Jaime Rangel-Mondragón

Part 3: Quandles, Inverting Triangles to Triangles and Inverting into Concentric Circles

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 built-in 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 well-known 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 six-element 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 two-element 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].

Inverting a Circle into Another

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.

Case 1. and Are External to Each Other

Theorem 1

Let and not intersect and be external to each other; say is to the left of and assume . Then is at a distance from along the line , is to the right of , and .
Additionally, is orthogonal to every circle tangent to both and , and is orthogonal to every circle orthogonal to and .

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.)

Case 2. Is inside

Theorem 2

Suppose and do not intersect, and let be inside . Then is at a distance from along the line joining the centers, and .
Additionally, is orthogonal to every circle orthogonal to and .

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 .

Case 3. and Intersect

Theorem 3

Let and intersect. Then there are two midcircles taking to , corresponding to the midcircles in theorems 1 and 2.

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.

Inverting Two Circles into Congruent Circles

Theorem 4

Every circle centered on the midcircle of and inverts and into two congruent circles.


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].

Inverting Two Circles into Concentric Circles

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

For and , let the point , where . Construct the line at perpendicular to the line AB. Then is the radical axis of and .

This checks that any point on has tangents to and of equal length.

Theorem 6

With as in theorem 5, the circle is orthogonal to and .

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

Two nonconcentric, nonintersecting circles can be inverted into two concentric circles [11].


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

The inverse of in or is .

Line and Circle into Concentric Circles

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

Let the circle and the nonintersecting line be such that on is the nearest point to , making perpendicular to the line . Then the circle , where , is orthogonal to both and . Either of the two intersections of with the line can serve as the center of a circle of inversion inverting and into concentric circles.


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. Rangel-Mondragon, Inversive Geometry: Part 1, The Mathematica Journal, 15, 2013. doi:10.3888/tmj.157.
[2] J. Rangel-Mondragon, Inversive Geometry: Part 2, The Mathematica Journal, 18, 2014. doi:10.3888/tmj.18-5.
[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. 197208. doi:10.4310/HHA.2005.v7.n1.a11.
[5] H. S. M. Coxeter, Inversive Geometry, Educational Studies in Mathematics, 3(3), 1971
pp. 310321. 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 MathWorldA Wolfram Web Resource. mathworld.wolfram.com/RadicalLine.html.
[9] S. R. Murthy. “Radical Axis and Radical Center” from the Wolfram Demonstrations ProjectA Wolfram Web Resource. demonstrations.wolfram.com/RadicalAxisAndRadicalCenter.
[10] J. Rangel-Mondragón, The Arbelos, The Mathematica Journal, 16, 2014. doi:10.3888/tmj.16-5.
[11] D. E. Blair, Inversion Theory and Conformal Mapping, Providence, RI: American Mathematical Society, 2000.
J. Rangel-Mondragón, Inversive Geometry: Part 3, The Mathematica Journal, 2017. dx.doi.org/doi:10.3888/tmj.19-4.

About the Author

Jaime Rangel-Mondragón 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 Rangel-Mondragón died in 2015.