Small sphenoverted ditrigonal trishecatonicosachoron

Small sphenoverted ditrigonal trishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Swav ditathi
Coxeter diagram	o5x3o5x5/3*b ()
Elements
Cells	120 icosidodecahedra, 120 quasitruncated small stellated dodecahedra, 120 great ditrigonal dodecicosidodecahedra
Faces	2400 triangles, 720+1440 pentagons, 1440 decagrams
Edges	3600+7200
Vertices	3600
Vertex figure	Wedge, edge lengths (1+√5)/2 (two edges of base rectangle and top edge), 1 (remaining base edges), and √(5–√5)/2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {11+{\sqrt {5}}}{2}}}\approx 2.57255$
Hypervolume	$5{\frac {1555+93{\sqrt {5}}}{4}}\approx 2203.69290$
Dichoral angles	Gidditdid–10/3–quit sissid: 144°
	Id–5–quit sissid: 144°
	Gidditdid–5–gidditdid: 144°
	Gidditdid–3–id: 120°
Related polytopes
Army	Srahi, edge length ${\frac {3-{\sqrt {5}}}{2}}$
Regiment	Swav ditathi
Conjugate	Great retrosphenoverted ditrigonal trishecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	129600
Euler characteristic	–1560
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small sphenoverted ditrigonal trishecatonicosachoron, or swav ditathi, is a nonconvex uniform polychoron that consists of 120 quasitruncated small stellated dodecahedra, 120 icosidodecahedra, and 120 great ditrigonal dodecicosidodecahedra. 1 icosidodecahedron, 2 quasitruncated small stellated dodecahedra, and 2 great ditrigonal dodecicosidodecahedra join at each vertex.

Vertex coordinates

Coordinates for the vertices of a small sphenoverted ditrigonal trishecatonicosachoron of edge length 1 are given by all permutations of:

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

together with all even permutations of:

• $\left(0,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {5}}{2}}\right),$
• $\left(0,\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{4}}\right),$
• $\left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm 1\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 1,\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 2,\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 1\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}}\right).$

Related polychora

The small sphenoverted ditrigonal trishecatonicosachoron is the colonel of a regiment of 7 members. Its other members include the great retrosphenoverted trishecatonicosachoron, small hecatonicosidishecatonicosachoron, small retrotrishecatonicosachoron, small small dishecatonicosachoron, dishecatonicosintercepted ditrigonal dishecatonicosachoron, and small hecatonicosintercepted trishecatonicosachoron.

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#235).

• Klitzing, Richard. "swav ditathi".