Dodecagonaltruncated icosahedral duoprism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Twati 

Coxeter diagram  x12o o5x3x () 

Elements 

Tera  12 pentagonaldodecagonal duoprisms, 20 hexagonaldodecagonal duoprisms, 12 truncated icosahedral prisms 

Cells  144 pentagonal prisms, 240 hexagonal prisms, 30+60 dodecagonal prisms, 12 truncated icosahedra 

Faces  360+720 squares, 144 pentagons, 240 hexagons, 60 dodecagons 

Edges  360+720+720 

Vertices  720 

Vertex figure  Digonal disphenoidal pyramid, edge lengths (1+√5)/2, √3, √3 (base triangle), √2+√3 (top), √2 (side edges) 

Measures (edge length 1) 

Circumradius  ${\sqrt {\frac {45+8{\sqrt {3}}+9{\sqrt {5}}}{8}}}\approx 3.14207$ 

Hypervolume  $3{\frac {250+125{\sqrt {3}}+86{\sqrt {5}}+43{\sqrt {15}}}{4}}\approx 619.00986$ 

Diteral angles  Tipe–ti–tipe: 150° 

 Pitwadip–twip–hitwadip: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$ 

 Hitwadip–twip–hitwadip: $\arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }$ 

 Pitwadip–pip–tipe: 90° 

 Hitwadip–hip–tipe: 90° 

Central density  1 

Number of external pieces  44 

Level of complexity  30 

Related polytopes 

Army  Twati 

Regiment  Twati 

Dual  Dodecagonalpentakis dodecahedral duotegum 

Conjugates  Dodecagrammictruncated icosahedral duoprism, Dodecagonaltruncated great icosahedral duoprism, Dodecagrammictruncated great icosahedral duoprism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{3}×I2(12), order 2880 

Convex  Yes 

Nature  Tame 

The dodecagonaltruncated icosahedral duoprism or twati is a convex uniform duoprism that consists of 12 truncated icosahedral prisms, 20 hexagonaldodecagonal duoprisms, and 12 pentagonaldodecagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 pentagonaldodecagonal duoprism, and 2 hexagonaldodecagonal duoprisms.
The vertices of a dodecagonaltruncated icosahedral duoprism of edge length 1 are given by all even permutations of the last three coordinates of:
 $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}}\right).$
A dodecagonaltruncated icosahedral duoprism has the following Coxeter diagrams:
 x12o o5x3x () (full symmetry)
 x6x o5x3x () (dodecagons as dihexagons)