G₃

G3
	(3D model); (OFF file)
Rank	3
Type	Acrohedron
Space	Spherical
Notation
Stewart notation	G3
Elements
Faces	1 + 6 triangles, 3 pentagons, 3 squares
Edges	3 + 3 + 3 + 3 + 6 + 6 = 24
Vertices	1 + 3 + 3 + 6 = 13
Abstract & topological properties
Flag count	96
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	No
Nature	Tame

G₃ is a non-convex regular faced polyhedron. It was the smallest known 5-4-3 acrohedron until the discovery of m*, which has one face fewer. It was named by Bonnie Stewart in Adventures Among the Toroids, although Stewart was not searching for acrohedra in particular.

Vertex coordinates[edit | edit source]

The vertex coordinates of a G₃ with unit edge length are given by:

$\left(\pm\frac{1+\sqrt5}{4},\,\pm\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),$
$\left(\pm\frac12,\,0,\,\frac{3+\sqrt5}{4}\right),$
$\left(0,\,\pm\frac{3+\sqrt5}{4},\,\frac12\right),$
$\left(\frac{3+\sqrt5}{4},\,\pm\frac12,\,0\right),$
$\left(\frac{\sqrt5-1}{4},\,\pm\frac12,\,0\right),$
$\left(-\frac12,\,0,\,\frac{\sqrt5-1}{4}\right).$