# G3

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

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

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

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