# Pentagonal tegum

Pentagonal tegum
Rank3
TypeCRF
Notation
Bowers style acronymPet
Coxeter diagramoxo5ooo&#xt
Elements
Faces10 triangles
Edges5+10
Vertices2+5
Vertex figures2 pentagons, edge length 1
5 rhombi, edge length 1
Measures (edge length 1)
Inradius${\displaystyle {\frac {5{\sqrt {3}}+{\sqrt {15}}}{30}}\approx 0.41777}$
Volume${\displaystyle {\frac {5+{\sqrt {5}}}{12}}\approx 0.60301}$
Dihedral angles3-3 pyramidal: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
3-3 equatorial: ${\displaystyle \arccos \left({\frac {4{\sqrt {5}}-5}{15}}\right)\approx 74.75474^{\circ }}$
Height${\displaystyle {\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\approx 1.05146}$
Central density1
Number of external pieces10
Level of complexity3
Related polytopes
ArmyPet
RegimentPet
DualSemi-uniform pentagonal prism
ConjugatePentagrammic tegum
Abstract & topological properties
Flag count60
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
ConvexYes
NatureTame

The pentagonal tegum also called a pentagonal bipyramid or pentagonal dipyramid, is one of the 92 Johnson solids (J13). It has 10 equilateral triangles as faces, with 2 order-5 and 5 order-4 vertices. It can be constructed by joining two pentagonal pyramids at their bases

It is one of three regular polygonal tegums to be CRF. The others are the regular octahedron (square tegum) and the triangular tegum.

## Vertex coordinates

A pentagonal tegum of edge length 1 has the following vertices:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0\right),}$
• ${\displaystyle \left(0,\,0,\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right).}$

## Representations

A pentagonal tegum has the following Coxeter diagrams:

• oxo5ooo&#xt
• yo ox5oo&#zx (y = ${\displaystyle {\sqrt {\frac {10-2{\sqrt {5}}}{5}}}}$)

## Variations

The pentagonal tegum can have the height of its pyramids varied while maintaining its full symmetry. These variations generally have 10 isosceles triangles for faces.

One notable variations can be obtained as the dual of the uniform pentagonal prism, which can be represented by m2m5o. In this variant the side edges are ${\displaystyle {\frac {5+{\sqrt {5}}}{5}}\approx 1.44721}$ times the length of the edges of the base pentagon, and all the dihedral angles are ${\displaystyle \arccos \left(-{\sqrt {\frac {11+4{\sqrt {5}}}{41}}}\right)\approx 119.10723^{\circ }}$. Each face has apex angle ${\displaystyle \arccos \left({\frac {{\sqrt {2}}+{\sqrt {10}}}{8}}\right)\approx 55.10590^{\circ }}$ and base angles ${\displaystyle \arccos \left({\frac {5-{\sqrt {5}}}{8}}\right)\approx 69.78820^{\circ }}$. If the base pentagon has edge length 1, its height is ${\displaystyle {\frac {5+3{\sqrt {5}}}{5}}\approx 2.34164}$.

A pentagonal tegum with base edges of length b and side edges of length l has volume given by ${\displaystyle {\frac {\sqrt {25+10{\sqrt {5}}}}{6}}b^{2}{\sqrt {l^{2}-b^{2}{\frac {5+{\sqrt {5}}}{10}}}}}$.

## Related polyhedra

A pentagonal prism can be inserted between the halves of the pentagonal bipyramid to produce the elongated pentagonal bipyramid. if a pentagonal antiprism is inserted instead, the result is the gyroelongated pentagonal bipyramid, better known as the regular icosahedron.