time .
The mean value and the standard deviation of the trajectories along the simulation time can also be computed and visualized. (This may take a few minutes.)
Figure 4. , the mean value of (red) with its standard deviation, (blue).
Parameter estimation is critical since it determines how well the model compares to the measurement data. The measurement process itself may also have serially uncorrelated errors due to the imperfect accuracy and precision of the measurement equipment.
Write the measurement equation as
(5) 
where . The voltage is assumed to be sampled between and at discrete time points , where , , with an additive measurement noise . To get , we consider a single realization and sample it at time points . Then we add white noise with .
Figure 5. A single realization of (brown) and the simulated measured values (black points).
As we have seen, the solutions to SDEs are stochastic processes that are described by probability distributions. This property allows for maximum likelihood estimation. Let the deviation of the measurements from the model be
(6) 
where . Assuming that the density function of can be approximated reasonably well by Gaussian density, the likelihood function to be maximized is
(7) 
For computation, we use its logarithm. Now we accept the model parameter and estimate and from the SDE model, employing the maximum likelihood method.
Here are the values of the .
This defines the logarithmic likelihood function.
The optimization of the likelihood function is not an easy task, since the objective function is often flat, with nondifferentiable terms and many local extrema. In addition, the model takes a long time to evaluate. Instead of using direct global optimization, first we compute the values of the objective function on a grid to use parallel computation to speed up the evaluation. We are looking for the parameters in the range . Let us create the grid points.
This computes the function values at the grid points in parallel.
Now apply interpolation for the grid point data and visualize the likelihood function.
Figure 6. The likelihood function of the parameter estimation problem.
Employing different global optimization methods, we compute the parameters. The third one generates a warning message that we suppress with Quiet.
There are many local maxima. Let us choose the global one.
Here is a new parameter list.
This computes 500 realizations.
This visualizes the result and the measurement data.
Figure 7. The simulated process with the estimated parameters as in Figure 4.
In the previous section, Parameter Estimation, we illustrated the technique of stochastic modeling. The measurement data was simulated with the correct model parameters , no assumed modeling error, existing system noise and measurement error . The model to be fitted had two free parameters, and , since the system noise was chosen for the model.
Now consider a different situation. Suppose that we have modeling error, since the measured values are simulated with ; however, in the model to be fitted, we use . In addition, let us increase the measurement error to . To compare the efficiencies of the deterministic and stochastic models, we should handle as a free parameter in the stochastic model to be fitted. So then we have three free parameters to be estimated: , and . Let us carry out the parameter estimation for different values of .
The results can be seen in Table 1; corresponds to the deterministic model. In order to demonstrate that the stochastic model can provide significant improvement compared with the deterministic model, the likelihoodratio test can be applied [3]. The likelihoodratio test has been used to compare two nested models. In our case, one of the models is the ODE model, with fewer parameters than the SDE model. The null hypothesis is that the two models are basically the same. The test statistic is
(8) 
where the subscripts and stand for the deterministic and stochastic models, respectively.
The distribution of is , where is the difference in the number of parameters between the two models; in our case, . Here is the critical value for at a confidence level of 95%.
If the value of is less than the critical value, then the null hypothesis can be accepted. In our case, . Therefore, we reject the null hypothesis, which means the SDE model can be considered as a different model providing a significant improvement compared with the ODE model.
For changing parameters to simulate the measured values, you should modify the content of the list in the Simulation section and the value of in the Measurement Values subsection. To change the parameter of the model to be fitted, change the value of in the list in the Likelihood Function subsection.
In reality, the three parameters should be handled simultaneously during the optimization process.
Assuming there is no modeling error, using the SDE model, we can separate the measurement error from the system noise represented by the estimated of the fitted model; see [4].
The advantage of using stochastic modeling is that the actual states in the model are predicted from data. This allows the prediction to stay close to the data even when the parameters in the model are imprecise. It has been demonstrated that Mathematica can significantly support a user carrying out stochastic modeling. There are many builtin functions that help in the computation of the trajectories of stochastic differential equations and their statistics, global optimization for parameter estimation and likelihoodratio test for model verification.
