Decagonal tegum

Decagonal tegum
Rank	3
Type	Uniform dual
Notation
Bowers style acronym	Det
Coxeter diagram	m2m10o
Elements
Faces	20 isosceles triangles
Edges	10+20
Vertices	2+10
Vertex figure	2 decagons, 10 squares
Measures (edge length 1)
Dihedral angle
Central density	1
Number of external pieces	20
Level of complexity	3
Related polytopes
Army	Det
Regiment	Det
Dual	Decagonal prism
Conjugate	Decagrammic tegum
Abstract & topological properties
Flag count	120
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	I2(10)×A1, order 40
Convex	Yes
Nature	Tame

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 ${\sqrt {50+22{\sqrt {5}}}}\approx 9.95959$ .