Dodecagonaltruncated dodecahedral duoprism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Twatid 

Coxeter diagram  x12o x5x3o () 

Elements 

Tera  20 triangulardodecagonal duoprisms, 12 decagonaldodecagonal duoprisms, 12 truncated dodecahedral prisms 

Cells  240 triangular prisms, 144 decagonal prisms, 30+60 dodecagonal prisms, 12 truncated dodecahedra 

Faces  240 triangles, 360+720 squares, 144 decagons, 60 dodecagons 

Edges  360+720+720 

Vertices  720 

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

Measures (edge length 1) 

Circumradius  ${\sqrt {\frac {53+8{\sqrt {3}}+15{\sqrt {5}}}{8}}}\approx 3.54255$ 

Hypervolume  $5{\frac {198+99{\sqrt {3}}+94{\sqrt {5}}+47{\sqrt {15}}}{4}}\approx 952.11705$ 

Diteral angles  Tiddip–tid–tiddip: 150° 

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

 Datwadip–twip–datwadip: $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$ 

 Titwadip–trip–tiddip: 90° 

 Datwadip–dip–tiddip: 90° 

Central density  1 

Number of external pieces  44 

Level of complexity  30 

Related polytopes 

Army  Twatid 

Regiment  Twatid 

Dual  Dodecagonaltriakis icosahedral duotegum 

Conjugates  Dodecagrammictruncated dodecahedral duoprism, Dodecagonalquasitruncated great stellated dodecahedral duoprism, Dodecagrammicquasitruncated great stellated dodecahedral duoprism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{3}×I2(12), order 2880 

Convex  Yes 

Nature  Tame 

The dodecagonaltruncated dodecahedral duoprism or twatid is a convex uniform duoprism that consists of 12 truncated dodecahedral prisms, 12 decagonaldodecagonal duoprisms and 20 triangulardodecagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangulardodecagonal duoprism, and 2 decagonaldodecagonal duoprisms.
The vertices of a dodecagonaltruncated dodecahedral duoprism of edge length 1 are given by all even permutations of the last three coordinates of:
 $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right).$
A dodecagonaltruncated dodecahedral duoprism has the following Coxeter diagrams:
 x12o x5x3o () (full symmetry)
 x6x x5x3o () (dodecagons as dihexagons)