The author is grateful for the EU FP7 International Research Staff Exchange Scheme, which provided the financial support that partly facilitated his stay at the Department of Mechanical Engineering of the University of Canterbury, NZ, where this article was completed.
[1]  E. Allen, Modeling with Itô Stochastic Differential Equations, Dordrecht, Netherlands: Springer, 2007. 
[2]  R. FitzHugh, “Impulses and Physiological States in Theoretical Models of Nerve Membrane,” Biophysics Journal, 1(6), 1961 pp. 445–466. doi:10.1016/S00063495(61)869026. 
[3]  H. Madsen and P. Thyregod, Introduction to General and Generalized Linear Models, Boca Raton: CRC Press, 2010. 
[4]  C. W. Tornøe, J. L. Jacobsen, O. Pedersen, T. Hansen and H. Madsen, “GreyBox Modelling of Pharmacokinetic/Pharmacodynamic Systems,” Journal of Pharmacokinetics and Pharmacodynamics, 31(5), 2004 pp. 401–417. doi.10.1007/s1092800483238. 
B. Paláncz, “Stochastic Simulation and Parameter Estimation of the FitzHugh–Nagumo Model,” The Mathematica Journal, 2016. dx.doi.org/doi:10.3888/tmj.186. 
Dr. Béla Paláncz is professor emeritus at the Department of Photogrammetry and Geoinformatics at the Faculty of Civil Engineering of the Budapest University of Economics and Technology, in Budapest, Hungary. His main interests are mathematical modeling and numericsymbolic computations. He is the coauthor of several books and more than a hundred papers.
Dr. Béla Paláncz:
Budapest University of Economics and Technology
1111 Műegytem rkp. 3
Budapest, Hungary
palancz@epito.bme.hu
This series of articles showcases a variety of applications around the theme of inversion, which constitutes a strategic way of manipulating configurations of circles and lines in the plane. This article includes the Riemann sphere, rings of four tangent circles, and inverting the Sierpinski sieve.
The story of inversion, propelled by the study of conic sections and stereographic projection, is quite complex [1]. Although Vietá (1540–1603) already spoke of mutually inverted points, the proper development started with a plethora of notable geometers including L’Huilier (1750–1840), Dandelin (1794–1847), Quetelet (1796–1874), Steiner (1796–1863), Magnus (1790–1861), Plücker (1801–1868), Bellavitis (1803–1880), and Simpson (1849–1924). Steiner, in a manuscript published in 1913, is considered to have been the first to formulate inversion as a method to systematically simplify the study of complex geometric figures where circles play a prominent role. This work culminated in 1855 with the studies of Möbius (1790–1868) (hence the choice of the letter for the inversive circle in this article) [2]. Peaucellier (1832–1913) also applied inversion to his famous linkage [3], and Lord Kelvin (1824–1907) applied inversion to elasticity.
Let denote either the line or the segment from to , depending on the context. The length of the segment is denoted by . The circle with center and radius is denoted by . Denote by the inverse of in the current circle of inversion , where is either a symbolic point or often ; in Manipulate results, is often shown as a red dashed circle.
The following functions encapsulate the basic properties of inversive geometry developed in the first part of this series [4].
The function squareDistance computes the square of the Euclidean distance between two given points. (It is more convenient to use the following definition than the builtin Mathematica function SquaredEuclideanDistance.)
The function collinearQ tests whether three given points are collinear. When exactly two of them are equal, it gives True, and when all three are equal, it gives False, because there is no unique line through them.
The function computes the circle passing through the points , , and . If the points are collinear, it gives the line through them; if all three points are the same, it returns an error message, as there is no meaningful definition of inversion in a circle of zero radius.
The function computes the inverse of in the circle . The object can be a point (including the special point that inverts to the center of ), a circle, or a line (specified by two points).
Here is an example.
Since a circle can be inverted into a line, define a generalized circle to be either an ordinary circle or a line, as in the first article in this series. A consequence of theorem 2 in the next section is that the set of generalized circles is closed under inversion in a circle. The following Manipulate shows that as , the circle tends to the line , and the inverse of the circle tends to its reflection in the line , the circle .
