# Replenished great icosahedron

Replenished great icosahedron
Rank3
Notation
Bowers style acronymRegi
Elements
Faces5+5+5 triangles, 1 pentagram
Edges5+5+5+10
Vertices1+5+5
Vertex figures1+5 pentagrams, edge length 1
5 crossed isosceles trapezoids, edge lengths 1, 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Volume${\displaystyle {\frac {25-9{\sqrt {5}}}{24}}\approx 0.20314}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$
5/2–3: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Related polytopes
ArmyGyepip, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentRegi
ConjugateGyroelongated pentagonal pyramid
Convex hullGyroelongated pentagonal pyramid
Abstract & topological properties
Flag count100
Euler characteristic1
OrientableYes
Genus1
Properties
SymmetryH2×I, order 10
ConvexNo
NatureTame

The replenished great icosahedron, or regi, is a nonconvex orbiform polyhedron and an edge-faceting of the small stellated dodecahedron. Its faces are 5+5+5 triangles and 1 pentagram. It can be formed from removing one vertex from the great icosahedron, by removing a pentagrammic pyramid.

## Vertex coordinates

Its vertex coordinates are the same as those of the small stellated dodecahedron with any 1 vertex removed.