# Heptagonal tegum

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Heptagonal tegum
Rank3
TypeUniform dual
Notation
Bowers style acronymHet
Coxeter diagramm2m7o ()
Elements
Faces14 isosceles triangles
Edges7+14
Vertices2+7
Vertex figure2 heptagons, 7 squares
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left({\frac {\sin ^{2}{\frac {\pi }{7}}-1}{\sin ^{2}{\frac {\pi }{7}}+1}}\right)\approx 133.08952^{\circ }}$
Central density1
Number of external pieces14
Level of complexity3
Related polytopes
ArmyHet
RegimentHet
DualHeptagonal prism
ConjugatesHeptagrammic tegum, Great heptagrammic tegum
Abstract & topological properties
Flag count84
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryI2(7)×A1, order 28
ConvexYes
NatureTame

The heptagonal tegum, also called a heptagonal bipyramid, is a tegum with a heptagon as the midsection, constructed as the dual of a heptagonal prism. It has 14 isosceles triangles as faces, with 2 order–7 and 7 order–4 vertices.

In the variant obtained as the dual of a uniform heptagonal prism, the side edges are ${\displaystyle {\frac {1}{2\sin ^{2}{\frac {\pi }{7}}}}\approx 2.65597}$ times the length of the edges of the base heptagon. Each face has apex angle ${\displaystyle \arccos \left(1-2\sin ^{4}{\frac {\pi }{7}}\right)\approx 21.70194^{\circ }}$ and base angles ${\displaystyle \arccos \left(\sin ^{2}{\frac {\pi }{7}}\right)\approx 79.14903^{\circ }}$. If the base heptagon has edge length 1, its height is ${\displaystyle {\frac {\cos {\frac {\pi }{7}}}{\sin ^{2}{\frac {\pi }{7}}}}\approx 4.78589}$.