**Spirals Based on the Fibonacci Sequence**
A polar coordinate system can be used to create spirals based on the Fibonacci sequence with *Mathematica*. The formulas used to determine the *n*th Fibonacci number yield complex results when *n* is a non-integer. The internal algorithm employed by the *Mathematica* command `Fibonacci[` *n*`]` drops the imaginary component of any complex results. As a result, *y* = `Fibonacci[` *x*`]` is a continuous function and generally increasing in the set of positive real numbers. Therefore, *r* = `Fibonacci[` `]` yields a spiral. The next program uses a parametric coordinate system to generate such a spiral.
Another method of generating spirals from the Fibonacci sequence is based on the arrangement of adjacent squares that increase
in size following the sequence of Fibonacci numbers. This method always yields logarithmic spirals. A line drawn from the
origin to any point on such a spiral will form a constant angle with the tangent line at that point. It is unknown whether
the spirals *r* = `Fibonacci[` `]` or spirals produced in a similar manner possess any of the properties of the logarithmic spirals.
By manipulating the equations for the spirals in a three-dimensional parametric coordinate system, it is possible to produce
graphic images that resemble shells.
Copyright © 2001 Wolfram Media, Inc. All rights reserved.[Article Index][Prev Page][Next Page] |