# Decagonal-snub dodecahedral duoprism

Decagonal-snub dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDasnid
Coxeter diagramx10o s5s3s ()
Elements
Tera20+60 triangular-decagonal duoprisms, 12 pentagonal-decagonal duoprisms, 10 snub dodecahedral prisms
Cells200+600 triangular prisms, 120 pentagonal prisms, 30+60+60 decagonal prisms, 10 snub dodecahedra
Faces200+600 triangles, 300+600+600 squares, 120 pentagons, 60 decagons
Edges300+600+600+600
Vertices600
Vertex figureMirror-symmetric pentagonal scalene, edge lengths 1, 1, 1, 1, (1+5)/2 (base pentagon), (5+5)/2 (top edge), 2 (side edges)
Measures (edge length 1)
Hypervolume≈ 289.43036
Sniddip–snid–sniddip: 144°
Central density1
Number of external pieces102
Level of complexity50
Related polytopes
ArmyDasnid
RegimentDasnid
DualDecagonal-pentagonal hexecontahedral duotegum
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3+×I2(10), order 1200
ConvexYes
NatureTame

The decagonal-snub dodecahedral duoprism or dasnid is a convex uniform duoprism that consists of 10 snub dodecahedral prisms, 12 pentagonal-decagonal duoprisms, and 80 triangular-decagonal duoprisms of two kinds. Each vertex joins 2 snub dodecahedral prisms, 4 triangular-decagonal duoprisms, and 1 pentagonal-decagonal duoprism.

## Vertex coordinates

The vertices of a decagonal-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:

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,{\frac {\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\xi \phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +\phi )+1}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\xi \phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +\phi )+1}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\xi \phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +\phi )+1}}}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,{\frac {\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +1)}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +1)}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +1)}}}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi {\sqrt {\xi +1-\phi }}}{2}},\,{\frac {\sqrt {\xi ^{2}(1+2\phi )-\phi }}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {\xi ^{2}\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi {\sqrt {\xi +1-\phi }}}{2}},\,{\frac {\sqrt {\xi ^{2}(1+2\phi )-\phi }}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi {\sqrt {\xi +1-\phi }}}{2}},\,{\frac {\sqrt {\xi ^{2}(1+2\phi )-\phi }}{2}}\right),}$

as well as all even permutations with an even number of sign changes of the last three coordinates of:

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}}{2\xi }}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}}{2\xi }}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}}{2\xi }}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,{\frac {\sqrt {\phi (\xi +2)+2}}{2}},\,{\frac {\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\xi {\sqrt {\xi (1+\phi )-\phi }}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {\sqrt {\phi (\xi +2)+2}}{2}},\,{\frac {\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\xi {\sqrt {\xi (1+\phi )-\phi }}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {\phi (\xi +2)+2}}{2}},\,{\frac {\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\xi {\sqrt {\xi (1+\phi )-\phi }}}{2}}\right),}$

where

• ${\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}},}$
• ${\displaystyle \xi ={\sqrt[{3}]{\frac {\phi +{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}+{\sqrt[{3}]{\frac {\phi -{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}.}$