# Dodecagonal prism

Dodecagonal prism Rank3
TypeUniform
SpaceSpherical
Bowers style acronymTwip
Info
Coxeter diagramx x12o
SymmetryI2(12)×A1, order 48
ArmyTwip
RegimentTwip
Elements
Vertex figureIsosceles triangle, edge lengths 2, 2, 2+3
Faces12 squares, 2 dodecagons
Edges12+24
Vertices24
Measures (edge length 1)
Circumradius$\frac{\sqrt{9+4\sqrt3}}{2} ≈ 1.99551$ Volume$3(2+\sqrt3) ≈ 11.19615$ Dihedral angles4–4: 150°
4–12: 90°
Height1
Central density1
Euler characteristic2
Number of pieces14
Level of complexity3
Related polytopes
DualDodecagonal tegum
ConjugateDodecagrammic prism
Properties
ConvexYes
OrientableYes
NatureTame

The dodecagonal prism, or twip, is a prismatic uniform polyhedron. It consists of 2 dodecagons and 12 squares. Each vertex joins one dodecagon and two squares. As the name suggests, it is a prism based on a dodecagon.

## Vertex coordinates

A dodecagonal prism of edge length 1 has vertex coordinates given by:

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

A dodecagonal prism has the following Coxeter diagrams:

## Semi-uniform variant

The dodecagonal prism has a semi-uniform variant of the form x y12o that maintains its full symmetry. This variant uses rectangles as its sides.

With base edges of length a and side edges of length b, its circumradius is given by $\sqrt{a^2(2+\sqrt3)+\frac{b^2}{4}}$ and its volume is given by $3(2+\sqrt3)a^2b$ .

A decagonal prism with base edges of length a and side edges of length b can be alternated to form a hexagonal antiprism with base edges of length $\frac{\sqrt2+\sqrt6}{2}a$ and side edges of lengths $\sqrt{a^2+b^2}$ . In particular if the side edges are $\sqrt{1+\sqrt3}$ times the length of the base edges this gives a uniform pentagonal antiprism.

## Variations

A dodecagonal prism has the following variations: