Compound of two squares

Compound of two squares
(OFF file)
Rank	2
Type	Regular
Notation
Bowers style acronym	Soc
Coxeter diagram	xo4ox
Schläfli symbol	{8/2}
Elements
Components	2 squares
Edges	8
Vertices	8
Vertex figure	Dyad, length √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Inradius	${\frac {1}{2}}=0.5$
Area	2
Angle	90°
Central density	2
Number of external pieces	16
Level of complexity	2
Related polytopes
Army	Oc, edge length ${\sqrt {\frac {2-{\sqrt {2}}}{2}}}$
Dual	Compound of two squares
Conjugate	Compound of two squares
Convex core	Octagon
Abstract & topological properties
Flag count	16
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(8), order 16
Flag orbits	1
Convex	No
Nature	Tame

The stellated octagon, or soc is a polygon compound composed of two squares. As such it has 8 edges and 8 vertices.

It is the first stellation of the octagon.

Its quotient prismatic equivalent is the square antiprism, which is three-dimensional.

Vertex coordinates

Coordinates for the vertices of a stellated octagon of edge length 1 centered at the origin are given by:

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

Variations

The stellated octagon can be varied by changing the angle between the two component squares from the usual 45°. These 2-square compounds generally have a ditetragon as their convex hull and remain uniform, but not regular, with square symmetry only.

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".