User:Tomruen/Flat toroid polyhedron

Further information: Regular_map_(graph_theory) § Flat_toroidal_polyhedra

Flat toroid polyhedron

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

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

Group: [4,4]⁺
_b,c, order 4(b²+c²):

Given rotation angles:

$\alpha _{1}=2\pi c/(b^{2}+c^{2})$

$\alpha _{2}=2\pi b/(b^{2}+c^{2})$

Generators:

$\sigma _{1}=\left[{\begin{smallmatrix}cos(\alpha _{1})&-sin(\alpha _{1})&0&0\\sin(\alpha _{1})&cos(\alpha _{1})&0&0\\0&0&cos(\alpha _{2})&-sin(\alpha _{2})\\0&0&sin(\alpha _{2})&cos(\alpha _{2})\end{smallmatrix}}\right]$

$\sigma _{2}=\left[{\begin{smallmatrix}cos(\alpha _{2})&sin(\alpha _{2})&0&0\\-sin(\alpha _{2})&cos(\alpha _{2})&0&0\\0&0&cos(\alpha _{1})&-sin(\alpha _{1})\\0&0&sin(\alpha _{1})&cos(\alpha _{1})\end{smallmatrix}}\right]$

Square forms

{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

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