Pentagonal-snub dodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym P esn id Coxeter diagram x5o s5s3s ( ) Elements Tera 20+60 triangular-pentagonal duoprisms , 12 pentagonal duoprisms , 5 snub dodecahedral prisms Cells 100+300 triangular prisms , 30+60+60+60 pentagonal prisms , 5 snub dodecahedra Faces 100+300 triangles , 150+300+300 squares , 60+60 pentagons Edges 150+300+300+300 Vertices 300 Vertex figure Mirror-symmetric pentagonal scalene , edge lengths 1, 1, 1, 1, (1+√5 )/2 (base pentagon), (1+√5 )/2 (top edge), √2 (side edges) Measures (edge length 1) Circumradius ≈ 2.31759 Hypervolume ≈ 64.71860 Diteral angles Trapedip–pip–trapedip: ≈ 164.17537° Trapedip–pip–pedip: ≈ 152.92992° Sniddip–snid–sniddip: 108° Trapedip–trip–sniddip: 90° Pedip–pip–sniddip: 90° Central density 1 Number of external pieces 97 Level of complexity 50 Related polytopes Army Pesnid Regiment Pesnid Dual Pentagonal-pentagonal hexecontahedral duotegum Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H3 +×H2 + , order 600Convex Yes Nature Tame
The pentagonal-snub dodecahedral duoprism or pesnid is a convex uniform duoprism that consists of 5 snub dodecahedral prisms , 12 pentagonal duoprisms , and 80 triangular-pentagonal duoprisms of two kinds. Each vertex joins 2 snub dodecahedral prisms, 4 triangular-pentagonal duoprisms, and 1 pentagonal duoprism.
The vertices of a pentagonal-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
,
5
+
5
10
,
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ξ
ϕ
3
−
ξ
2
2
,
ϕ
ξ
(
ξ
+
ϕ
)
+
1
2
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\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),}
(
±
1
+
5
4
,
5
−
5
40
,
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ξ
ϕ
3
−
ξ
2
2
,
ϕ
ξ
(
ξ
+
ϕ
)
+
1
2
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\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),}
(
±
1
2
,
−
5
+
2
5
20
,
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ξ
ϕ
3
−
ξ
2
2
,
ϕ
ξ
(
ξ
+
ϕ
)
+
1
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\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
,
5
+
5
10
,
ϕ
3
−
ξ
2
2
,
ξ
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
,
ϕ
ξ
(
ξ
+
1
)
2
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\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),}
(
±
1
+
5
4
,
5
−
5
40
,
ϕ
3
−
ξ
2
2
,
ξ
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
,
ϕ
ξ
(
ξ
+
1
)
2
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\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),}
(
±
1
2
,
−
5
+
2
5
20
,
ϕ
3
−
ξ
2
2
,
ξ
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
,
ϕ
ξ
(
ξ
+
1
)
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\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
,
5
+
5
10
,
ξ
2
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ϕ
ξ
+
1
−
ϕ
2
,
ξ
2
(
1
+
2
ϕ
)
−
ϕ
2
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\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),}
(
±
1
+
5
4
,
5
−
5
40
,
ξ
2
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ϕ
ξ
+
1
−
ϕ
2
,
ξ
2
(
1
+
2
ϕ
)
−
ϕ
2
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\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),}
(
±
1
2
,
−
5
+
2
5
20
,
ξ
2
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ϕ
ξ
+
1
−
ϕ
2
,
ξ
2
(
1
+
2
ϕ
)
−
ϕ
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\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
,
5
+
5
10
,
ξ
2
ϕ
3
−
ξ
2
2
,
ξ
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ϕ
2
ξ
(
ξ
+
ϕ
)
+
1
2
ξ
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\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),}
(
±
1
+
5
4
,
5
−
5
40
,
ξ
2
ϕ
3
−
ξ
2
2
,
ξ
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ϕ
2
ξ
(
ξ
+
ϕ
)
+
1
2
ξ
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\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),}
(
±
1
2
,
−
5
+
2
5
20
,
ξ
2
ϕ
3
−
ξ
2
2
,
ξ
ϕ
ϕ
(
ξ
−
1
−
1
ξ
)
2
,
ϕ
2
ξ
(
ξ
+
ϕ
)
+
1
2
ξ
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\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
,
5
+
5
10
,
ϕ
(
ξ
+
2
)
+
2
2
,
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
,
ξ
ξ
(
1
+
ϕ
)
−
ϕ
2
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\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),}
(
±
1
+
5
4
,
5
−
5
40
,
ϕ
(
ξ
+
2
)
+
2
2
,
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
,
ξ
ξ
(
1
+
ϕ
)
−
ϕ
2
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\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),}
(
±
1
2
,
−
5
+
2
5
20
,
ϕ
(
ξ
+
2
)
+
2
2
,
ϕ
1
−
ξ
+
1
+
ϕ
ξ
2
,
ξ
ξ
(
1
+
ϕ
)
−
ϕ
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\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}}}.}