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.

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:


 * x x12o (full symmetry)
 * x x6x (generally a dihexagonal prism)
 * s2s12x (generally a dihexagonal trapezoprism)
 * xx12oo&#x (bases esen separately)
 * xx6xx&#x

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:


 * Dihexagonal prism - prism with dipentagons as bases, and 2 types of rectangles
 * Dihexagonal trapezoprism - isogonal with trapezoid sides
 * Dodecagonal frustum
 * dihexagonal frustum