Therefore, it makes sense to define inversion in a line to be reflection.
Here is an example.
The function squareDistanceToLine computes the square of the distance of a given point to a given line.
The function exCircles computes the four circles tangent to the lines forming the sides of a triangle having given vertices.
The function inCircle computes the incircle of a triangle having given vertices as the excircle of smallest radius.
The function redPoint is used to mark a given point in red.
The function draws an arc of radius r and center b from the line ab to the line bc.
The function intersections computes the possible points of intersection of a given line and circle. (There may be zero, one, or two points.)
The following Manipulate shows several of the functions introduced so far.
The concept of orthogonality between circles plays an important role in inversive geometry. Two intersecting circles are orthogonal if they have perpendicular tangents at either point of intersection. It is rather amusing to consider orthogonality of circles without mentioning any notion of perpendicularity [3, 4], as is done in the following definition.
Definition
The following Manipulate shows the central circle is orthogonal to the black one by showing the other two circles (blue). Some values of the positions of the black circle or the radii of the blue circles make it impossible to close the three circles, so adjustments to close them are done on the fly.
Theorems 1–11 review the basic properties of inversion and introduce some of its remarkable properties. Consider all inversions to be with respect to the circle .
Theorem 1
The following Manipulate illustrates this property and shows that triangles and are similar. The gray circle is orthogonal to .
The following interesting variation of theorem 1 was presented at the Swiss Mathematical Contest
in 1999 [5, 6]:
Theorem 2
For instance, invert the circle in .
The set of circles and lines, called generalized circles, is therefore closed under inversion.
Theorem 3
Theorem 4
Theorem 5
Theorem 6
Proof
By the definition of inversion, . Then . Since , the result follows. □
Theorem 7
To verify this result, notice that in the following expression (the condition implies does not pass through and thus indeed inverts into a circle).
Theorem 8
.
Subtracting the first points of the following lines obtained by inversion gives the result.
Theorem 9
The following Manipulate shows the terms mentioned in theorem 9.
Proof
Let and be the ends of the diameter of shown in the figure above. The point is outside the segment , as otherwise would not be defined. Then
and the result follows. □
For instance, consider inverting in .
Then , which we verify as follows.
In the case of theorem 7, this is the corresponding comparison.
In order to verify this result, assume that and , with and (otherwise would not be a circle).
From this theorem, it follows that the product of the lengths of the tangents from to and
must be .
Theorem 10
Proof
Invert , , and in the circle . Then and (by theorem 6). Substituting this and similar results for , , , and in the inequality gives
which reduces to ; this always applies in the triangle . Equality holds if and only if the points , , and are collinear, that is, if and only if points , , , and are concyclic or collinear. □
Ptolemy (circa 127 AD) compiled much of what today is the pseudoscience of astrology. His Earthcentered universe held sway for 1500 years. In the words of Carl Sagan, in the first episode of his TV series Cosmos, “… showing that intellectual brilliance is no guarantee against being dead wrong.” You can find fascinating applications of Ptolemy’s theorem in [9, 10].
Theorem 11
This result is useful when inverting to concentric circles elsewhere. In order to verify it, assume without loss of generality that the two circles are and , with , and that the orthogonal circle is . Then the following quantity is independent of , and it is equal to the position of one of the two fixed points mentioned.
The following provides a framework with which to interpret inversion. Consider a sphere of unit radius centered at the origin. Draw a line from the north pole to a point on the sphere. The point at which intersects the  plane defines a onetoone correspondence between points on the sphere and points in the plane; is called stereographic projection. The image of the south pole is the origin. The image of the north pole is not defined, but is introduced as a new point to serve as the image of ; this makes the mapping continuous and onetoone. In the context of stereographic projection, the sphere is referred to as the Riemann sphere after the prominent mathematician Bernhard Riemann (1826–1866), who studied under Steiner and earned his PhD degree under Gauss [11, 12].
