Prismatodishecatonicosachoron

Prismatodishecatonicosachoron
	(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Padohi
Coxeter diagram	x5/2o5o3x ()
Elements
Cells	1200 triangular prisms, 720 pentagrammic prisms, 120 icosahedra, 120 small stellated dodecahedra
Faces	2400 triangles, 3600 squares, 1440 pentagrams
Edges	3600+3600
Vertices	1440
Vertex figure	Pentagonal antipodium, edge lengths 1 (large base), (√5–1)/2 (small base), and √2 (sides)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral angles	Stip–4–trip:
	Sissid–5/2–stip: 162°
	Ike–3–trip: 150°
Central density	4
Number of external pieces	22080
Related polytopes
Army	Semi-uniform Tex, edge lengths (icosahedra), (surrounded by truncated tetrahedra)
Regiment	Padohi
Conjugate	Quasiprismatodishecatonicosachoron
Abstract & topological properties
Flag count	115200
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The prismatodishecatonicosachoron, or padohi, is a nonconvex uniform polychoron that consists of 120 icosahedra, 120 small stellated dodecahedra, 1200 triangular prisms, and 720 pentagrammic prisms. 1 icosahedron, 1 small stellated dodecahedron, 5 triangular prisms, and 5 pentagrammic prisms join at each vertex. It is the result of expanding the cells of either a small stellated hecatonicosachoron or a faceted hexacosichoron outwards.

The prismatodishecatonicosachoron contains the vertices of a prismatorhombisnub icositetrachoron.

Blending 10 prismatodishecatonicosachora results in the small difusiprismatosnub diprismatosnub disdishexacosichoron, which is scaliform.

Vertex coordinates[edit | edit source]

The vertices of a prismatodishecatonicosachoron of edge length 1 are given by all permutations of:

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

plus all even permutations of:

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

Related polychora[edit | edit source]

The prismatodishecatonicosachoron is the colonel of a regiment that includes 81 uniform members, as well as 78 fissary uniforms, 18 normal and 59 fissary scaliforms, and 1 scaliform compound.

o5/2o5o3o truncations
Name	OBSA	CD diagram
Small stellated hecatonicosachoron	sishi	x5/2o5o3o ()
(degenerate, quadruple cover of hi)		x5/2x5o3o ()
Rectified small stellated hecatonicosachoron	rasishi	o5/2x5o3o ()
Small hecatonicosihecatonicosachoron	shihi	o5/2x5x3o ()
Rectified faceted hexacosichoron	rofix	o5/2o5x3o ()
Truncated faceted hexacosichoron	tiffix	o5/2o5x3x ()
Faceted hexacosichoron	fix	o5/2o5o3x ()
Small rhombated small stellated hecatonicosachoron	sirsashi	x5/2o5x3o ()
(degenerate, sirdhi + 720(10/2))		x5/2x5x3o ()
Small rhombated faceted hexacosichoron	sirfix	o5/2x5o3x ()
Great rhombated faceted hexacosichoron	girfix	o5/2x5x3x ()
Prismatodishecatonicosachoron	padohi	x5/2o5o3x ()
(degenerate, six fipady + 720 double covers of pip)		x5/2x5o3x ()
Prismatorhombated stellated hecatonicosachoron	pirshi	x5/2o5x3x ()
(degenerate, sripady + 720 double covers of pip)		x5/2x5x3x ()