# Dodecagonal duoprism

Dodecagonal duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx12o x12o ()
Elements
Cells24 dodecagonal prisms
Faces144 squares, 24 dodecagons
Edges288
Vertices144
Vertex figureTetragonal disphenoid, edge lengths (2+6)/2 (bases) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle 1+{\sqrt {3}}\approx 2.73205}$
Inradius${\displaystyle {\frac {2+{\sqrt {3}}}{2}}\approx 1.86603}$
Hypervolume${\displaystyle 9(7+4{\sqrt {3}})\approx 125.35383}$
Dichoral anglesTwip–12–twip: 150°
Twip–4–twip: 90°
Central density1
Number of external pieces24
Level of complexity3
Related polytopes
DualDodecagonal duotegum
ConjugateDodecagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(12)≀S2, order 1152
ConvexYes
NatureTame

The dodecagonal duoprism or twaddip, also known as the dodecagonal-dodecagonal duoprism, the 12 duoprism or the 12-12 duoprism, is a noble uniform duoprism that consists of 24 dodecagonal prisms, with 4 joining at each vertex. It is also the 24-11 gyrochoron. It is the first in an infinite family of isogonal dodecagonal dihedral swirlchora and also the first in an infinite family of isochoric dodecagonal hosohedral swirlchora.

This polychoron can be alternated into a hexagonal duoantiprism, although it cannot be made uniform. Twelve of the dodecagons can also be alternated into long ditrigons to create a hexagonal-hexagonal prismantiprismoid, or it can be subsymmetrically faceted into a square triswirlprism or a triangular tetraswirlprism, which are nonuniform.

## Vertex coordinates

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

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

## Variations

A dodecagonal duoprism has the following Coxeter diagrams:

• x12o x12o () (full symmetry)
• x6x x12o () (one dodecagon as dihexagon)
• x6x x6x () (both dodecagons as dihexagons)