β°' is the hemi-di-4n+2-gon.
β° is the hemi-4n+2-hosohedron.
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| Α | n ≥ 0, no. of edges is odd |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| α | { 2n+1, 4n+2 }2 | 1, 2, 2n+1 | 4n+2,2n+1 | n |
| α' | { 4n+2, 2n+1 }2 | 2, 1, 2n+1 | 2n+1,4n+2 |
| α'° | { 2, 2n+1 }4n+2 | 2, 2n+1, 2n+1 | 2n+1,1 | 0 |
| α'°' | { 2n+1, 2 }4n+2 | 2n+1, 2, 2n+1 | 1,2n+1 |
| α°' | { 4n+2, 2 }2n+1 | 2n+1, 1, 2n+1 | 1,4n+2 | 1 |
| α° | { 2, 4n+2 }2n+1 | 1, 2n+1, 2n+1 | 4n+2,1 |
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α'° is the 2n+1-hosohedron.
α'°' is the di-2n+1-gon.
α°' is the hemi-di-4n+2-gon.
α° is the hemi-4n+2-hosohedron.
γ° is the 2n+1-hosohedron.
γ°' is the di-2n+1-gon.
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| Β | n > 0, no. of edges is even |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| β' | { 4n, 2 }4n | 2n, 1, 2n | n | 1 |
| β | { 2, 4n }4n | 1, 2n, 2n | 4n,1 |
| β° | { 4n, 4n }2 | 1, 1, 2n | 4n,4n | n |
|
| Γ | n ≥ 0, no. of edges is even |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| γ' | { 2n+2, 2 }2n+2 | 2n+2, 2, 2n+2 | 1,2n+2 | 0 |
| γ | { 2, 2n+2 }2n+2 | 2, 2n+2, 2n+2 | 2n+2,1 |
| γ° | { 2n+2, 2n+2 }2 | 2, 2, 2n+2 | 2n+2,2n+2 | n |
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|
| Δ | n = ±1 modulo 6, no. of edges is odd |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| δ | { 6, 3n }2n | 2, n, 3n | 3n,3 | n |
| δ' | { 3n, 6 }2n | n, 2, 3n | 3,3n |
| δ'° | { 2n, 6 }3n | n, 3, 3n | 3,n | 2n-1 |
| δ'°' | { 6, 2n }3n | 3, n, 3n | n,3 |
| δ°' | { 3n, 2n }6 | 3, 2, 3n | n,3n | (n-1)*3/2 |
| δ° | { 2n, 3n }6 | 2, 3, 3n | 3n,n |
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| Ε | n = ±2 modulo 6, no. of edges is even |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| ε | { 6, 3n }n | 2, n, 3n | 3n,3 | n |
| ε' | { 3n, 6 }n | n, 2, 3n | 3,3n |
| ε'° | { n, 6 }3n | n, 6, 3n | 3,n/2 | n-2 |
| ε'°' | { 6, n }3n | 6, n, 3n | n/2,3 |
| ε°' | { 3n, n }6 | 6, 2, 3n | n/2,3n | (n-2)*3/2 |
| ε° | { n, 3n }6 | 2, 6, 3n | 3n,n/2 |
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| Ζ | n odd, no. of edges a multiple of 4 but not 8 |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| ζ | { 4, 2n }4n | 4, 2n, 4n | n,2 | n |
| ζ' | { 2n, 4 }4n | 2n, 4, 4n | 2,n |
| ζ'° | { 4n, 4 }2n | 2n, 2, 4n | 2,4n | n |
| ζ'°' | { 4, 4n }2n | 2, 2n, 4n | 4n,2 |
| ζ°' | { 2n, 4n }4 | 2, 4, 4n | 4n,n | 2n |
| ζ° | { 4n, 2n }4 | 4, 2, 4n | n,4n |
| |
| Η | n even and positive, no. of edges a multiple of 8 |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV, mF | Genus |
| η | { 4, 4n }4n | 2, 2n, 4n | 4n, 2 | n |
| η' | { 4n, 4 }4n | 2n, 2, 4n | 2,4n |
| η° | { 4n, 4n }4 | 2, 2, 4n | 4n,4n | 2n-1 |
