# Dodecagonal-great rhombicosidodecahedral duoprism

Dodecagonal-great rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwagrid
Coxeter diagramx12o x5x3x ()
Elements
Tera30 square-dodecagonal duoprisms, 20 hexagonal-dodecagonal duoprisms, 12 decagonal-dodecagonal duoprisms, 12 great rhombicosidodecahedral prisms
Cells360 cubes, 240 hexagonal prisms, 144 decagonal prisms, 60+60+60 dodecagonal prisms, 12 great rhombicosidodecahedra
Faces360+720+720+720 squares, 240 hexagons, 144 decagons, 120 dodecagons
Edges720+720+720+1440
Vertices1440
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, (5+5)/2 (base triangle), 2+3 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {39+4{\sqrt {3}}+12{\sqrt {5}}}}{2}}\approx 4.26500}$
Hypervolume${\displaystyle 15(38+19{\sqrt {3}}+20{\sqrt {1}}+10{\sqrt {15}})\approx 2315.40238}$
Diteral anglesSitwadip–twip–hitwadip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Griddip–grid–griddip: 150°
Sitwadip–twip–datwadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Hitwadip–twip–datwadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
HeightsGriddip atop griddip: ${\displaystyle 2+{\sqrt {3}}\approx 3.73205}$
Sitwadip atop sitwadip: ${\displaystyle {\sqrt {29+12{\sqrt {5}}}}\approx 7.47214}$
Hitwadip atop hitwadip: ${\displaystyle {\sqrt {27+12{\sqrt {5}}}}\approx 7.33708}$
Datwadip atop datwadip: ${\displaystyle {\sqrt {5(5+2{\sqrt {5}})}}\approx 6.88191}$
Central density1
Number of external pieces74
Level of complexity60
Related polytopes
ArmyTwagrid
RegimentTwagrid
DualDodecagonal-disdyakis triacontahedral duotegum
ConjugatesDodecagrammic-great rhombicosidodecahedral duoprism, Dodecagonal-great quasitruncated icosidodecahedral duoprism, Dodecagrammic-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(12), order 2880
ConvexYes
NatureTame

The dodecagonal-great rhombicosidodecahedral duoprism or twagrid is a convex uniform duoprism that consists of 12 great rhombicosidodecahedral prisms, 12 decagonal-dodecagonal duoprisms, 20 hexagonal-dodecagonal duoprisms, and 30 square-dodecagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-dodecagonal duoprism, 1 hexagonal-dodecagonal duoprism, and 1 decagonal-dodecagonal duoprism.

This polyteron can be alternated into a hexagonal-snub dodecahedral duoantiprism, although it cannot be made uniform. The dodecagons can also be alternated into long ditrigons to create a snub dodecahedral-hexagonal prismantiprismoid, which is also nonuniform.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),}$

along with all even permutations of the last three coordinates of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right).}$

## Representations

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

• x12o x5x3x () (full symmetry)
• x6x x5x3x () (dodecagons as dihexagons)