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Showing posts from March, 2014

A honeycomb in hyperbolic space

Image
Originally shared by John Baez

A honeycomb in hyperbolic space

Hyperbolic space is a 3d space where the angles of a triangle add up to less than 180 degrees.  A new kind of space lets you imagine new kinds of patterns - mathematical art that doesn't quite fit into our universe!

This drawing by Roice Nelson shows what you'd see if you lived in hyperbolic space and filled this space with a {6,3,3} honeycomb.

Hyperbolic space is curved, but each hexagon here lies on a flat plane inside hyperbolic space.  Each of these plane is tiled with hexagons in the usual way, 3 meeting at each corner. 

However, each edge in this picture has 3 different planes like this going through it!   Planes of hexagons, 3 hexagons meeting at a corner, with each edge lying on 3 planes - that's the reason for the symbol {6,3,3}.

For much more about this honeycomb and its relatives, visit my Visual Insight blog:

http://blogs.ams.org/visualinsight/2014/03/15/633-honeycomb/

It contains a puzzle for true math fan…