geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an octagram is an eight-angled

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

. The name ''octagram'' combine a Greek

numeral prefix Numeral or number prefixes are prefixes derived from Numeral (linguistics), numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1 ...

, '' octa-'', with the

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line".

Detail

In general, an octagram is any self-intersecting

octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...

(8-sided

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

). The regular octagram is labeled by the

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

, 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 The Rub-el-Hizb ( ar, ربع الحزب, '), also known as the Islamic Star, is an Islamic symbol. It is in the shape of an octagram, represented as two overlapping squares. It has been found on a number of emblems and flags. The main purpose of ...

is a

Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...

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=. A quasitruncated square, inverted as , is an octagram, t=.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'',

Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentstar polyhedron In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave p ...

stellated truncated hexahedron, t'=t 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 In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin di ...

(quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t_0,2.

Star polygon compounds

There are two regular octagrammic star figures (compounds) of the form , the first constructed as two squares =2, and second as four degenerate

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visua ...

s, =4. There are other isogonal and isotoxal compounds including rectangular and rhombic forms. or 2, like

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

s + , can be seen as the 2D equivalent of the 3D

compound of cube and octahedron The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction The 14 Cartesian coordinates of the vertices of the compound are. : 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2) : ...

, + , 4D

compound of tesseract and 16-cell In 4-dimensional geometry, the tesseract 16-cell compound is a polytope compound composed of a regular tesseract and its dual, the regular 16-cell. Its convex hull is the regular 24-cell, which is self-dual. A '' compound polytope'' is a figure t ...

, + and 5D

compound of 5-cube and 5-orthoplex In 5-dimensional geometry, the 5-cube 5-orthoplex compound is a polytope compound composed of a regular 5-cube and dual regular 5-orthoplex.Coxeter, ''Regular Polytopes (book), Regular Polytopes'', (3rd edition, 1973), Dover edition, A ''compoun ...

; that is, the compound of a

n-cube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...

and cross-polytope in their respective dual positions.

Other uses

* In

, the "Eight Spoked Asterisk" symbol ✳ is U+2733.

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 John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...

, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

External links

* {{Polygons 08 8 (number)