# Semicupolaically-faceted great icosahedron

Semicupolaically-faceted great icosahedron
Rank3
TypeOrbiform
Notation
Bowers style acronymScufgi
Elements
Faces1+6 triangles, 3 pentagrams
Edges3+3+6+6
Vertices3+3+3
Vertex figures3 nonconvex pentagons, edge lengths 1, 1, (5–1)/2, 1, (5–1)/2
3 butterflies, edge lengths 1 and (5–1)/2
3 isosceles triangles, edge lengths 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Dihedral angles3-5/2: ${\displaystyle \arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 79.18768^{\circ }}$
3-3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$
Related polytopes
ArmyTeddi, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
ConjugateSemicupolaically-faceted icosahedron
Abstract & topological properties
Flag count72
Euler characteristic1
OrientableNo
Genus1
Properties
SymmetryA2×I, order 6
ConvexNo
NatureTame

The semicupolaically-faceted great icosahedron, or scufgi, is an orbiform polyhedron. It consists of 7 triangles and 3 pentagrams. As its name suggests, it is a faceting of the great icosahedron, and thus also of the small stellated dodecahedron. It can also be obtained by blending together three pentagrammic pyramids, the three of them all sharing a triangle.

It appears as a cell of the disnub disicositetrachoron.

## Vertex coordinates

The vertices of a semicupolaically-faceted great icosahedron of edge length 1 are given by:

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

and all sign changes of none or one of the nonzero coordinates of:

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