Decagonal prism

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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
Volume
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[edit | edit source]

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

  • ,
  • ,
  • .

Representations[edit | edit source]

A decagonal prism has the following Coxeter diagrams:

Semi-uniform variant[edit | edit source]

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 and its volume is given by .

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 and side edges of lengths . In particular if the side edges are times the length of the base edges this gives a uniform pentagonal antiprism.

Variations[edit | edit source]

A decagonal prism has the following variations:

Related polyhedra[edit | edit source]

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.

External links[edit | edit source]