# Decagonal-dodecagonal duoprism

Decagonal-dodecagonal duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx10o x12o (       )
Elements
Cells12 decagonal prisms, 10 dodecagonal prisms
Faces120 squares, 12 decagons, 10 dodecagons
Edges120+120
Vertices120
Vertex figureDigonal disphenoid, edge lengths (5+5)/2 (base 1), (2+6)/2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\sqrt {\frac {7+2{\sqrt {3}}+{\sqrt {5}}}{2}}}\approx 2.51994$ Hypervolume${\frac {15{\sqrt {35+20{\sqrt {3}}+14{\sqrt {5}}+8{\sqrt {15}}}}}{2}}\approx 86.14553$ Dichoral anglesTwip–12–twip: 144°
Dip–10–dip: 150°
Twip–4–dip: 90°
Central density1
Number of external pieces22
Level of complexity6
Related polytopes
DualDecagonal-dodecagonal duotegum
ConjugatesDecagonal-dodecagrammic duoprism, Decagrammic-dodecagonal duoprism, Decagrammic-dodecagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(10)×I2(12), order 480
ConvexYes
NatureTame

The decagonal-dodecagonal duoprism or datwadip, also known as the 10-12 duoprism, is a uniform duoprism that consists of 10 dodecagonal prisms and 12 decagonal prisms, with two of each joining at each vertex.

This polychoron can be alternated into a pentagonal-hexagonal duoantiprism, although it cannot be made uniform. The dodecagons can also be alternated into long ditrigons to create a pentagonal-hexagonal prismantiprismoid, which is also nonuniform.

## Vertex coordinates

The coordinates of a decagonal-dodecagonal duoprism of edge length 1, centered at the origin, are given by:

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

A decagonal-dodecagonal duoprism has the following Coxeter diagrams:

• x10o x12o (full symmetry)
• x5x x12o (decagons as dipentagons)
• x6x x10o (dodecagons as dihexagons)
• x5x x6x (both of these applied)