In geometry, there are a sequence of regular polytopes and honeycombs, {4,3,p}, with cubic cells. The first is the finite tesseract in 4-dimensional space. The second is the cubic honeycomb that tessellates Euclidean 3-space. The next two tessellate hyperbolic 3-space.
| {4,3,p} regular honeycombs | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Space | S3 | E3 | H3 | ||||||||
| Form | Finite | Affine | Compact | Paracompact | Noncompact | ||||||
Name    
|
{4,3,3}
|
{4,3,4}    
|
{4,3,5}
|
{4,3,6}    
|
{4,3,7}
|
{4,3,8} 
|
...
{4,3,∞}
| ||||
| Image |
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|
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Vertex figure
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 {3,3}
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 {3,4}
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 {3,5}
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 {3,6}
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 {3,7}
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 {3,8}
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 {3,∞}
| ||||
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2015-04-02 at the Wayback Machine) Table III
- N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130. [1]
In geometry, there are a sequence of regular polytopes and honeycombs, {4,3,p}, with cubic cells. The first is the finite tesseract in 4-dimensional space. The second is the cubic honeycomb that tessellates Euclidean 3-space. The next two tessellate hyperbolic 3-space.
| {4,3,p} regular honeycombs | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Space | S3 | E3 | H3 | ||||||||
| Form | Finite | Affine | Compact | Paracompact | Noncompact | ||||||
| Name
|
{4,3,3}
|
{4,3,4}
|
{4,3,5}
|
{4,3,6}
|
{4,3,7}
|
{4,3,8}
|
...
{4,3,∞}
| ||||
| Image |
|
|
|
|
|
|
| ||||
|
Vertex figure
|
{3,3}
|
{3,4}
|
{3,5}
|
{3,6}
|
{3,7}
|
{3,8}
|
{3,∞}
| ||||
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2015-04-02 at the Wayback Machine) Table III
- N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130. [1]