Doyle Spirals and Möbius Transformations

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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




"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].


[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)

[3] A. Sutcliffe, "Doyle Spiral Circle Packings Animated." (Feb 2, 2017)

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