Octagram on arvau

Wikipedia / Octagram

In geometry, an octagram is an eight-angled star polygon.

The name octagram combine a Greek numeral prefix, octa-, with the Greek suffix -gram. The -gram suffix derives from γραμμή (grammḗ) meaning "line".

Detail

In general, an octagram is any self-intersecting octagon (8-sided polygon).

The regular octagram is labeled by the Schläfli symbol {8/3}, which means an 8-sided star, connected by every third point.

Variations

These variations have a lower dihedral, Dih4, symmetry:The symbol Rub el Hizb is a Unicode glyph ۞ at U+06DE.

As a quasitruncated square

Deeper truncations of the square can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated square is an octagon, t{4}={8}. A quasitruncated square, inverted as {4/3}, is an octagram, t{4/3}={8/3}.

The uniform star polyhedron stellated truncated hexahedron, t'{4,3}=t{4/3,3} has octagram faces constructed from the cube in this way. It may be considered for this reason as a three-dimensional analogue of the octagram.Another three-dimensional version of the octagram is the nonconvex great rhombicuboctahedron (quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t0,2{4/3,3}.

Star polygon compounds

There are two regular octagrammic star figures (compounds) of the form {8/k}, the first constructed as two squares {8/2}=2{4}, and second as four degenerate digons, {8/4}=4{2}. There are other isogonal and isotoxal compounds including rectangular and rhombic forms.{8/2} or 2{4}, like Coxeter diagrams + , can be seen as the 2D equivalent of the 3D compound of cube and octahedron, + , 4D compound of tesseract and 16-cell, + and 5D compound of 5-cube and 5-orthoplex; that is, the compound of a n-cube and cross-polytope in their respective dual positions.

Other uses

One of the oldest depictions of several variations of the octagram nested within each other was found in a cave in Jordan. Dating back 6,000 years and associated with the Ghassulian culture, it is speculated to have represented some kind of calendar system.
In Unicode, the "Eight Spoked Asterisk" symbol ✳ is U+2733.
The unicode for Dingir, a Sumerian cuneiform eight-pointed ideogram meaning "sky" and later a logogram for a deity is U+1202D .

The 8-pointed diffraction spikes of the star images from the James Webb Space Telescope are due to the diffraction caused by the hexagonal shape of the mirror sections and the struts holding the secondary mirror.
Used as a parol or star for the 2010 ABS-CBN Christmas Station ID Ngayong Pasko Magniningning Ang Pilipino () due to the usage of a sun from the Philippine flag, making it also a nationalism and patriotism-themed song aside from being a Christmas song.

References

Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), .
Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp.43–70.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26. pp.404: Regular star-polytopes Dimension 2)