Similar triangles give
By using these formulas, it is possible to prove that the stereographic projection of a circle on is a generalized circle in the plane. (Circles through go to lines and other circles go to circles.) So inversion in the unit circle in the plane induces a map from circles to circles on .
This defines stereographic projection.
An inversive pair of points maps to points reflected in the  plane.
Moreover, as the following Manipulate shows, an inversive pair of generalized circles maps to circles on that are reflections across the  plane. You can vary the radius and center of the control circle in the  plane. The circle inverts to in the unit circle, which is the equator of . You can show the stereographic cone joining to through its stereographic projection and the stereographic cone joining to through its stereographic projection , which is the reflection of in the  plane [13].
Amusing applications related to stereographic projection are found in [14, 15], and interesting Demonstrations in [16–18].
Consider a ring of four cyclically externally tangent circles as in the following Manipulate. The four points of tangency are concyclic (pink circle), even when some of the circles are internally tangent. You can drag the four points around this circle to alter the shape of the arrangement. To see many other similar patterns, see [19]. In contrast to the case of three tangent circles, the circle through the points of tangency is not necessarily orthogonal to the other four circles.
The function perturb slightly varies three points that are coincident or collinear.
Let us apply inversion to deduce the concyclic property. Invert in a circle centered at one of the tangent points, for instance, the red dashed circle in the following Manipulate. That inversion transforms the circles centered at and into two parallel lines and two circles in between these lines. These four inversions are sequentially tangent in three points. The problem then reduces to show that the three points of tangency lie on the same line (shown in blue).
Let us take a closer look at the arrangement formed by the inversion of the chain of circles, starting from a different point of view. The following Manipulate shows an arrangement of two parallel lines, one slanted line, and two circles with the tangencies indicated. The two orange disks indicate adjustable tangency points and of the circles and the parallel lines. It is easy to show that the slanted line always passes through the tangency point of the circles regardless of the positions of and and the value of one of the radii, say, of the lower circle. In fact, the sum of the radii remains constant for fixed positions of and . For some values of the radii there exists a third circle also tangent to the parallel lines and the circles. Selecting the appropriate option in the Manipulate, the radii are adjusted according to the position of the slanted line for this third (blue) circle to exist. When the slanted line is vertical and the circles are congruent, three of those third circles exist.
The next Manipulate inverts the previous arrangement. It shows that the tangent points of the set of (blue) pairwise tangent circles are concyclic. It also shows that there exist up to three circles tangent to the circles of , and sets of up to four circles orthogonal to the circles of . You can vary the center of the inversive circle. Similar examples can be found at [20, 21].
The quadrilateral joining the centers of the circles forming the ring has an inscribed circle. This is easily seen using Pitot’s theorem (1695–1771): a convex quadrilateral with consecutive side lengths , , , is tangential, that is, it has an inscribed circle, if and only if . Oddly enough, in the case of the quadrilateral, its incircle does not necessarily coincide with the circle passing through the four points of tangency, as the following result shows.
The Sierpinski sieve (or Sierpinski gasket, or Sierpinski triangle) [5, 6], named after the prolific Polish mathematician Waclaw Sierpinski (1882–1969), is a selfsimilar subdivision of a triangle [22, 23]. The function Sierpinski constructs the iteration corresponding to the recursive definition of this subdivision.
Inverting the vertices of the triangles from the first four iterations in the unit circle gives rise to stunning patterns.
The following Manipulate lets you vary the radius and center of the inverting circle, the number of iterations, and the type of transformation applied to the vertices forming the triangles forming the Sierpinski sieve.
I would like to thank the anonymous referee whose thoughtful and detailed comments on this article greatly improved its presentation, and also Queretaro’s Institute of Technology, which provided essential support for the completion of this second part.
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[2]  B. C. Patterson, “The Origins of the Geometric Principle of Inversion,” Isis, 19(1), 1933 pp. 154–180. www.jstor.org/stable/225190. 
[3]  J. RangelMondragón. “Inversive Geometry II: The Peaucellier Inversor Mechanism” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/InversiveGeometryIIThePeaucellierInversorMechanism. 
