Heptagonal-dodecagonal duoprismatic prism Rank 5 Type Uniform Notation Bowers style acronym Hetwip Coxeter diagram x x7o x12o Elements Tera 12 square-heptagonal duoprisms , 7 square-dodecagonal duoprisms , 2 heptagonal-dodecagonal duoprisms Cells 84 cubes , 7+14 dodecagonal prisms , 12+24 heptagonal prisms , 84 cubes Faces 84+84+168 squares , 24 heptagons , 14 dodecagons Edges 84+168+168 Vertices 168 Vertex figure Digonal disphenoidal pyramid , edge lengths 2cos(π/7) (disphenoid base 1), √2+√3 (disphenoid base 2), √2 (remaining edges)Measures (edge length 1) Circumradius
9
+
4
3
+
1
sin
2
π
7
2
≈
2.30435
{\displaystyle {\frac {\sqrt {9+4{\sqrt {3}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.30435}
Hypervolume
21
2
+
3
4
tan
π
7
≈
40.68584
{\displaystyle 21{\frac {2+{\sqrt {3}}}{4\tan {\frac {\pi }{7}}}}\approx 40.68584}
Diteral angles Squahedip–hep–squahedip: 150° Sitwadip–twip–sitwadip:
5
π
7
≈
128.57143
∘
{\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}
Sitwadip–cube–squahedip: 90° Hetwadip–hep–squahedip: 90° Sitwadip–twip–hetwadip: 90° Height 1 Central density 1 Number of external pieces 21 Level of complexity 30 Related polytopes Army Hetwip Regiment Hetwip Dual Heptagonal-dodecagonal duotegmatic tegum Conjugates Heptagonal-dodecagrammic duoprismatic prism , Heptagrammic-dodecagonal duoprismatic prism , Heptagrammic-dodecagrammic duoprismatic prism , Great heptagrammic-dodecagonal duoprismatic prism , Great heptagrammic-dodecagrammic duoprismatic prism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry I2 (7)×I2 (12)×A1 , order 672Convex Yes Nature Tame
The heptagonal-dodecagonal duoprismatic prism or hetwip , also known as the heptagonal-dodecagonal prismatic duoprism , is a convex uniform duoprism that consists of 2 heptagonal-dodecagonal duoprisms , 7 square-dodecagonal duoprisms , and 12 square-heptagonal duoprisms . Each vertex joins 2 square-heptagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 heptagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron .
The vertices of a heptagonal-dodecagonal duoprismatic prism of edge length 2sin(π/7) are given by all permutations of the third and fourth coordinates of:
(
1
,
0
,
±
(
1
+
3
)
sin
π
7
,
±
(
1
+
3
)
sin
π
7
,
±
sin
π
7
)
,
{\displaystyle \left(1,\,0,\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
(
1
+
3
)
sin
π
7
,
±
(
1
+
3
)
sin
π
7
,
±
sin
π
7
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}
(
1
,
0
,
±
sin
π
7
,
±
(
2
+
3
)
sin
π
7
,
±
sin
π
7
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
sin
π
7
,
±
(
2
+
3
)
sin
π
7
,
±
sin
π
7
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}
where j = 2, 4, 6.
A heptagonal-dodecagonal duoprismatic prism has the following Coxeter diagrams :
x x7o x12o (full symmetry)
x x7o x6x (dodecagons as dihexagons)
xx7oo xx12oo&#x (heptagonal-dodecagonal duoprism atop heptagonal-dodecagonal duoprism)
xx7oo xx6xx&#x 
