# Decagonal prism

(Redirected from Dip)
Decagonal prism
Rank3
TypeUniform
Notation
Bowers style acronymDip
Coxeter diagramx x10o ()
Conway notationP10
Elements
Faces10 squares, 2 decagons
Edges10+20
Vertices20
Vertex figureIsosceles triangle, edge lengths 2, 2, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+2{\sqrt {5}}}}{2}}\approx 1.69353}$
Volume${\displaystyle 5{\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 7.69421}$
Dihedral angles4–4: 144°
4–10: 90°
Height1
Central density1
Number of external pieces12
Level of complexity3
Related polytopes
ArmyDip
RegimentDip
DualDecagonal tegum
ConjugateDecagrammic prism
Abstract & topological properties
Flag count120
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
SkeletonGP(10,1)
Properties
SymmetryI2(10)×A1, order 40
ConvexYes
NatureTame

The decagonal prism, or dip, is a prismatic uniform polyhedron. It consists of 2 decagons and 10 squares. Each vertex joins one decagon and two squares. As the name suggests, it is a prism based on a decagon.

It is the highest convex polygonal prism to occur as cells in uniform polychora.

## Vertex coordinates

A decagonal prism of edge length 1 has vertex coordinates given by:

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

## Representations

A decagonal prism has the following Coxeter diagrams:

## Semi-uniform variant

The decagonal prism has a semi-uniform variant of the form x y10o that maintains its full symmetry. This variant uses rectangles as its sides.

With base edges of length a and side edges of length b, its circumradius is given by ${\displaystyle {\sqrt {a^{2}{\frac {3+{\sqrt {5}}}{2}}+{\frac {b^{2}}{4}}}}}$ and its volume is given by ${\displaystyle 5{\frac {\sqrt {5+2{\sqrt {5}}}}{2}}a^{2}b}$.

A decagonal prism with base edges of length a and side edges of length b can be alternated to form a pentagonal antiprism with base edges of length ${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{2}}}a}$ and side edges of lengths ${\displaystyle {\sqrt {a^{2}+b^{2}}}}$. In particular if the side edges are ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ times the length of the base edges this gives a uniform pentagonal antiprism.

## Variations

A decagonal prism has the following variations:

## Related polyhedra

A number of Johnson solids can be formed by attaching various configurations of pentagonal cupolas and pentagonal rotundas to the bases of the decagonal prism:

The rhombisnub dodecahedron is a uniform polyhedron compound composed of 6 decagonal prisms.