# Decagonal tegum

Decagonal tegum Rank3
TypeUniform dual
Notation
Bowers style acronymDet
Coxeter diagramm2m10o
Elements
Faces20 isosceles triangles
Edges10+20
Vertices2+10
Vertex figure2 decagons, 10 squares
Measures (edge length 1)
Dihedral angle$\arccos \left(-{\frac {15+4{\sqrt {5}}}{29}}\right)\approx 145.65592^{\circ }$ Central density1
Number of external pieces20
Level of complexity3
Related polytopes
ArmyDet
RegimentDet
DualDecagonal prism
ConjugateDecagrammic tegum
Abstract & topological properties
Flag count120
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryI2(10)×A1, order 40
ConvexYes
NatureTame

The decagonal tegum, also called a decagonal bipyramid, is a tegum with a decagon as the midsection, constructed as the dual of a decagonal prism. It has 20 isosceles triangles as faces, with 2 order–10 and 10 order–4 vertices.

In the variant obtained as the dual of a uniform decagonal prism, the side edges are $3+{\sqrt {5}}\approx 5.23607$ times the length of the edges of the base decagon. Each face has apex angle $\arccos \left(3{\frac {3+{\sqrt {5}}}{16}}\right)\approx 10.95922^{\circ }$ and base angles $\arccos \left({\frac {3-{\sqrt {5}}}{8}}\right)\approx 84.52039^{\circ }$ . If the base decagon has edge length 1, its height is ${\sqrt {50+22{\sqrt {5}}}}\approx 9.95959$ .