# Octagram

Octagram
Rank2
TypeRegular
Notation
Bowers style acronymOg
Coxeter diagramx8/3o ()
Schläfli symbol{8/3}
Elements
Edges8
Vertices8
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {2-{\sqrt {2}}}{2}}}\approx 0.54120}$
Inradius${\displaystyle {\frac {{\sqrt {2}}-1}{2}}\approx 0.20711}$
Area${\displaystyle 2({\sqrt {2}}-1)\approx 0.82843}$
Angle45°
Central density3
Number of external pieces16
Level of complexity2
Related polytopes
ArmyOc, edge length ${\displaystyle {\sqrt {2}}-1}$
DualOctagram
ConjugateOctagon
Convex coreOctagon
Abstract & topological properties
Flag count16
Euler characteristic0
OrientableYes
Properties
SymmetryI2(8), order 16
ConvexNo
NatureTame

The octagram is a non-convex polygon with 8 sides. It's created by stellating the octagon. A regular octagram has equal sides and equal angles.

This is the second stellation of the octagon, and the only one that is not a compound. The only other polygons with a single non-compound stellation are the pentagon, the decagon, and the dodecagon.

The octagram is the uniform quasitruncation of the square, and as such is the only regular star polygon to regularly appear in non-prismatic uniform polytopes in 5 dimensions and higher.

## Vertex coordinates

Coordinates for an octagram of unit edge length, centered at the origin, are all permutations of

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

## Representations

An octagram has the following Coxeter diagrams:

• x8/3o (full symmetry)
• x4/3x () (B2 symmetry)

## Related polytopes

The Octagram is one of 12 regular octagons in Euclidean space.

Octagons in Euclidean space
Name Extended Schläfli symbol Dimensions
octagon ${\displaystyle \left\{{\dfrac {8}{1}}\right\}}$ 2
octagram ${\displaystyle \left\{{\dfrac {8}{3}}\right\}}$ 2
octagonal-square coil ${\displaystyle \left\{{\dfrac {8}{1,2}}\right\}}$ 4
octagonal-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,3}}\right\}}$ 4
skew octagon ${\displaystyle \left\{{\dfrac {8}{1,4}}\right\}}$ 3
square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{2,3}}\right\}}$ 4
skew octagram ${\displaystyle \left\{{\dfrac {8}{3,4}}\right\}}$ 3
octagonal-square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,2,3}}\right\}}$ 6
skew octagonal-square coil ${\displaystyle \left\{{\dfrac {8}{1,2,4}}\right\}}$ 5
skew octagonal-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,3,4}}\right\}}$ 5
skew square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{2,3,4}}\right\}}$ 5
skew octagonal-square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,2,3,4}}\right\}}$ 7