# Pentagonal antitegum

Pentagonal antitegum
Rank3
TypeUniform dual
Notation
Bowers style acronymPat
Coxeter diagramp2p10o ()
Conway notationdA5
Elements
Faces10 kites
Edges10+10
Vertices2+10
Vertex figure2 pentagons, 10 triangles
Measures (edge lengths 1, ${\displaystyle {\frac {3+{\sqrt {5}}}{2}}}$)
Dihedral angle${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces10
Level of complexity4
Related polytopes
ArmyPat
RegimentPat
DualPentagonal antiprism
ConjugatePentagrammic retroantitegum
Abstract & topological properties
Flag count80
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(10)×A1)/2, order 20
ConvexYes
NatureTame

The pentagonal antitegum, also known as the pentagonal trapezohedron, is an antitegum based on the pentagon, constructed as the dual of a pentagonal antiprism. It has 10 kites as faces, with 2 order–5 and 10 order–3 vertices.

Each face of this polyhedron is a kite with its longer edges ${\displaystyle {\frac {3+{\sqrt {5}}}{2}}\approx 2.61803}$ times the length of its shorter edges. These kites have 3 angles measuring 108° and 1 measuring 36°.

The pentagonal antitegum can be constructed from the regular dodecahedron by augmenting tall pyramids on two opposite faces.

In tabletop games, it is a popular design for a 10-sided die or "d10." A standard set of dice used in role-playing games comprises the five Platonic solids and two d10s, one marked with the numbers 0-9 and the other with the multiples of 10 from 00-90.

The pentagonal antitegum (with theoretical edge length 1) is the vertex figure of the triangular-gyroprismatic hexacosichoron, which cannot be made uniform.