More fun with honeycombs in hyperbolic 3-space... Rectification
More fun with honeycombs in hyperbolic 3-space... Rectification
Check out the four compact regular H³ honeycombs rectified. Rectification is a special case of truncation, where you truncate all the way to the midpoints of edges.
Regular H³ honeycombs contain one type of polyhedron, but rectifying them yields a new honeycomb composed of two different types of polyhedra. One is the original polyhedron rectified, and the other is the vertex figure of the parent honeycomb.
Because rectified honeycombs have two cell types, they are not regular, but they still have a great deal of symmetry and are also called uniform or Archimedean honeycombs.
Let's see what two polyhedra live in a rectified {5,3,4}, the cover image of this post. The parent honeycomb is built from dodecahedra, and the the vertex figure is an octahedron. So the rectified honeycomb contains rectified dodecahedra (a.k.a. icosidodecahedra) and octahedra, all fitting neatly together.
(aside: you can read off the vertex figure of a regular honeycomb by looking at the last two numbers in the Schläfli symbol. In this case, {3,4} denotes the octahedron.)
It's fun to predict what polyhedra the rectification operation will produce using the honeycomb Schläfli symbols, and to check your reasoning against the images. Try it with the {4,3,5}, {5,3,5}, and {3,5,3}!
Links for further study
Visual Insight post by John Baez describing the {5,3,5} honeycomb and Schläfli symbols:
http://blogs.ams.org/visualinsight/2014/02/01/535-honeycomb/
Wiki article on rectification:
http://en.wikipedia.org/wiki/Rectification_(geometry)
Images of regular hyperbolic honeycombs:
https://plus.google.com/112844794913554774416/posts/ejoqNvGnD5U
Check out the four compact regular H³ honeycombs rectified. Rectification is a special case of truncation, where you truncate all the way to the midpoints of edges.
Regular H³ honeycombs contain one type of polyhedron, but rectifying them yields a new honeycomb composed of two different types of polyhedra. One is the original polyhedron rectified, and the other is the vertex figure of the parent honeycomb.
Because rectified honeycombs have two cell types, they are not regular, but they still have a great deal of symmetry and are also called uniform or Archimedean honeycombs.
Let's see what two polyhedra live in a rectified {5,3,4}, the cover image of this post. The parent honeycomb is built from dodecahedra, and the the vertex figure is an octahedron. So the rectified honeycomb contains rectified dodecahedra (a.k.a. icosidodecahedra) and octahedra, all fitting neatly together.
(aside: you can read off the vertex figure of a regular honeycomb by looking at the last two numbers in the Schläfli symbol. In this case, {3,4} denotes the octahedron.)
It's fun to predict what polyhedra the rectification operation will produce using the honeycomb Schläfli symbols, and to check your reasoning against the images. Try it with the {4,3,5}, {5,3,5}, and {3,5,3}!
Links for further study
Visual Insight post by John Baez describing the {5,3,5} honeycomb and Schläfli symbols:
http://blogs.ams.org/visualinsight/2014/02/01/535-honeycomb/
Wiki article on rectification:
http://en.wikipedia.org/wiki/Rectification_(geometry)
Images of regular hyperbolic honeycombs:
https://plus.google.com/112844794913554774416/posts/ejoqNvGnD5U
In the style of the polygloss, o5x3o4o, o4x3o5o, and o5x3o5o. The edge2 are 0.618, 0.8944 and 1.5124.
ReplyDeleteThese are gorgeous, Roice! I wish we could fly through them in our WebGL app.
ReplyDeleteThanks Scott! There are close ties between H3 and S3 reorientations when H3 is in the ball model and S3 is stereographically projected. So there is potential for adaption in our web app!
ReplyDeleteThere's just so much data here (about 1 million edges in each picture), a huge hurdle to do in realtime. I wish there was hardware accelerated ray tracing.
Indeed, but even with just simple lighting these would be amazing, and I'm sure WebGL could handle it. I've been talking with Henry Segerman about doing a WebGL renderer for H3, so he can fly through his ideal triangulations. I need to get started on that, and we'll kill two birds with one stone. I trust you'll be able to help us with camera transformations, etc.
ReplyDeleteYou're really making a habit of it. Cool!
ReplyDelete