Truncated small ditrigonary hexacosihecatonicosachoron

Truncated small ditrigonary hexacosihecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Tissidtixhi
Coxeter diagram	x3x3o3o5/2*b ()
Elements
Cells	600 truncated tetrahedra, 120 small ditrigonary icosidodecahedra, 120 truncated great icosahedra
Faces	2400 triangles, 1440 pentagrams, 2400 hexagons
Edges	1200+7200
Vertices	2400
Vertex figure	Ditrigonal pyramid, base edge lengths 1 and (√5–1)/2, side edge length √3
Measures (edge length 1)
Circumradius	${\sqrt {7}}\approx 2.64575$
Hypervolume	$5{\frac {794{\sqrt {5}}-405}{4}}\approx 1713.04747$
Dichoral angles	Tiggy–6–tut: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Sidtid–5/2–tiggy: 144°
	Sidtid–3–tut: $\arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }$
Central density	9
Number of external pieces	15960
Level of complexity	66
Related polytopes
Army	Semi-uniform Sidpixhi, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (dodecahedra), ${\sqrt {5}}-2$ (tetrahedra)
Regiment	Tissidtixhi
Conjugate	Truncated great ditrigonary hexacosihecatonicosachoron
Convex core	Semi-uniform truncated hexacosichoron
Abstract & topological properties
Flag count	115200
Euler characteristic	–600
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The truncated small ditrigonary hexacosihecatonicosachoron, or tissidtixhi, is a nonconvex uniform polychoron that consists of 600 truncated tetrahedra, 120 small ditrigonary icosidodecahedra, and 120 truncated great icosahedra. 1 small ditrigonary icosidodecahedron, 3 truncated great icosahedra, and 3 truncated tetrahedra join at each vertex. As the name suggests, it can be obtained by truncating the small ditrigonary hexacosihecatonicosachoron.

Cross-sections

Vertex coordinates

The vertices of a truncated small ditrigonary hexacosihecatonicosachoron of edge length 1 are all permutations of:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {5}{2}}\right),$
• $\left(\pm 1,\,\pm 1,\,\pm 1,\,\pm 2\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {3}{2}}\right),$

along with the even permutations of:

• $\left(0,\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5}{2}}\right),$
• $\left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 2\right),$
• $\left(0,\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 1,\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {3}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 1\right),$

Related polychora

The truncated small ditrigonary hexacosihecatonicosachoron is the colonel of a regiment that also includes the truncated ditrigonary dishecatonicosachoron and the truncated great ditrigonary hexacosihecatonicosachoron. This regiment also includes the truncated dodecahedronary hexacosichoron, a compound of 25 truncated icositetrachora.

External links

• Bowers, Jonathan. "Category 2: Truncates" (#31).

• Klitzing, Richard. "tissidtixhi".