Octagonal-dodecagonal duoprism

Octagonal-dodecagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Otwadip
Coxeter diagram	x8o x12o ()
Elements
Cells	12 octagonal prisms, 8 dodecagonal prisms
Faces	96 squares, 12 octagons, 8 dodecagons
Edges	96+96
Vertices	96
Vertex figure	Digonal disphenoid, edge lengths √2+√2 (base 1), (√2+√6)/2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {6+{\sqrt {2}}+2{\sqrt {3}}}{2}}}\approx 2.33220$
Hypervolume	$6(2+2{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}})\approx 54.05981$
Dichoral angles	Op–8–op: 150°
	Twip–12–twip: 135°
	Op–4–twip: 90°
Central density	1
Number of external pieces	20
Level of complexity	6
Related polytopes
Army	Otwadip
Regiment	Otwadip
Dual	Octagonal-dodecagonal duotegum
Conjugates	Octagonal-dodecagrammic duoprism, Octagrammic-dodecagonal duoprism, Octagrammic-dodecagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(8)×I2(12), order 384
Convex	Yes
Nature	Tame

The octagonal-dodecagonal duoprism or otwadip, also known as the 8-12 duoprism, is a uniform duoprism that consists of 8 dodecagonal prisms and 12 octagonal prisms, with two of each joining at each vertex.

This polychoron can be alternated into a square-hexagonal duoantiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a hexagonal-square prismantiprismoid or the dodecagons into long ditrigons to create a square-hexagonal prismantiprismoid, or it can be subsymmetrically faceted into a digonal-triangular tetraswirlprism, which are nonuniform.

Vertex coordinates

The coordinates of an octagonal-dodecagonal duoprism of edge length 1, centered at the origin, are given by:

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

Representations

An octagonal-dodecagonal duoprism has the fllowing Coxeter diagrams:

• x8o x12o (full symmetry)
• x4x x12o (octagons as ditetragons)
• x6x x8o (dodecagons as dihexagons)
• x4x x6x (both of these applied)

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Klitzing, Richard. "otwadip".