Semicupolaically-faceted great icosahedron

Semicupolaically-faceted great icosahedron
(OFF file)
Rank	3
Type	Orbiform
Notation
Bowers style acronym	Scufgi
Elements
Faces	1+6 triangles, 3 pentagrams
Edges	3+3+6+6
Vertices	3+3+3
Vertex figures	3 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	${\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779$
Dihedral angles	3-5/2: $\arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 79.18768^{\circ }$
	3-3: $\arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }$
Related polytopes
Army	Teddi, edge length ${\frac {{\sqrt {5}}-1}{2}}$
Conjugate	Semicupolaically-faceted icosahedron
Abstract & topological properties
Flag count	72
Euler characteristic	1
Orientable	No
Genus	1
Properties
Symmetry	A2×I, order 6
Convex	No
Nature	Tame

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:

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

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

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

Gallery

External links

• Klitzing, Richard. "scufgi".
• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#12 under sissid).