# Dodecagonal duoprism

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.

Dodecagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTwaddip
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${\displaystyle 1+\sqrt3 ≈ 2.73205}$
Inradius${\displaystyle \frac{2+\sqrt3}{2} ≈ 1.86603}$
Hypervolume${\displaystyle 9(7+4\sqrt3) ≈ 125.35383}$
Dichoral anglesTwip–12–twip: 150°
Twip–4–twip: 90°
Central density1
Number of external pieces24
Level of complexity3
Related polytopes
ArmyTwaddip
RegimentTwaddip
DualDodecagonal duotegum
ConjugateDodecagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(12)≀S2, order 1152
ConvexYes
NatureTame

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:

• ${\displaystyle \left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{2+\sqrt3}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac{2+\sqrt3}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt3}{2},\,±\frac12,\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt3}{2},\,±\frac12,\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt3}{2},\,±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac12\right).}$

## 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)