https://www.shadertoy.com/view/cd3fWs

Here's a GLSL script hosted in Shadertoy that extends the concept of Steiner chains from simple circles to Generalized Circles, and renders them in real-time with user controls.

The code permits specification of any containing circle or half-plane you like, specify the center of the inner circle (or half-plane) around which the chain circles are arranged, and specify the contact angle of the chain itself, with reference to either the containing or inner circle.

This demo chooses the configuration of the containing circle, the user controls the center of the inner circle with the mouse.

Click-and-drag the mouse to move the center of the white circle; use the arrow keys to fix the orientation of the key chain circle (the dark red one) to a particular direction. Various config parameters start around line 90.

I'd love feedback and suggestions, let me know!

Under the hood it's all based on finding just the right FLT to map a trivially-constructed concentric chain of length n within the unit circle to the desired config.

Hardcore fans may recall that a pre-rendered version of this was my first ever post : https://mathstodon.xyz/@KleinianArborist/110674924302552811

#glsl #steinerchains #flt #shadertoy

Some more Steiner Chains, where one of the circles has twice the radius of the others.

#codeart #mathart #steinerchains #mastoart

I wanted to know how to specify and create all possible Steiner chains in the extended complex plane.
This was made much easier by the fact that Steiner chains are closed under mobius transformations.

Attached is an animation illustrating the result. Also available at

https://www.youtube.com/watch?v=u-_vr7cPUt

I used these sources heavily while putting this together:

[1] Pedoe, D. (1970) Geometry, a Comprehensive Course, Cambridge University Press, Cambridge.

[2] https://en.wikipedia.org/wiki/Steiner_chain

[3] https://en.wikipedia.org/wiki/Generalised_circle

[4] https://en.wikipedia.org/wiki/M%C3%B6bius_transformation

#steinerchains #steinersporism #mobius #mobiustransforms

#introduction

Hi, I'm Sônia (she/her), and this is my hobbyist computational-math account. I decided to invest some time into _Indra's Pearls_ by Mumford, Series, Wright a little over a decade ago, and a quest that started out with "do all the assignments" has turned into a personal research and artistic program.

I spend a lot of time thinking about sets of tangent circles. So much so that if I got a "revive an ancient mathematician for a day" boon, I'd spend it on Apollonius. I'm pretty sure that my guy from Perga would a) be fascinated by the beauty of Kleinian double-cusp groups and b) tell me we were cheating by using calculations instead of construction. Sorry, fans of Euclid, Brahmagupta, and person-who-invented-zero. #academicnecromancy

This account is intended to be primarily used to post artwork/videos and links to interactive demos, explanations of the techniques used to render them, and engaging with any discussions prompted by the preceding.

#kleiniangroups #steinerchains #fractionallineartransforms #mobius #fractals #R #glsl #shadertoy #indraspearls #mobiustransforms