# Gyroelongated pentagonal pyramid

Gyroelongated pentagonal pyramid
Rank3
TypeCRF
Notation
Bowers style acronymGyepip
Coxeter diagramoxo5oox&#xt
Elements
Faces
Edges5+5+5+10
Vertices1+5+5
Vertex figures1+5 pentagons, edge length 1
5 trapezoids, edge lengths 1, 1, 1, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Volume${\displaystyle {\frac {25+9{\sqrt {5}}}{24}}\approx 1.88019}$
Dihedral angles3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Central density1
Number of external pieces16
Level of complexity10
Related polytopes
ArmyGyepip
RegimentGyepip
DualOrder-5 monotruncated pentagonal trapezohedron
ConjugateReplenished great icosahedron
Abstract & topological properties
Flag count100
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
Flag orbits10
ConvexYes
NatureTame

The gyroelongated pentagonal pyramid (OBSA: gyepip) is one of the 92 Johnson solids (J11). It consists of 5+5+5 triangles and 1 pentagon. It can be constructed by attaching a pentagonal antiprism to the base of the pentagonal pyramid.

Alternatively, it can be constructed by diminishing one vertex from the regular icosahedron, which is why this polyhedron can also be called the diminished icosahedron. This means the icosahedron can also be thought of as a gyroelongated pentagonal bipyramid.

## Vertex coordinates

A gyroelongated pentagonal pyramid of edge length 1 has the following vertices:

• ${\displaystyle \left(0,\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right)}$,
• ${\displaystyle \pm \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right)}$,
• ${\displaystyle \pm \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right)}$,
• ${\displaystyle \pm \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right)}$.

Alternative coordinates can be obtained by removing one vertex from the regular icosahedron:

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