Trireplenished great icosahedron

Trireplenished great icosahedron
(OFF file)
Rank	3
Type	Orbiform
Notation
Bowers style acronym	Targi
Elements
Faces	1+1+3 triangles, 3 pentagrams
Edges	3+3+3+6
Vertices	3+3+3
Vertex figures	3 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	${\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779$
Volume	${\frac {7{\sqrt {5}}-15}{24}}\approx 0.027186$
Dihedral angles	5/2–5/2: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	3-3: $\arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }$
	3-5/2: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }$
Related polytopes
Army	Adjacent tridiminished ike
Regiment	Targi
Conjugate	Tridiminished icosahedron
Abstract & topological properties
Flag count	60
Orientable	Yes
Properties
Symmetry	A2×I, order 6
Convex	No
Nature	Tame

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:

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

In vertex figures

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

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#11 under sissid).

• Klitzing, Richard. "targi".