# Decagonal tegum

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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${\displaystyle \arccos \left(-{\frac {15+4{\sqrt {5}}}{29}}\right)\approx 145.65592^{\circ }}$
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 ${\displaystyle 3+{\sqrt {5}}\approx 5.23607}$ times the length of the edges of the base decagon. Each face has apex angle ${\displaystyle \arccos \left(3{\frac {3+{\sqrt {5}}}{16}}\right)\approx 10.95922^{\circ }}$ and base angles ${\displaystyle \arccos \left({\frac {3-{\sqrt {5}}}{8}}\right)\approx 84.52039^{\circ }}$. If the base decagon has edge length 1, its height is ${\displaystyle {\sqrt {50+22{\sqrt {5}}}}\approx 9.95959}$.