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4.6.4
Isometries in Poincare's disk model
This is my hometown, Jinan — a city with nearly ten million residents, yet with a surprisingly low presence on the internet. It’s famous for its springs, with hundreds of them scattered throughout the very heart of the city. The three largest are Baotu Spring, Five Dragon Pool, and Black Tiger Spring. These springs feed into the city moat and Daming Lake. Because the spring water is incredibly clear, you’ll find here perhaps the clearest moat in the world — something remarkably rare for a city of this size.
The Kontorovich–Nakamura crystallographic sphere packings made with a Python script:
https://sites.math.rutgers.edu/~alexk/crystallographic/
The attached figures are:
+ Bi(7), with clusters = {3, 4}. Integral.
+ Bi(14), with clusters = {1, 8}, non-integral
I also implemented a simple version of Vinberg’s algorithm - it's not rigorous, but it just about manages to compute the root systems of reflections group for the quadratic forms \(-p x_0^2 + x_1^2 + \cdots + x_n^2\). I'm planning to try rendering the 3D sphere packing corresponding to such quadratic forms. (Append to my long long to-do list)
@matthewarcus@mathstodon.xyz hasn’t been active for a long time, and I’m a bit worried about him. Does anyone know if he’s doing okay? Maybe something came up in his life and he had to step away for a bit. Either way, I just hope he’s doing well!
My two recent shadertoy works on reproducing figures from Indra's pearls:
https://www.shadertoy.com/view/WcsBzH
https://www.shadertoy.com/view/tcSyWK
The 2/5 double cusps group and the limit set of the limit of 1/n double cusps group:
Shadertoy animation about the mapping class group of the torus and Sturmian words:
https://www.shadertoy.com/view/3cXyDS
It’s interactive—you can input a rational slope \(p/q\) and a twist matrix in \(\mathrm{SL}(2,Z)\).
This twist acts on the slope, mapping \(p/q\) to another slope \(p′/q′\). On the torus, it shows the effect of the matrix twisting the torus, as well as the geodesic corresponding to \(p′/q′\).
If I understand correctly, this is exactly the so called Dehn twist on the torus.
A few periodic planar solution of the three-body problem.
The simulation is done with vispy +glsl.
Source code: https://github.com/neozhaoliang/pywonderland/blob/master/src/shader-playground/3body.py