Dodecagonal antiprism

Dodecagonal antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Twap
Coxeter diagram	s2s24o ()
Elements
Faces	24 triangles, 2 dodecagons
Edges	24+24
Vertices	24
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, (√2+√6)/2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {10+3{\sqrt {6}}+{\sqrt {98+40{\sqrt {6}}}}}{8}}}\approx 1.97951$
Volume	$2{\sqrt {-5+12{\sqrt {2}}-3{\sqrt {3}}+7{\sqrt {6}}}}\approx 9.78179$
Dihedral angles	3–3: $\arccos \left({\frac {1-{\sqrt {2}}-{\sqrt {6}}}{3}}\right)\approx 162.66285^{\circ }$
	12–3: $\arccos \left(-{\frac {3-3{\sqrt {2}}+2{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 94.35923^{\circ }$
Height	${\sqrt {\frac {-6+3{\sqrt {6}}+{\sqrt {98-40{\sqrt {6}}}}}{2}}}\approx 0.86352$
Central density	1
Number of external pieces	26
Level of complexity	4
Related polytopes
Army	Twap
Regiment	Twap
Dual	Dodecagonal antitegum
Conjugates	Dodecagrammic antiprism, dodecagrammic retroprism
Abstract & topological properties
Flag count	192
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(24)×A1)/2, order 48
Convex	Yes
Nature	Tame

The dodecagonal antiprism, or twap, is a prismatic uniform polyhedron. It consists of 24 triangles and 2 dodecagons. Each vertex joins one dodecagon and three triangles. It is an antiprism based on a dodecagon.

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,H\right)$,
• $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,H\right)$,
• $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,H\right)$,
• $\left(0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,-H\right)$,
• $\left(\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,0,\,-H\right)$,
• $\left(\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,-H\right)$,
• $\left(\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,-H\right)$,

where $H={\sqrt {\frac {-6+3{\sqrt {6}}+{\sqrt {98-40{\sqrt {6}}}}}{8}}}$ is the distance between the antiprism's center and the center of one of its bases.

Representations

A dodecagonal antiprism has the following Coxeter diagrams:

• s2s24o () (alternated icositetragonal prism)
• s2s12s () (alternated didodecagonal prism)
• xo12ox&#x (bases considered separately)

External links

• Klitzing, Richard. "n-ap".
• Wikipedia contributors. "Dodecagonal antiprism".