Small disprismatodishecatonicosachoron

Small disprismatodishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Siddapady
Coxeter diagram	x5/2o3o5x ()
Elements
Cells	720 pentagonal prisms, 720 pentagrammic prisms, 120 dodecahedra, 120 great stellated dodecahedra
Faces	3600 squares, 1440 pentagons, 1440 pentagrams
Edges	3600+3600
Vertices	2400
Vertex figure	Triangular antipodium, edge lengths (√5–1)/2 (small base), (1+√5)/2 (large base), and √2 (sides)
Measures (edge length 1)
Circumradius	2
Hypervolume	${\frac {2025}{2}}=1012.5$
Dichoral angles	Gissid–5/2–stip: 162°
	Stip–4–pip: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	Doe-5–pip: 126°
Central density	20
Number of external pieces	64440
Level of complexity	203
Related polytopes
Army	Sidpixhi, edge length ${\frac {3-{\sqrt {5}}}{2}}$
Regiment	Siddapady
Conjugate	Small disprismatodishecatonicosachoron
Abstract & topological properties
Flag count	115200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small disprismatodishecatonicosachoron, or siddapady, is a nonconvex uniform polychoron that consists of 120 regular dodecahedra, 120 great stellated dodecahedra, 720 pentagonal prisms, and 720 pentagrammic prisms. 1 dodecahedron, 1 great stellated dodecahedron, 3 pentagonal prisms, and 3 pentagrammic prisms join at each vertex. It is the result of expanding the cells of either a grand hecatonicosachoron or a great stellated hecatonicosachoron outwards.

Gallery

Vertex coordinates

The vertices of a small disprismatodishecatonicosachoron of edge length 1 are all permutations of:

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

along with the even permutations of:

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

Related polychora

The small disprismatodishecatonicosachoron is the colonel of a regiment of 7 members. The other members include the hecatonicosafaceted prismatodishecatonicosachoron, medial hecatonicosafaceted prismatodishecatonicosachoron, trishecatonicosachoron, disprismatohecatonicosachoron, small spinoprismatohecatonicosihecatonicosachoron, and great spinoprismatohecatonicosihecatonicosachoron.

This polychoron can blend with quit sishi or tigaghi to form sabbadipady and gabbadipady respectively.

External links

• Bowers, Jonathan. "Category 11: Antipodiumverts" (#461).

• Klitzing, Richard. "siddapady".