Honeycombs!

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 these honeycombs with physical models:
Hyperbolic honeycombs

An short intro post on honeycombs:
http://roice3.blogspot.com/2013/09/the-dual-534-and-435.html

Wikipedia list of regular honeycombs:
https://en.wikipedia.org/wiki/List_of_regular_polytopes#Tessellations_of_hyperbolic_3-space

Poincare Ball Model:
https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model














Comments

  1. John BaezDecember 27, 2013 at 9:48 AM

    WOW!  Can I use some of these in Visual Insight?   They're great!

    ReplyDelete
  2. John BaezDecember 27, 2013 at 12:29 PM

    Thanks!  Okay, I'll send you some of those AMS forms.

    ReplyDelete
  3. Tom RuenDecember 27, 2013 at 2:45 PM

    Spectacular work Roice!

    ReplyDelete
  4. Sonaam RanaDecember 27, 2013 at 5:18 PM

    Thank you for sharing.

    ReplyDelete
  5. wendy kriegerDecember 27, 2013 at 7:34 PM

    Tom Reun is a very good artist. I have had many conversations with him on thr email about these sprt of things.

    The lead piccie looks like {5,3,6}, with a resolution of something near 194.

    ReplyDelete
  6. annarita rubertoDecember 29, 2013 at 11:23 AM

    Roice Nelson Fantastic indeed!

    ReplyDelete
  7. Tiberiu IgrisanDecember 29, 2013 at 12:25 PM

    jawdropping

    ReplyDelete
  8. Tonydakota AlexanderApril 3, 2014 at 2:10 PM

    Nice to, ic.

    ReplyDelete

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