# G3

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.

G3
Rank3
TypeAcrohedron
Notation
Stewart notationG3
Elements
Faces1 + 6 triangles, 3 pentagons, 3 squares
Edges3 + 3 + 3 + 3 + 6 + 6 = 24
Vertices1 + 3 + 3 + 6 = 13
Abstract & topological properties
Flag count96
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexNo
NatureTame

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,0,\,{\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left({\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,0,\,{\frac {{\sqrt {5}}-1}{4}}\right).}$