### New things: Hyperbolic honeycombs.

**Originally shared by Henry Segerman**

New things: Hyperbolic honeycombs. These sculptures are joint work with Roice Nelson and show various tilings of 3-dimensional hyperbolic space.

http://www.youtube.com/watch?v=YzzJGeiucNg

Perfect presentation!

ReplyDeleteAt the back one sees {6,3,4}, and in the front, {4,3,5} and its dual {5,3,4}. The {4,3,5} contains the vertices of the {5,3,4}, but the cubes are the same size as one might inscribe in the dodecahedra of {5,3,4}. The order of resolution is about 16.

ReplyDeleteHi Wendy. The one in the back is a {6,3,3}. I'm curious, what do you mean by "the order of resolution"?

ReplyDeleteYou'e right about {6,3,3}. The order of resolution is a measure of the edge detail, something to do with the mean horocyclic diameter squared, in terms of sextants. They're ideas from my Hyperbolic geometry without the tears.

ReplyDeleteThanks, can you point me to your hyperbolic geometry without the tears?

ReplyDeleteAt the moment, no. It,s based on the idea that if you use a euclidean ruler in hyperbolic space, you can do euclidean geometry. But the intersect between euclidean and hyperbolic geometries is a circle. So most of the trick is to draw a chain of circles. The actual rulers measure the square of distances, which greatly simplifies the maths.

ReplyDelete