Dodecagonal duoprism

Dodecagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Twaddip
Coxeter diagram	x12o x12o ()
Elements
Cells	24 dodecagonal prisms
Faces	144 squares, 24 dodecagons
Edges	288
Vertices	144
Vertex figure	Tetragonal disphenoid, edge lengths (√2+√6)/2 (bases) and √2 (sides)
Measures (edge length 1)
Circumradius	$1+{\sqrt {3}}\approx 2.73205$
Inradius	${\frac {2+{\sqrt {3}}}{2}}\approx 1.86603$
Hypervolume	$9(7+4{\sqrt {3}})\approx 125.35383$
Dichoral angles	Twip–12–twip: 150°
	Twip–4–twip: 90°
Central density	1
Number of external pieces	24
Level of complexity	3
Related polytopes
Army	Twaddip
Regiment	Twaddip
Dual	Dodecagonal duotegum
Conjugate	Dodecagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(12)≀S2, order 1152
Convex	Yes
Nature	Tame

The dodecagonal duoprism or twaddip, also known as the dodecagonal-dodecagonal duoprism, the 12 duoprism or the 12-12 duoprism, is a noble uniform duoprism that consists of 24 dodecagonal prisms, with 4 joining at each vertex. It is also the 24-11 gyrochoron. It is the first in an infinite family of isogonal dodecagonal dihedral swirlchora and also the first in an infinite family of isochoric dodecagonal hosohedral swirlchora.

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

Vertex coordinates

The vertices of a dodecagonal duoprism of edge length 1, centered at the origin, are given by:

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

Variations

A dodecagonal duoprism has the following Coxeter diagrams:

• x12o x12o () (full symmetry)
• x6x x12o () (one dodecagon as dihexagon)
• x6x x6x () (both dodecagons as dihexagons)

External links

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

• Klitzing, Richard. "twaddip".