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| Θ | n even, no. of edges a multiple of 8 |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV, mF | Genus |
| θ | { 4, 2n }2n | 4, 2n, 4n | n, 2 | n-1 |
| θ' | { 2n, 4 }2n | 2n, 4, 4n | 2, n |
| θ° | { 2n, 2n }4 | 4, 4, 4n | n, n | 2n-3 |
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| |
| Ι | n > 0, no. of edges a multiple of 16 |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV, mF | Genus |
| ι | { 4, 8n }4 | 4, 8n, 16n | 4n, 1 | 4n-1 |
| ι' | { 8n, 4 }4 | 8n, 4, 16n | 1, 4n |
| ι° | { 4, 4 }8n | 4, 4, 16n | 4n, 4n | 8n-3 |
| |
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| Κ | n odd, no. of edges is twice an odd square |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| κ' |
{ 4, 2n }4 |
2n, n2, 2n2 |
2,1 |
n2-2n+2 |
| κ |
{ 2n, 4 }4 |
n2, 2n,2n2 |
1,2 |
| κ° |
{ 4, 4 }2n |
n2, n2, 2n2 |
1,1 |
1 |
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| Λ | n>0, n even, no. of edges is twice an even square |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| λ' |
{ 4, 2n }4 |
2n, n2, 2n2 |
2,1 |
n2/2-n+1 |
| λ |
{ 2n, 4 }4 |
n2, 2n,2n2 |
1,2 |
| λ° |
{ 4, 4 }2n |
n2, n2, 2n2 |
1,1 |
1 |
|
|
κ° are those {4,4}
tesellations of the
torus with the
designation (a,0),
a odd.
μ° are those {4,4}
tesellations of the
torus with the
designation (a,a).
|
| Μ | n > 0, no. of edges is 4 × a square |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| μ' |
{ 4, 2n }4 |
4n, 2n2, 4n2 |
1,1 |
(n-1)2 |
| μ |
{ 2n, 4 }4 |
2n2, 4n, 4n2 |
1,1 |
| μ° |
{ 4, 4 }2n |
2n2, 2n2, 4n2 |
1,1 |
1 |
|
λ° are those {4,4}
tesellations of the
torus with the
designation (a,0),
a even.
|
|
|
| Ν | n > 0, no. of edges modulo 6 is 0 |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV, mF | Genus |
| ν |
{ 4, 3n }3n |
4, 3n, 6n |
n,2 |
3n-2 |
| ν' |
{ 3n, 4 }3n |
3n, 4, 6n |
2,n |
| ν° |
{ 3n, 3n }4 |
4, 4, 6n |
n, n |
3n-3 |
| |
|
| Ξ | n > 0, no. of edges is 3 × a square |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| ξ | { 6, 3 }2n | 2n2, n2, 3n2 | 1,1 | 1 |
| ξ' | { 3,6 }2n | n2, 2n2, 3n2 | 1,1 |
| ξ'° | { 2n, 6 }3 | n2, 3n, 3n2 | 1,1 | 2n2-3n+2 |
| ξ'°' | { 6, 2n }3 | 3n, n2, 3n2 | 1,1 |
| ξ°' | { 3, 2n }6 | 3n, 2n2, 3n2 | 1,1 | n(n-3)/2+1 |
| ξ° | { 2n,3 }6 | 2n2, 3n, 3n2 | 1,1 |
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| Ο | n > 0, no. of edges is 9 × a square |
|---|
| name | Schläfli formula | Vertices, Faces, Edges | mV,mF | Genus |
| ο | { 6, 3 }6n | 6n2, 3n2, 9n2 | 1,1 | 1 |
| ο' | { 3,6 }6n | 3n2, 6n2, 9n2 | 1,1 |
| ο'° | { 6n, 6 }3 | 3n2, 3n, 9n2 | 1,3 | 6n2-3n+2 |
| ο'°' | { 6, 6n }3 | 3n, 3n2, 9n2 | 3,1 |
| ο°' | { 3, 6n }6 | 3n, 6n2, 9n2 | 3,1 | 3n(n-1)/2+1 |
| ο° | { 6n,3 }6 | 6n2, 3n, 9n2 | 1,3 |
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