Doyle Spirals and Möbius Transformations

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spiral
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4
Q
30
type of graphic
basic
Möbius
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colors
item
circle
disk
sphere
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Q
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.

Details

Controls
"spiral"
"
P
" = number of spiral arms
"
Q
" = number of steps (circles) per spiral revolution
"type of graphic"
"basic": spiral with parameters
P
and
Q
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"Möbius": basic spiral under Möbius transformation
"
P
-
Q
graph": spiral elements along
P
and
Q
axes in basic spiral
"colors"
select the color bar to use
"item"
visualization of the circle packing using a circle, disk or sphere
"
P
", "
Q
"
switch to use the same color along
P
or
Q
axis (see "
P
-
Q
graph")
A logarithmic spiral starts at the origin and winds around the origin at an ever-increasing distance. The Möbius transformation
z↦(z-1)/(z+1)
maps the points
(0,1,∞)
of the real axis of the complex plane to the points
(-1,0,1)
on the real axis. Because Möbius transformations preserve circles, we get a new circle packing in the shape of a double spiral centered at
+1
and
-1
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.

External Links

Doyle Spirals
Logarithmic Spiral (Wolfram MathWorld)

Permanent Citation

Dieter Steemann
​
​"Doyle Spirals and Möbius Transformations"​
​http://demonstrations.wolfram.com/DoyleSpiralsAndMobiusTransformations/​
​Wolfram Demonstrations Project​
​Published: February 3, 2017