Unscheduled Post: Hyperbolic Polydron
What happens when you put seven equilateral triangles round a point?
Inspired by sumidiot who was himself inspired by division by zero who were making paper hyperbolic soccer balls, I thought I would put up some pictures of a similar project with polydron. The construction is simple. Attach seven triangles round a vertex this gives 7 outside vertices. Add triangles so these all have exactly seven around them and repeat. The surface flexes in most interesting ways.
Interestingly it is still an open question whether, assuming zero thickness and perfect hinges, this construction can be continued for ever, embedding a model of the hyperbolic plane in 3 dimensions. If you take eight triangles round a vertex I believe it is possible as you can regularly fold things up and down.
David Richter has some more thoughts and discussion.