# Decagonal prism

(Redirected from Dip)
Decagonal prism Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymDip
Coxeter diagramx x10o (     )
Elements
Faces10 squares, 2 decagons
Edges10+20
Vertices20
Vertex figureIsosceles triangle, edge lengths 2, 2, (5+5)/2
Measures (edge length 1)
Circumradius$\frac{\sqrt{7+2\sqrt5}}{2} ≈ 1.69353$ Volume$5\frac{\sqrt{5+2\sqrt5}}{2} ≈ 7.69421$ 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
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:

• $\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$ • $\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),$ • $\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right).$ ## 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 $\sqrt{a^2\frac{3+\sqrt5}{2}+\frac{b^2}{4}}$ and its volume is given by $5\frac{\sqrt{5+2\sqrt5}}{2}a^2b$ .

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 $\sqrt{\frac{5+\sqrt5}{2}}a$ and side edges of lengths $\sqrt{a^2+b^2}$ . In particular if the side edges are $\frac{1+\sqrt5}{2}$ 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.