# Trireplenished great icosahedron

Trireplenished great icosahedron
Rank3
TypeOrbiform
Notation
Bowers style acronymTargi
Elements
Faces1+1+3 triangles, 3 pentagrams
Edges3+3+3+6
Vertices3+3+3
Vertex figures3 crossed isosceles trapezoids, edge length 1, 1, 1, (5-1)/2
3+3 isosceles triangles, edge lengths 1, (5-1)/2, (5-1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Volume${\displaystyle {\frac {7{\sqrt {5}}-15}{24}}\approx 0.027186}$
Dihedral angles5/2–5/2: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
3-3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$
3-5/2: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Related polytopes
RegimentTargi
ConjugateTridiminished icosahedron
Abstract & topological properties
Flag count60
OrientableYes
Properties
SymmetryA2×I, order 6
ConvexNo
NatureTame

The trireplenished great icosahedron, or targi, is a nonconvex orbiform polyhedron. It consists of 1+1+3 triangles and 3 pentagrams. It can be constructed by removing 3 mutually non-adjacent vertices from a regular great icosahedron.

## Vertex coordinates

A trireplenished icosahedron of edge length 1 has the following vertices:

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

## In vertex figures

The trireplenished great icosahedron is the vertex figure of the uniform retrosnub disicositetrachoron.