Hexagonal-decagonal duoprismatic prism

Hexagonal-decagonal duoprismatic prism
Rank5
TypeUniform
Notation
Coxeter diagramx x6o x10o
Elements
Tera10 square-hexagonal duoprisms, 6 square-decagonal duoprisms, 2 hexagonal-decagonal duoprisms
Cells60 cubes, 6+12 decagonal prisms, 10+20 hexagonal prisms
Faces60+60+120 squares, 20 hexagons, 12 decagons
Edges60+120+120
Vertices120
Vertex figureDigonal disphenoidal pyramid, edge lengths 3 (disphenoid base 1), (5+5)/2 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+2{\sqrt {5}}}}{2}}\approx 1.96673}$
Hypervolume${\displaystyle {\frac {15{\sqrt {15+6{\sqrt {5}}}}}{4}}\approx 19.99014}$
Diteral anglesShiddip–hip–shiddip: 144°
Height1
Central density1
Number of external pieces18
Level of complexity30
Related polytopes
DualHexagonal-decagonal duotegmatic tegum
ConjugateHexagonal-decagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(10)×A1, order 480
ConvexYes
NatureTame

The hexagonal-decagonal duoprismatic prism or hadip, also known as the hexagonal-decagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal-decagonal duoprisms, 6 square-decagonal duoprisms, and 10 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-decagonal duoprisms, and 1 hexagonal-decagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a triangular-pentagonal duoantiprismatic antiprism, although it cannot be made uniform.

Vertex coordinates

The vertices of a hexagonal-decagonal duoprismatic prism of edge length 1 are given by:

• ${\displaystyle \left(0,\,\pm 1,\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).}$

Representations

A hexagonal-decagonal duoprismatic prism has the following Coxeter diagrams:

• x x6o x10o (full symmetry)
• x x3x x10o (hexagons as ditrigons)
• x x6o x5x (decagons as dipentagons)
• x x3x x5x
• xx6oo xx10oo&#x (hexagonal-decagonal duoprism atop hexagonal-decagonal duoprism)
• xx3xx xx10oo&#x
• xx6oo xx5xx&#x
• xx3xx xx5xx&#x