Hexagonal-snub dodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hasnid Coxeter diagram x6o s5s3s Elements Tera 20+60 triangular-hexagonal duoprisms , 12 pentagonal-hexagonal duoprisms , 6 snub dodecahedral prisms Cells 120+360 triangular prisms , 72 pentagonal prisms , 30+60+60 hexagonal prisms , 6 snub dodecahedra Faces 120+360 triangles , 180+360+360 squares , 72 pentagons , 60 hexagons Edges 180+360+360+360 Vertices 360 Vertex figure Mirror-symmetric pentagonal scalene , edge lengths 1, 1, 1, 1, (1+√5 )/2 (base pentagon), √3 (top edge), √2 (side edges) Measures (edge length 1) Circumradius ≈ 2.37648 Hypervolume ≈ 97.73092 Diteral angles Thiddip–hip–thiddip: ≈ 164.17537° Thiddip–hip–phiddip: ≈ 152.92992° Sniddip–snid–sniddip: 120° Thiddip–trip–sniddip: 90° Phiddip–pip–sniddip: 90° Central density 1 Number of external pieces 98 Level of complexity 50 Related polytopes Army Hasnid Regiment Hasnid Dual Hexagonal-pentagonal hexecontahedral duotegum Conjugates Hexagonal-great snub icosidodecahedral duoprism , Hexagonal-great inverted snub icosidodecahedral duoprism , Hexagonal-great inverted retrosnub icosidodecahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H3 +×G2 , order 720Convex Yes Nature Tame
The hexagonal-snub dodecahedral duoprism or hasnid is a convex uniform duoprism that consists of 6 snub dodecahedral prisms , 12 pentagonal-hexagonal duoprisms , and 80 triangular-hexagonal duoprisms of two kinds. Each vertex joins 2 snub dodecahedral prisms, 4 triangular-hexagonal duoprisms, and 1 pentagonal-hexagonal duoprism.
The vertices of a hexagonal-snub dodecahedral duoprism of edge length 1 are given by all even permutations with an odd number of sign changes of the last three coordinates of:
(
0
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±
1
,
ϕ
ϕ
(
ξ
−
1
−
1
ξ
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2
,
ξ
ϕ
3
−
ξ
2
2
,
ϕ
ξ
(
ξ
+
ϕ
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+
1
2
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,
{\displaystyle \left(0,\,\pm 1,\,{\frac {\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\xi \phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +\phi )+1}}}{2}}\right),}
(
±
3
2
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±
1
2
,
ϕ
ϕ
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ξ
−
1
−
1
ξ
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2
,
ξ
ϕ
3
−
ξ
2
2
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ϕ
ξ
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ξ
+
ϕ
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+
1
2
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,
{\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\xi \phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +\phi )+1}}}{2}}\right),}
(
0
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±
1
,
ϕ
3
−
ξ
2
2
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ξ
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
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ϕ
ξ
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ξ
+
1
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2
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,
{\displaystyle \left(0,\,\pm 1,\,{\frac {\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +1)}}}{2}}\right),}
(
±
3
2
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±
1
2
,
ϕ
3
−
ξ
2
2
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ξ
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
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ϕ
ξ
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ξ
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{\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +1)}}}{2}}\right),}
(
0
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±
1
,
ξ
2
ϕ
ϕ
(
ξ
−
1
−
1
ξ
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2
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ξ
+
1
−
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2
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ξ
2
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1
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ϕ
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−
ϕ
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{\displaystyle \left(0,\,\pm 1,\,{\frac {\xi ^{2}\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi {\sqrt {\xi +1-\phi }}}{2}},\,{\frac {\sqrt {\xi ^{2}(1+2\phi )-\phi }}{2}}\right),}
(
±
3
2
,
±
1
2
,
ξ
2
ϕ
ϕ
(
ξ
−
1
−
1
ξ
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2
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ϕ
ξ
+
1
−
ϕ
2
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ξ
2
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1
+
2
ϕ
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−
ϕ
2
)
,
{\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi {\sqrt {\xi +1-\phi }}}{2}},\,{\frac {\sqrt {\xi ^{2}(1+2\phi )-\phi }}{2}}\right),}
as well as all even permutations with an even number of sign changes of the last three coordinates of:
(
0
,
±
1
,
ξ
2
ϕ
3
−
ξ
2
2
,
ξ
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
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ϕ
2
ξ
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ξ
+
ϕ
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+
1
2
ξ
)
,
{\displaystyle \left(0,\,\pm 1,\,{\frac {\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}}{2\xi }}\right),}
(
±
3
2
,
±
1
2
,
ξ
2
ϕ
3
−
ξ
2
2
,
ξ
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ϕ
2
ξ
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ξ
+
ϕ
)
+
1
2
ξ
)
,
{\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}}{2\xi }}\right),}
(
0
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±
1
,
ϕ
(
ξ
+
2
)
+
2
2
,
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
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ξ
ξ
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1
+
ϕ
)
−
ϕ
2
)
,
{\displaystyle \left(0,\,\pm 1,\,{\frac {\sqrt {\phi (\xi +2)+2}}{2}},\,{\frac {\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\xi {\sqrt {\xi (1+\phi )-\phi }}}{2}}\right),}
(
±
3
2
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±
1
2
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ϕ
(
ξ
+
2
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+
2
2
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ϕ
1
−
ξ
+
1
+
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ξ
2
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ξ
ξ
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1
+
ϕ
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−
ϕ
2
)
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{\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {\phi (\xi +2)+2}}{2}},\,{\frac {\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\xi {\sqrt {\xi (1+\phi )-\phi }}}{2}}\right),}
where
ϕ
=
1
+
5
2
,
{\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}},}
ξ
=
ϕ
+
ϕ
−
5
27
2
3
+
ϕ
−
ϕ
−
5
27
2
3
.
{\displaystyle \xi ={\sqrt[{3}]{\frac {\phi +{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}+{\sqrt[{3}]{\frac {\phi -{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}.}