[4]  J. RangelMondragón, “Inversive Geometry: Part 1. Inverting Generalized Circles, Ellipses, Polygons, and Tilings,” The Mathematica Journal, 15, 2013. doi:10.3888/tmj.157. 
[5]  R. Todev, Geometry Problems and Solutions from Mathematical Olympiads, MathOlymps, 2010. 
[6]  D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium, 2nd ed., New York: Springer, 2011. 
[7]  D. E. Blair, Inversion Theory and Conformal Mapping, Providence, RI: The American Mathematical Society, 2000. 
[8]  H. Fukagawa and D. Pedoe, Japanese Temple Geometry Problems: San Gaku, Winnipeg, Canada: The Charles Babbage Research Centre, 1989. 
[9]  A. S. Posamentier, Advanced Euclidean Geometry, Emeryville, CA: Key College Publishing, 2002. 
[10]  E. W. Weisstein. “Ptolemy’s Theorem” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PtolemysTheorem.html. 
[11]  D. Pedoe, Geometry, A Comprehensive Course, New York: Dover Publications Inc., 1988. 
[12]  E. W. Weisstein. “Riemann Sphere” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/RiemannSphere.html. 
[13]  T. Needham, Visual Complex Analysis, Oxford: Claredon Press, 1997. 
[14]  D. Gehrig, “The Orloj,” The Mathematica Journal, 7(4), 2000. www.mathematicajournal.com/issue/v7i4/features/gehrig. 
[15]  P. W. Kuchel, “Spatial Inversion: Reflective Anamorphograms,” The Mathematical Journal, 9(2), 2004. www.mathematicajournal.com/issue/v9i2/SpatialInversion.html. 
[16]  J. RangelMondragón. “A General Cone” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/AGeneralCone. 
[17]  P. W. Kuchel, “Anamorphoscopes: A Visual Aid for Circle Inversion,” Mathematical Gazette, 63(424), 1979 pp. 82–89. 
[18]  E. Mahieu. “Inverse Stereographic Projection of Simple Geometric Shapes” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/InverseStereographicProjectionOfSimpleGeometricShapes. 
[19]  A. Akopyan, Geometry in Figures, CreateSpace Independent Publishing Platform, 2011. 
[20]  J. RangelMondragon. “Problems on Circles IV: Circles Tangent to Two Others with a Given Radius” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ProblemsOnCirclesIVCirclesTangentToTwoOthersWithAGivenRadius. 
[21]  J. RangelMondragon. “Problems on Circles X: Tangent Circles Generate Ellipses” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ProblemsOnCirclesXTangentCirclesGenerateEllipses. 
[22]  E. W. Weisstein. “Sierpiński Sieve” from MathWorld–A Wolfram Web Resource. mathworld.wolfram.com/SierpinskiSieve.html. 
[23]  A. MarquezRaygoza. “Oftenpaper.net: The Sierpinski Triangle Page to End Most Sierpinski Triangle Pages.” (Dec 5, 2016) www.oftenpaper.net/sierpinski.htm. 
J. RangelMondragón, “Inversive Geometry,” The Mathematica Journal, 2016. dx.doi.org/doi:10.3888/tmj.185. 
Jaime RangelMondragón received M.Sc. and Ph.D. degrees in Applied Mathematics and Computation from the School of Mathematics and Computer Science at the University College of North Wales in Bangor, UK. He was a visiting scholar at Wolfram Research, Inc. and held positions in the Faculty of Informatics at UCNW, the Center of Literary and Linguistic Studies at the College of Mexico, the Department of Electrical Engineering at the Center of Research and Advanced Studies, the Center of Computational Engineering (of which he was director) at the Monterrey Institute of Technology, the Department of Mechatronics at the Queretaro Institute of Technology and the Autonomous University of Queretaro in Mexico, where he was a member of the Department of Informatics and in charge of the Academic Body of Algorithms, Computation, and Networks. His research included combinatorics, the theory of computing, computational geometry, and recreational mathematics. Jaime RangelMondragón died in 2015.
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