Roice's G+ Posts

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 p...

Originally shared by Scott Vorthmann Drawing Heptagons Using Integers Would it surprise you to learn that the two figures below were completely generated by manipulating integers ? http://nbviewer.ipython.org/github/vorth/ipython/blob/master/heptagons/HeptagonNumbers.ipynb

Honeycombs! Here is a complete set of images of all 15 regular honeycombs in hyperbolic 3-space. Thanks to Tom Ruen for encouraging me to make these for wikipedia. A honeycomb is when you take a bunch of polyhedra and pack them together with no gaps. The polyhedra in a honeycomb are called "cells". The background colors group the honeycombs as follows: Teal:  Cells are finite Blue:  Cells have "ideal" vertices (vertices that live at infinity) Green:  Cells have an infinite number of facets Cyan:  Cells have ideal vertices and an infinite number of facets All the images show the honeycombs in the Poincare Ball model, with the camera placed either at the origin or on the boundary of the ball. They were generated with custom C# code and rendered with POV-Ray. Links for further study "Regular Honeycombs in Hyperbolic Space", Coxeter: http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf YouTube video by Henry Segerman explaining a few of ...

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

Originally shared by Erno Rubik A 5-dimensional Analogue of the Rubik's Cube You can now try to solve a 5-dimensional Rubik's Cube. How many permutations of the 3x3 cube in 5-dimensions?  Approx. 7.0 x 10^560 Good luck. Visit here to download the program and learn more: http://www.gravitation3d.com/magiccube5d/index.html #mathematics   #geometry   #rubikscube

A more accurate rendering of the {6,3,3}.  This shows the geodesic edges more properly, as arcs orthogonal to the "plane at infinity" of hyperbolic space (vs. straight lines).

Originally shared by John Baez This the {6,3,3} honeycomb , as drawn by Roice Nelson. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3d analogue of a tiling of the plane.  But besides honeycombs in ordinary Euclidean space, we can also have them in hyperbolic space .  This is a curved 3d space.  The {6,3,3} honeycomb lives in hyperbolic space.  That's why it looks weirdly distorted.  Actually all the hexagons are the same size... but we have to warp hyperbolic space to draw it in ordinary space. You can learn more about all these concepts by going to my new blog, Visual Insight : http://blogs.ams.org/visualinsight/2013/09/15/633-honeycomb-in-upper-half-space/ But let me just answer one obvious question: why is it called the {6,3,3} honeycomb?   {6,3,3} is a Schläfli symbol.    The symbol for the hexagon is {6}. The symbol for the hexagonal tiling of the plane is {6,3} because 3 hexagons meet at each vertex. S...