Dodecagonal-great rhombicuboctahedral duoprism

Dodecagonal-great rhombicuboctahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Twagirco
Coxeter diagram	x12o x4x3x ()
Elements
Tera	12 square-dodecagonal duoprisms, 8 hexagonal-dodecagonal duoprisms, 6 octagonal-dodecagonal duoprisms, 12 great rhombicuboctahedral prisms
Cells	144 cubes, 96 hexagonal prisms, 72 octagonal prisms, 24+24+24 dodecagonal prisms, 12 great rhombicuboctahedra
Faces	144+288+288+288 squares, 96 hexagons, 72 octagons, 48 dodecagons
Edges	288+288+288+576
Vertices	576
Vertex figure	Mirror-symmetric pentachoron, edge lengths √2, √3, √2+√2 (base triangle), √2+√3 (top edge), √2 (side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {21+6{\sqrt {2}}+4{\sqrt {3}}}}{2}}\approx 3.01718$
Hypervolume	$6(22+14{\sqrt {2}}+11{\sqrt {3}}+7{\sqrt {6}})\approx 467.98786$
Diteral angles	Gircope–girco–gircope: 150°
	Sitwadip–twip–hitwadip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Sitwadip–twip–otwadip: 135°
	Hitwadip–twip–otwadip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Sitwadip–cube–gircope: 90°
	Hitwadip–hip–gircope: 90°
	Otwadip–op–gircope: 90°
Central density	1
Number of external pieces	38
Level of complexity	60
Related polytopes
Army	Twagirco
Regiment	Twagirco
Dual	Dodecagonal-disdyakis dodecahedral duotegum
Conjugates	Dodecagrammic-great rhombicuboctahedral duoprism, Dodecagonal-quasitruncated cuboctahedral duoprism, Dodecagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(12), order 1152
Convex	Yes
Nature	Tame

The dodecagonal-great rhombicuboctahedral duoprism or twagirco is a convex uniform duoprism that consists of 12 great rhombicuboctahedral prisms, 6 octagonal-dodecagonal duoprisms, 8 hexagonal-dodecagonal duoprisms, and 12 square-dodecagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-dodecagonal duoprism, 1 hexagonal-dodecagonal duoprism, and 1 octagonal-dodecagonal duoprism.

This polyteron can be alternated into a hexagonal-snub cubic duoantiprism, although it cannot be made uniform. The dodecagons can also be edge-snubbed to create a snub cubic-hexagonal prismantiprismoid or the great rhombicuboctahedra to create a hexagonal-pyritohedral prismantiprismoid, which are also both nonuniform.

Vertex coordinates

The vertices of a dodecagonal-great rhombicuboctahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

Representations

A dodecagonal-great rhombicuboctahedral duoprism has the following Coxeter diagrams:

• x12o x4x3x () (full symmetry)
• x6x x4x3x () (dodecagons as dihexagons)