Dodecagonal prism

Dodecagonal prism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Twi
Coxeter diagram	x2x12o ()
Conway notation	P12
Elements
Faces	12 squares, 2 dodecagons
Edges	12+24
Vertices	24
Vertex figure	Isosceles triangle, edge lengths √2, √2, √2+√3
Measures (edge length 1)
Circumradius	${\frac {\sqrt {9+4{\sqrt {3}}}}{2}}\approx 1.99551$
Volume	$3(2+{\sqrt {3}})\approx 11.19615$
Dihedral angles	4–4: 150°
	4–12: 90°
Height	1
Central density	1
Number of external pieces	14
Level of complexity	3
Related polytopes
Army	Twip
Regiment	Twip
Dual	Dodecagonal tegum
Conjugate	Dodecagrammic prism
Abstract & topological properties
Flag count	144
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Skeleton	GP(12,1)
Properties
Symmetry	I2(12)×A1, order 48
Convex	Yes
Nature	Tame

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:

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

Representations

A dodecagonal prism has the following Coxeter diagrams:

• x2x12o () (full symmetry)
• x2x6x () (generally a dihexagonal prism)
• s2s12x () (generally a dihexagonal trapezoprism)
• xx12oo&#x (bases separate)
• 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 ${\sqrt {a^{2}(2+{\sqrt {3}})+{\frac {b^{2}}{4}}}}$ and its volume is given by $3(2+{\sqrt {3}})a^{2}b$.

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

Variations

A dodecagonal prism has the following variations:

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

External links

• Klitzing, Richard. "twip".
• Wikipedia contributors. "Dodecagonal prism".