# Skew pure dodecahedron

Skew pure dodecahedron
Rank3
Dimension4
TypeRegular
Notation
Schläfli symbol${\displaystyle \left\{{\frac {5}{1,2}},3:{\frac {10}{1,3}}\right\}}$
Elements
Faces12 pentagonal-pentagrammic coils
Edges30
Vertices20
Vertex figureTriangle
Petrie polygons6 decagonal-decagrammic coils ${\displaystyle \left\{{\dfrac {10}{1,3}}\right\}}$
Related polytopes
ArmyBamid
Petrie dualPetrial skew pure dodecahedron
κ ?Hemidodecahedron (4-dimensional)
Abstract & topological properties
Flag count120
Euler characteristic2
Schläfli type{5,3}
OrientableYes
Genus0
Properties
SymmetryA4+×2, order 120
Flag orbits1
ChiralYes
ConvexNo
Dimension vector(2,2,2)

The skew pure dodecahedron is a regular skew polyhedron in 4-dimensional Euclidean space. It is abstractly equivalent to the regular dodecahedron in 3-dimensional Euclidean space. It is pure, so even though it is a skew embedding of the dodecahedron it can not be decomposed into the blend of two other polyhedra.

## Vertex coordinates

The vertex coordinates of the skew pure dodecahedron with an edge length of 1 are the same as those of the biambodecachoron, where the 30 smaller edges are sized to be length 1.

## Related polytopes

The skew pure dodecahedron is a pure realization of the abstract regular polytope {5,3}. In total there are 28 faithful symmetric realizations of this polytope, of which 3 are pure. There are an additional 4 degenerate symmetric realizations, two of which are pure.