Template:Tesseract family
In geometry, this family of uniform 4-polytopes has diploid hexadecachoric symmetry,[1] [4,3,3], of order 24*16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.
| B4 symmetry polytopes | |||||||||||
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| Name | tesseract | rectified tesseract |
truncated tesseract |
cantellated tesseract |
runcinated tesseract |
bitruncated tesseract |
cantitruncated tesseract |
runcitruncated tesseract |
omnitruncated tesseract | ||
| Coxeter diagram |
   
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| Schläfli symbol |
{4,3,3} | t1{4,3,3} r{4,3,3} |
t0,1{4,3,3} t{4,3,3} |
t0,2{4,3,3} rr{4,3,3} |
t0,3{4,3,3} | t1,2{4,3,3} 2t{4,3,3} |
t0,1,2{4,3,3} tr{4,3,3} |
t0,1,3{4,3,3} | t0,1,2,3{4,3,3} | ||
| Schlegel diagram |
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| B4 | 
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| Name | 16-cell | rectified 16-cell |
truncated 16-cell |
cantellated 16-cell |
runcinated 16-cell |
bitruncated 16-cell |
cantitruncated 16-cell |
runcitruncated 16-cell |
omnitruncated 16-cell | ||
| Coxeter diagram |
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| Schläfli symbol |
{3,3,4} | t1{3,3,4} r{3,3,4} |
t0,1{3,3,4} t{3,3,4} |
t0,2{3,3,4} rr{3,3,4} |
t0,3{3,3,4} | t1,2{3,3,4} 2t{3,3,4} |
t0,1,2{3,3,4} tr{3,3,4} |
t0,1,3{3,3,4} | t0,1,2,3{3,3,4} | ||
| Schlegel diagram |
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| B4 | 
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References
[edit]- ^ Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups