# Decagonal-dodecagonal duoprism

Decagonal-dodecagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Info
Coxeter diagramx10o x12o
SymmetryI2(10)×I2(12), order 480
Elements
Vertex figureDigonal disphenoid, edge lengths (5+5)/2 (base 1), (2+6)/2 (base 2), and 2 (sides)
Cells12 decagonal prisms, 10 dodecagonal prisms
Faces120 squares, 12 decagons, 10 dodecagons
Edges120+120
Vertices120
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{7+2\sqrt3+\sqrt5}{2}} ≈ 2.51994}$
Hypervolume${\displaystyle \frac{15\sqrt{35+20\sqrt3+14\sqrt5+8\sqrt{15}}}{2} ≈ 86.14553}$
Dichoral anglesTwip–12–twip: 144°
Dip–10–dip: 150°
Twip–4–dip: 90°
Central density1
Euler characteristic0
Number of pieces22
Level of complexity6
Related polytopes
DualDecagonal-dodecagonal duotegum
ConjugatesDecagonal-dodecagrammic duoprism, Decagrammic-dodecagonal duoprism, Decagrammic-dodecagrammic duoprism
Properties
ConvexYes
OrientableYes
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:

• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt3}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt3}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac12\right).}$

## 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)