# Hyperbolic Spidrons

A Spidron is
an alternating sequence of isosceles triangles of decreasing
size that assemble into a spiral arrangement. Two Spidrons can be
joined by a half-turn to create an attractive S-shaped double spiral.

If a Spidron is constructed with the correct angles, a number of them
can be arranged around a regular polygon. Alternatively, *n*
Spidrons can be constructed inside a regular *n*-gon by
drawing a decreasing sequence of inscribed regular and star polygons.

This regular polygon construction adapts itself naturally to other
spaces. Most notably, it applies to polygons in the hyperbolic
plane. And in the hyperbolic plane, there are infinitely many regular
tilings to work with!

I've done some experiments drawing hyperbolic Spidron arrangements.
The results have the characteristic look of all hyperbolic pictures,
but are fun to look at. Note how different colourings bring out
the different geometric features of the tilings. One other thing
to observe is that these Spidron arrangements give you a sense of
infinity in two different ways. There's the infinity of ever smaller
Spidrons as we approach the edge of the bounding circle, the infinity
that Escher sought to capture on a finite page. But within each
ring of Spidrons, there's also an infinitely decreasing sequence of
triangles that make up each spiral. The tiling as a whole is an
infinite collection of infinite objects.

## Gallery

## Other resources

Spidrons are the work of
Dániel Erdély. The interested reader should
consult the papers by Erdély and Marc Pelletier in the
2005 and 2006 Bridges Conferences.
All images are copyright 2007 by Craig S. Kaplan. You are
free to use them for **personal** and **non-commercial** purposes.
Please check with me about any other uses.