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
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
WOW! Can I use some of these in Visual Insight? They're great!
ReplyDeleteThanks! Okay, I'll send you some of those AMS forms.
ReplyDeleteSpectacular work Roice!
ReplyDeleteThank you for sharing.
ReplyDeleteTom Reun is a very good artist. I have had many conversations with him on thr email about these sprt of things.
ReplyDeleteThe lead piccie looks like {5,3,6}, with a resolution of something near 194.
Roice Nelson Fantastic indeed!
ReplyDeletejawdropping
ReplyDeleteNice to, ic.
ReplyDelete