Heptagonal-octagonal duoprismatic prism Rank 5 Type Uniform Notation Bowers style acronym Heop Coxeter diagram x x7o x8o Elements Tera 8 square-heptagonal duoprisms , 7 square-octagonal duoprisms , 2 heptagonal-octagonal duoprisms Cells 56 cubes , 7+14 octagonal prisms , 8+16 heptagonal prisms Faces 56+56+112 squares , 16 heptagons , 14 octagons Edges 56+112+112 Vertices 112 Vertex figure Digonal disphenoidal pyramid , edge lengths 2cos(π/7) (disphenoid base 1), √2+√2 (disphenoid base 2), √2 (remaining edges)Measures (edge length 1) Circumradius ${\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.81248$ Hypervolume $7{\frac {1+{\sqrt {2}}}{2\tan {\frac {\pi }{7}}}}\approx 17.54608$ Diteral angles Squahedip–hep–squahedip: 135° Sodip–op–sodip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ Sodip–cube–squahedip: 90° Heodip–hep–squahedip: 90° Sodip–op–heodip: 90° Height 1 Central density 1 Number of external pieces 17 Level of complexity 30 Related polytopes Army Heop Regiment Heop Dual Heptagonal-octagonal duotegmatic tegum Conjugates Heptagonal-octagrammic duoprismatic prism , Heptagrammic-octagonal duoprismatic prism , Heptagrammic-octagrammic duoprismatic prism , Great heptagrammic-octagonal duoprismatic prism , Great heptagrammic-octagrammic duoprismatic prism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry I_{2} (7)×I_{2} (8)×A_{1} , order 448Convex Yes Nature Tame

The heptagonal-octagonal duoprismatic prism or heop , also known as the heptagonal-octagonal prismatic duoprism , is a convex uniform duoprism that consists of 2 heptagonal-octagonal duoprisms , 7 square-octagonal duoprisms , and 8 square-heptagonal duoprisms . Each vertex joins 2 square-heptagonal duoprisms, 2 square-octagonal duoprisms, and 1 heptagonal-octagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron .

The vertices of a heptagonal-octagonal duoprismatic prism of edge length 2sin(π/7) are given by all permutations of the third and fourth coordinates of:

$\left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
$\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
where j = 2, 4, 6.

A heptagonal-octagonal duoprismatic prism has the following Coxeter diagrams :

x x7o x8o (full symmetry)
x x7o x4x (octagons as ditetragons)
xx7oo xx8oo&#x (heptagonal-octagonal duoprism atop heptagonal-octagonal duoprism)
xx7oo xx4xx&#x