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
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tesseract
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rectified tesseract
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truncated tesseract
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cantellated tesseract
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runcinated tesseract
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bitruncated tesseract
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cantitruncated tesseract
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runcitruncated tesseract
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omnitruncated tesseract
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Coxeter diagram
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Template:CDD
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Template:CDD = Template:CDD
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Template:CDD
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Template:CDD
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Template:CDD
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Template:CDD = Template:CDD
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Template:CDD
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Template:CDD
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Template:CDD
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Schläfli symbol
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{4,3,3}
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t1{4,3,3} r{4,3,3}
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t0,1{4,3,3} t{4,3,3}
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t0,2{4,3,3} rr{4,3,3}
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t0,3{4,3,3}
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t1,2{4,3,3} 2t{4,3,3}
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t0,1,2{4,3,3} tr{4,3,3}
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t0,1,3{4,3,3}
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t0,1,2,3{4,3,3}
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Schlegel diagram
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B4
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Name
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16-cell
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rectified 16-cell
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truncated 16-cell
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cantellated 16-cell
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runcinated 16-cell
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bitruncated 16-cell
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cantitruncated 16-cell
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runcitruncated 16-cell
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omnitruncated 16-cell
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Coxeter diagram
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Template:CDD = Template:CDD
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Template:CDD = Template:CDD
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Template:CDD = Template:CDD
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Template:CDD = Template:CDD
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Template:CDD
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Template:CDD = Template:CDD
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Template:CDD = Template:CDD
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Template:CDD
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Template:CDD
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Schläfli symbol
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{3,3,4}
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t1{3,3,4} r{3,3,4}
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t0,1{3,3,4} t{3,3,4}
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t0,2{3,3,4} rr{3,3,4}
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t0,3{3,3,4}
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t1,2{3,3,4} 2t{3,3,4}
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t0,1,2{3,3,4} tr{3,3,4}
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t0,1,3{3,3,4}
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t0,1,2,3{3,3,4}
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Schlegel diagram
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B4
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- ↑ Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups