Hyperbolic Forms
There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin diagrams for each family.
From these 9 families, there are a total of 76 unique honeycombs generated:
- : - 9 forms
- : - 15 forms
- : - 9 forms
- : - 11 forms (7 overlap with family, 4 are unique)
- : - 9 forms
- : - 6 forms
- : - 9 forms
- : - 9 forms
- : - 6 forms
The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is in the {3,5,3} family.
There are also 23 noncompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity:
| 7 | , |
|---|---|
| 7 | , ,, |
| 6 | , |
| 3 | , |
Read more about this topic: Convex Uniform Honeycomb
Famous quotes containing the word forms:
“Pervading nationalism imposes its dominion on man today in many different forms and with an aggressiveness that spares no one.... The challenge that is already with us is the temptation to accept as true freedom what in reality is only a new form of slavery.”
—Pope John Paul II (b. 1920)