Dodecagonal-icosidodecahedral duoprism

Dodecagonal-icosidodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Twid
Coxeter diagram	x12o o5x3o ()
Elements
Tera	20 triangular-dodecagonal duoprisms, 12 pentagonal-dodecagonal duoprisms, 12 icosidodecahedral prisms
Cells	240 triangular prisms, 144 pentagonal prisms, 60 dodecagonal prisms, 12 icosidodecahedra
Faces	240 triangles, 720 squares, 144 pentagons, 30 dodecagons
Edges	360+720
Vertices	360
Vertex figure	Rectangular scalene, edge lengths 1, (1+√5)/2, 1, (1+√5)/2 (base rectangle), √2+√3 (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {7+2{\sqrt {3}}+{\sqrt {5}}}{2}}}\approx 2.51994$
Hypervolume	${\frac {90+45{\sqrt {3}}+34{\sqrt {5}}+17{\sqrt {15}}}{2}}\approx 154.90466$
Diteral angles	Iddip–id–iddip: 150°
	Titwadip–twip–pitwadip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	Titwadip–trip–iddip: 90°
	Pitwadip–pip–iddip: 90°
Central density	1
Number of external pieces	44
Level of complexity	20
Related polytopes
Army	Twid
Regiment	Twid
Dual	Dodecagonal-rhombic triacontahedral duotegum
Conjugates	Dodecagrammic-icosidodecahedral duoprism, Dodecagonal-great icosidodecahedral duoprism, Dodecagrammic-great icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×I2(12), order 2880
Convex	Yes
Nature	Tame

The dodecagonal-icosidodecahedral duoprism or twid is a convex uniform duoprism that consists of 12 icosidodecahedral prisms, 12 pentagonal-dodecagonal duoprisms, and 20 triangular-dodecagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-dodecagonal duoprisms, and 2 pentagonal-dodecagonal duoprisms.

Vertex coordinates

The vertices of a dodecagonal-icosidodecahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

as well as all even permutations of the last three coordinates of:

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

Representations

A dodecagonal-icosidodecahedral duoprism has the following Coxeter diagrams:

• x12o o5x3o () (full symmetry)
• x6x o5x3o () (H₃×G₂ symmetry, dodecagons as dihexagons)

External links

Klitzing, Richard. "n-id-dip".