# Dodecagonal prism

(Redirected from Twip)
Dodecagonal prism
Rank3
TypeUniform
Notation
Bowers style acronymTwi
Coxeter diagramx2x12o ()
Conway notationP12
Elements
Faces12 squares, 2 dodecagons
Edges12+24
Vertices24
Vertex figureIsosceles triangle, edge lengths 2, 2, 2+3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {9+4{\sqrt {3}}}}{2}}\approx 1.99551}$
Volume${\displaystyle 3(2+{\sqrt {3}})\approx 11.19615}$
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
Flag count144
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
SkeletonGP(12,1)
Properties
SymmetryI2(12)×A1, order 48
ConvexYes
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:

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

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

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 ${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}a}$ and side edges of lengths ${\displaystyle {\sqrt {a^{2}+b^{2}}}}$. In particular if the side edges are ${\displaystyle {\sqrt {1+{\sqrt {3}}}}}$ times the length of the base edges this gives a uniform pentagonal antiprism.

## Variations

A dodecagonal prism has the following variations: