# Hexagonal tegum

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Hexagonal tegum
Rank3
TypeUniform dual
Notation
Bowers style acronymHat
Coxeter diagramm2m6o
Elements
Faces12 isosceles triangles
Edges6+12
Vertices2+6
Vertex figure2 hexagons, 6 squares
Measures (edge lengths 1, ${\displaystyle {\sqrt {3}}}$)
Dihedral angle${\displaystyle \arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }}$
Central density1
Number of external pieces12
Level of complexity3
Related polytopes
ArmyHat
RegimentHat
DualHexagonal prism
ConjugateHexagonal tegum
Abstract & topological properties
Flag count72
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryG2×A1, order 24
ConvexYes
NatureTame

The hexagonal tegum, also called a hexagonal bipyramid, is a tegum with a hexagon as the midsection, constructed as the dual of a hexagonal prism. It has 12 isosceles triangles as faces, with 2 order–6 and 6 order–4 vertices. The variant with equilateral triangles is flat, and is not considered to be a Johnson solid.

In the variant obtained as the dual of a uniform hexagonal prism, the side edges are exactly 2 times the length of the edges of the base hexagon. Each face has apex angle ${\displaystyle \arccos \left({\frac {7}{8}}\right)\approx 28.95502^{\circ }}$ and base angles ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$. If the base hexagon has edge length 1, its height is ${\displaystyle 2{\sqrt {3}}\approx 3.46410}$.