Pentagonaloctagonal duoprismatic prism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Pop 

Coxeter diagram  x x5o x8o 

Elements 

Tera  8 squarepentagonal duoprisms, 5 squareoctagonal duoprisms, 2 pentagonaloctagonal duoprisms 

Cells  40 cubes, 5+10 octagonal prisms, 8+16 pentagonal prisms 

Faces  40+40+80 squares, 16 pentagons, 10 octagons 

Edges  40+80+80 

Vertices  80 

Vertex figure  Digonal disphenoidal pyramid, edge lengths (1+√5)/2 (disphenoid base 1), √2+√2 (disphenoid base 2), √2 (remaining edges) 

Measures (edge length 1) 

Circumradius  ${\sqrt {\frac {35+2{\sqrt {5}}+10{\sqrt {2}}}{20}}}\approx 1.63729$ 

Hypervolume  ${\frac {\sqrt {75+50{\sqrt {2}}+30{\sqrt {5}}+20{\sqrt {10}}}}{2}}\approx 8.30720$ 

Diteral angles  Squipdip–pip–squipdip: 135° 

 Sodip–op–sodip: 108° 

 Sodip–cube–squipdip: 90° 

 Podip–pip–squipdip: 90° 

 Sodip–op–podip: 90° 

Height  1 

Central density  1 

Number of external pieces  15 

Level of complexity  30 

Related polytopes 

Army  Pop 

Regiment  Pop 

Dual  Pentagonaloctagonal duotegmatic tegum 

Conjugates  Pentagonaloctagrammic duoprismatic prism, Pentagrammicoctagonal duoprismatic prism, Pentagrammicoctagrammic duoprismatic prism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{2}×I_{2}(8)×A_{1}, order 320 

Convex  Yes 

Nature  Tame 

The pentagonaloctagonal duoprismatic prism or pop, also known as the pentagonaloctagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonaloctagonal duoprisms, 5 squareoctagonal duoprisms, and 8 squarepentagonal duoprisms. Each vertex joins 2 squarepentagonal duoprisms, 2 squareoctagonal duoprisms, and 1 pentagonaloctagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.
The vertices of a pentagonaloctagonal duoprismatic prism of edge length 1 are given by all permutations of the third and fourth coordinates of:
 $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right),$
 $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right),$
 $\left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right).$
A pentagonaloctagonal duoprismatic prism has the following Coxeter diagrams:
 x x5o x8o (full symmetry)
 x x5o x4x (octagons as ditetragons)
 xx5oo xx8oo&#x (pentagonaloctagonal duoprism atop pentagonaloctagonal duoprism)
 xx5oo xx4xx