# Doyle Spirals and Möbius Transformations

Initializing live version

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Doyle spirals are special logarithmic spirals of touching circles in which every circle is surrounded by a corona of six touching circles. A linear fractional transformation (or Möbius transformation) is applied to map such spirals (in particular, circle packings) into double spirals.

Contributed by: Dieter Steemann (February 2017)
Open content licensed under CC BY-NC-SA

## Details

Controls

"spiral" "" = number of spiral arms "" = number of steps (circles) per spiral revolution

"type of graphic" "basic": spiral with parameters and "Möbius": basic spiral under Möbius transformation "- graph": spiral elements along and axes in basic spiral

"colors" select the color bar to use

"item" visualization of the circle packing using a circle, disk or sphere

"", "" switch to use the same color along or axis (see "- graph")

A logarithmic spiral starts at the origin and winds around the origin at an ever-increasing distance. The Möbius transformation maps the points of the real axis of the complex plane to the points on the real axis. Because Möbius transformations preserve circles, we get a new circle packing in the shape of a double spiral centered at and on the real axis.

This Demonstration was inspired by [1] and artistic images in [2]. More about this subject can be found at [3].

References

[1] D. Mumford, C. Series and D. Wright, Indra's Pearls: The Vision of Felix Klein, Cambridge: Cambridge University Press, 2006 pp. 62.

[2] J. Leys, "Hexagonal Circle Packings and Doyle Spirals." (Feb 2, 2017) www.josleys.com/articles/HexCirclePackings.pdf.

[3] A. Sutcliffe, "Doyle Spiral Circle Packings Animated." (Feb 2, 2017) archive.bridgesmathart.org/2008/bridges2008-131.html.

## Permanent Citation

Dieter Steemann

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send