Tesseractihexadecachoron
Tesseractihexadecachoron | |
---|---|
Rank | 4 |
Type | Uniform |
Notation | |
Bowers style acronym | Tah |
Coxeter diagram | o4x3x3o () |
Elements | |
Cells | 16 truncated tetrahedra, 8 truncated octahedra |
Faces | 32 triangles, 24 squares, 64 hexagons |
Edges | 96+96 |
Vertices | 96 |
Vertex figure | Digonal disphenoid, edge lengths 1 (base 1), √2 (base 2) and √3 (sides) |
Measures (edge length 1) | |
Circumradius | |
Hypervolume | |
Dichoral angles | Toe–6–tut: 120° |
Tut–3–tut: 120° | |
Toe–4–toe: 90° | |
Central density | 1 |
Number of external pieces | 24 |
Level of complexity | 6 |
Related polytopes | |
Army | Tah |
Regiment | Tah |
Dual | Disphenoidal enneacontahexachoron |
Conjugate | None |
Abstract & topological properties | |
Flag count | 2304 |
Euler characteristic | 0 |
Orientable | Yes |
Properties | |
Symmetry | B4, order 384 |
Convex | Yes |
Nature | Tame |
The tesseractihexadecachoron, or tah, also commonly called the bitruncated tesseract, is a convex uniform polychoron that consists of 16 truncated tetrahedra and 8 truncated octahedra. 2 truncated tetrahedra and 2 truncated octahedra join at each vertex. It is the medial stage of the truncation series between a tesseract and its dual hexadecachoron. As such is could also be called a bitruncated 16-cell.
Gallery[edit | edit source]
-
B4 Coxeter plane projection
-
Alternate view, centered on truncated octahedron
-
Stereographic projection
-
Net
-
Card with vertex figure, cell counts, and cross-sections
Vertex coordinates[edit | edit source]
The vertices of a tesseractihexadecachoron of edge length 1 are all permutations of:
- .
Alternatively it can be given under D4 symmetry as all even sign changes of:
- .
Representations[edit | edit source]
The tesseractihexadecachoron has the following Coxeter diagrams:
- o4x3x3o () (full symmetry)
- x3x3x *b3o () (D4 symmetry, great rhombated demitesseract)
- s4o3x3x () (as runcicantic tesseract)
- s4x3x3o () (also as snub)
- ooqoo4xuxux3xooox&#xt (B3 axial, truncated octahedron-first)
- xxuxoo3xuxxux3ooxuxx&#xt (A3 axial, truncated tetrahedron-first)
- Qqo ooq4xux3xoo&#zx (B3×A1 symmetry)
Semi-uniform variant[edit | edit source]
The tesseractihexadecachoron has a semi-uniform variant of the form o4x3y3o that maintains its full symmetry. This variant uses 16 semi-uniform truncated tetrahedra of form x3y3o and 8 semi-uniform truncated octahedra of form o4x3y as cells, with 2 edge lengths.
With edges of length a (of square faces) and b (of triangular faces), its circumradius is given by and its hypervolume is given by .
It has coordinates given by all permutations of:
- .
There is also a semi-uniform variant with D4 symmetry known as the great rhombated demitesseract.
Related polychora[edit | edit source]
Uniform polychoron compounds composed of tesseractihexadecachora include:
- Bitruncated demidistesseract (2)
- Bitruncated great icositetrachoron (3)
- Bitruncated great stellated tetracontoctachoron (6)
- Bitruncated dodecahedronary hexacosichoron (75)
External links[edit | edit source]
- Bowers, Jonathan. "Category 7: Bitruncates" (#299).
- Bowers, Jonathan. "Tessic Isogonals".
- Klitzing, Richard. "tah".
- Quickfur. "The Bitruncated Tesseract".
- Wikipedia contributors. "Bitruncated tesseract".