User:Tomruen/Flat toroid polyhedron
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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.
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]1,0 | 2,0 | 3,0 | 4,0 |
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1,1 | 2,2 = 2(1,1) | 3,3 = 3(1,1) | 4,4 = 4(1,1) |
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2,1 | 3,1 | 3,2 | 4,1 | 4,2 = 2(2,1) | 4,3 |
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χ | g | Schläfli | n | Vert. | Edges | Faces | Graph1 | Graph2 | Pattern |
---|---|---|---|---|---|---|---|---|---|
0 | 1 | {4,4}1,0 | 1 | 1 | 2 | 1 | ![]() |
![]() Projection onto torus |
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0 | 1 | {4,4}2,0 | 4 | 4 | 8 | 4 | ![]() |
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0 | 1 | {4,4}3,0 | 9 | 9 | 18 | 9 | ![]() |
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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 | ![]() |
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0 | 1 | {4,4}2,2 | 8 | 8 | 16 | 8 | ![]() |
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0 | 1 | {4,4}3,3 | 18 | 18 | 36 | 18 | ![]() |
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0 | 1 | {4,4}4,4 | 32 | 32 | 64 | 32 | ![]() |
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0 | 1 | {4,4}2,1 | 5 | 5 | 10 | 5 | ![]() |
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0 | 1 | {4,4}3,1 | 10 | 10 | 20 | 10 | ![]() |
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0 | 1 | {4,4}3,2 | 13 | 13 | 26 | 13 | ![]() |
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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 | ![]() |
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0 | 1 | {4,4}4,3 | 25 | 25 | 50 | 25 | ![]() |
File:Regular map 4-4 4-3-rect.png |
Hexagonal forms
[edit]χ | g | Schläfli | t | Vert. | Edges | Faces | Graph | Pattern |
---|---|---|---|---|---|---|---|---|
0 | 1 | {3,6}1,0 | 1 | 1 | 3 | 2 | ![]() |
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0 | 1 | {3,6}1,1 | 3 | 3 | 9 | 6 | ![]() |
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0 | 1 | {3,6}2,0 | 4 | 4 | 12 | 8 | ![]() |
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0 | 1 | {3,6}2,1 | 7 | 7 | 21 | 14 | ![]() | |
0 | 1 | {3,6}2,2 | 12 | 12 | 36 | 24 | ![]() |
χ | g | Schläfli | t | Vert. | Edges | Faces | Graph | Pattern | Realization |
---|---|---|---|---|---|---|---|---|---|
0 | 1 | {6,3}1,0 | 1 | 2 | 3 | 1 | ![]() |
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0 | 1 | {6,3}1,1 | 3 | 6 | 9 | 3 | ![]() |
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0 | 1 | {6,3}2,0 | 4 | 8 | 12 | 4 | ![]() |
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![]() Petrial cube |
0 | 1 | {6,3}2,1 | 7 | 14 | 21 | 7 | ![]() |
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0 | 1 | {6,3}2,2 | 12 | 24 | 36 | 12 | ![]() |
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χ | g | Schläfli | t | Vert. | Edges | Faces | Graph | Pattern | Realization |
---|---|---|---|---|---|---|---|---|---|
0 | 1 | r{6,3}1,0 | 1 | 3 | 6 | 3 | ![]() |
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0 | 1 | r{6,3}1,1 | 4 | 9 | 18 | 9 | ![]() |
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0 | 1 | r{6,3}2,0 | 4 | 12 | 24 | 12 | ![]() |
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![]() Octahemioctahedron |