Dodecagonal prism

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.

Dodecagonal prism
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTwip
Coxeter diagramx x12o ()
Elements
Faces12 squares, 2 dodecagons
Edges12+24
Vertices24
Vertex figureIsosceles triangle, edge lengths 2, 2, 2+3
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{9+4\sqrt3}}{2} ≈ 1.99551}$
Volume${\displaystyle 3(2+\sqrt3) ≈ 11.19615}$
Dihedral angles4–4: 150°
4–12: 90°
Height1
Central density1
Number of external pieces14
Level of complexity3
Related polytopes
ArmyTwip
RegimentTwip
DualDodecagonal tegum
ConjugateDodecagrammic prism
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryI2(12)×A1, order 48
ConvexYes
NatureTame

Vertex coordinates

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

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

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 ${\displaystyle \sqrt{a^2(2+\sqrt3)+\frac{b^2}{4}}}$  and its volume is given by ${\displaystyle 3(2+\sqrt3)a^2b}$ .

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 ${\displaystyle \frac{\sqrt2+\sqrt6}{2}a}$  and side edges of lengths ${\displaystyle \sqrt{a^2+b^2}}$ . In particular if the side edges are ${\displaystyle \sqrt{1+\sqrt3}}$  times the length of the base edges this gives a uniform pentagonal antiprism.

Variations

A dodecagonal prism has the following variations: