Heptagonal-truncated dodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hetid Coxeter diagram x7o x5x3o Elements Tera 20 triangular-heptagonal duoprisms , 12 heptagonal-decagonal duoprisms , 7 truncated dodecahedral prisms Cells 140 triangular prisms , 30+60 heptagonal prisms , 84 decagonal prisms , 7 truncated dodecahedra Faces 140 triangles , 210+420 squares , 60 heptagons , 84 decagons Edges 210+420+420 Vertices 420 Vertex figure Digonal disphenoidal pyramid , edge lengths 1, √(5+√5 )/2 , √(5+√5 )/2 (base triangle), cos(π/7) (top), √2 (side edges)Measures (edge length 1) Circumradius
37
+
15
5
+
2
sin
2
π
7
8
≈
3.18522
{\displaystyle {\sqrt {\frac {37+15{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 3.18522}
Hypervolume
35
99
+
47
5
48
tan
π
7
≈
309.02670
{\displaystyle 35{\frac {99+47{\sqrt {5}}}{48\tan {\frac {\pi }{7}}}}\approx 309.02670}
Diteral angles Theddip–hep–hedadip:
arccos
(
−
5
+
2
5
15
)
≈
142.62263
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}
Tiddip–tid–tiddip:
5
π
7
≈
128.57143
∘
{\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}
Hedadip–hep–hedadip:
arccos
(
−
5
5
)
≈
116.56505
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}
Theddip–trip–tiddip: 90° Hedadip–dip–tiddip: 90° Central density 1 Number of external pieces 39 Level of complexity 30 Related polytopes Army Hetid Regiment Hetid Dual Heptagonal-triakis icosahedral duotegum Conjugates Heptagrammic-truncated dodecahedral duoprism , Great heptagrammic-truncated dodecahedral duoprism , Heptagonal-quasitruncated great stellated dodecahedral duoprism , Heptagrammic-quasitruncated great stellated dodecahedral duoprism , Great heptagrammic-quasitruncated great stellated dodecahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H3 ×I2(7) , order 1680Convex Yes Nature Tame
The heptagonal-truncated dodecahedral duoprism or hetid is a convex uniform duoprism that consists of 7 truncated dodecahedral prisms , 12 heptagonal-decagonal duoprisms , and 20 triangular-heptagonal duoprisms . Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-heptagonal duoprism, and 2 heptagonal-decagonal duoprisms.
The vertices of a heptagonal-truncated dodecahedral duoprism of edge length 2sin(π/7) are given by all even permutations of the last three coordinates of:
(
1
,
0
,
0
,
±
sin
π
7
,
±
(
5
+
3
5
)
sin
π
7
2
)
,
{\displaystyle \left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}
(
1
,
0
,
±
sin
π
7
,
±
(
3
+
5
)
sin
π
7
2
,
±
(
3
+
5
)
sin
π
7
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}
(
1
,
0
,
±
(
3
+
5
)
sin
π
7
2
,
±
(
1
+
5
)
sin
π
7
,
±
(
2
+
5
)
sin
π
7
)
,
{\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
0
,
±
sin
π
7
,
±
(
5
+
3
5
)
sin
π
7
2
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
sin
π
7
,
±
(
3
+
5
)
sin
π
7
2
,
±
(
3
+
5
)
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 {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
(
3
+
5
)
sin
π
7
2
,
±
(
1
+
5
)
sin
π
7
,
±
(
2
+
5
)
sin
π
7
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}
where j = 2, 4, 6.