Jump to content

User:Tomruen/Flat toroid polyhedron

From Wikipedia, the free encyclopedia

Flat toroid polyhedron

[edit]

Regular maps of the form {4,4}m,0 can be represented as the finite regular skew polyhedron {4,4 | m}, seen as the square faces of a m×m duoprism in 4-dimensions.

Regular maps with zero Euler characteristic[1]
χ g Schläfli Vert. Edges Faces Group Order Notes
0 1 {4,4}b,0
n=b2
n 2n n [4,4](b,0) 8n Flat toroidal polyhedra
0 1 {4,4}b,b
n=2b2
n 2n n [4,4](b,b) 8n Flat toroidal polyhedra
0 1 {4,4}b,c
n=b2+c2
n 2n n [4,4]+
(b,c)
4n Flat chiral toroidal polyhedra
0 1 {3,6}b,0
t=b2
t 3t 2t [3,6](b,0) 12t Flat toroidal polyhedra
0 1 {3,6}b,b
t=2b2
t 3t 2t [3,6](b,b) 12t Flat toroidal polyhedra
0 1 {3,6}b,c
t=b2+bc+c2
t 3t 2t [3,6]+
(b,c)
6t Flat chiral toroidal polyhedra
0 1 {6,3}b,0
t=b2
2t 3t t [3,6](b,0) 12t Flat toroidal polyhedra
0 1 {6,3}b,b
t=2b2
2t 3t t [3,6](b,b) 12t Flat toroidal polyhedra
0 1 {6,3}b,c
t=b2+bc+c2
2t 3t t [3,6]+
(b,c)
6t Flat chiral toroidal polyhedra

Generators

[edit]

Group: [4,4]+
b,c
, order 4(b2+c2):

Given rotation angles:

Generators:

Square forms

[edit]
{4,4} class 1
1,0 2,0 3,0 4,0
{4,4} class 2
1,1 2,2 = 2(1,1) 3,3 = 3(1,1) 4,4 = 4(1,1)
{4,4} class 3
2,1 3,1 3,2 4,1 4,2 = 2(2,1) 4,3
Square toroidal polyhedra
χ g Schläfli n Vert. Edges Faces Graph1 Graph2 Pattern
0 1 {4,4}1,0 1 1 2 1
Projection onto torus
0 1 {4,4}2,0 4 4 8 4
0 1 {4,4}3,0 9 9 18 9
0 1 {4,4}4,0 16 16 32 16
Projected onto torus

{4,4|4}
0 1 {4,4}1,1 2 2 4 2
0 1 {4,4}2,2 8 8 16 8
0 1 {4,4}3,3 18 18 36 18
0 1 {4,4}4,4 32 32 64 32
0 1 {4,4}2,1 5 5 10 5
0 1 {4,4}3,1 10 10 20 10
0 1 {4,4}3,2 13 13 26 13
0 1 {4,4}4,1 17 17 34 17 File:Regular map 4-4 4-1-rect.png
0 1 {4,4}4,2 20 20 40 20
0 1 {4,4}4,3 25 25 50 25 File:Regular map 4-4 4-3-rect.png

Hexagonal forms

[edit]
Triangular toroidal polyhedra
χ g Schläfli t Vert. Edges Faces Graph Pattern
0 1 {3,6}1,0 1 1 3 2
0 1 {3,6}1,1 3 3 9 6
0 1 {3,6}2,0 4 4 12 8
0 1 {3,6}2,1 7 7 21 14
0 1 {3,6}2,2 12 12 36 24
Hexagonal toroidal polyhedra
χ g Schläfli t Vert. Edges Faces Graph Pattern Realization
0 1 {6,3}1,0 1 2 3 1
0 1 {6,3}1,1 3 6 9 3
0 1 {6,3}2,0 4 8 12 4
Petrial cube
0 1 {6,3}2,1 7 14 21 7
0 1 {6,3}2,2 12 24 36 12
Uniform Hexagonal toroidal polyhedra
χ g Schläfli t Vert. Edges Faces Graph Pattern Realization
0 1 r{6,3}1,0 1 3 6 3
0 1 r{6,3}1,1 4 9 18 9
0 1 r{6,3}2,0 4 12 24 12
Octahemioctahedron
  1. ^ Coxeter and Moser, Generators and Relations for Discrete Groups, 1957, Chapter 8, Regular maps, 8.3 Maps of type {4,4} on a torus, 8.4 Maps of type {3,6} or {6,3} on a torus