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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'.^[1]

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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[edit]

These variations have a lower dihedral, Dih₄, symmetry:

The symbol Rub el Hizb is a Unicode glyph ۞ at U+06DE.

As a quasitruncated square[edit]

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}.^[2]

The uniform star polyhedronstellated 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, t_0,2{4/3,3}.

Star polygon compounds[edit]

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 presentations of an octagonal star[edit]

An octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines.

Other uses[edit]

• In Unicode, the 'Eight Spoked Asterisk' symbol ✳ is U+2733.

See also[edit]

Usage

• Rub el Hizb – Islamic character
• Star of Ishtar - symbol of the ancient Sumerian goddess Inanna and her East Semitic counterpart Ishtar and Roman Venus.
• Star of Lakshmi – Indian character
• Surya Majapahit – usage during Majapahit times in Indonesia to represent the Hindu gods of the directions
• Compass rose – usage in compasses to represent the cardinal directions for the eight principal winds
• Auseklis – usage of regular octagram by Latvians
• Guñelve – representation of Venus in Mapuche iconography.
• Selburose – usage of regular octagram in Norwegian design

Stars generally

References[edit]

1. ^γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
2. ^The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

• Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), ISBN0-7167-1193-1.
• 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-Strass, The Symmetries of Things 2008, ISBN978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

External links[edit]

• Weisstein, Eric W.'Octagram'. MathWorld.

• Sunday 04.11.2021 at 10:00 AM ET
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