favorite theorem click HERE
The interaction of probability and geometry was at the heart of my PhD Thesis.
it is very useful to twist up an octahedron.
In fact, I'd highly recommend
turning the octahedron
inside out.
In order to find out why, click
here.
This utilizes the notion of Scissors Congruence.
see it cleaned up,
see the mess...
Watch the the relationship between the two as the mess cleans up its act.
Explore the mysterious property of hyperbolic volume used to perform this clean up.
construct the "empty sphere" pattern,
connect the vertices of empty spheres,
and see the random Delaunay triangulation!
Here are a few more.
See how to use random triangulation to understand conformal geometry.
The pattern is Penrose Tilling.
This is a random Penrose tillling.