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 ...

, a generalized polygon can be called a polygram, and named specifically by its number of sides. All

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 ...

s are polygrams, but can also include disconnected sets of edges, called a

compound polygon In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...

. For example, a regular

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...

, , has 5 sides, and the regular

hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...

, or 2, has 6 sides divided into two triangles. A regular polygram can either be in a set of

regular star polygons 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 ...

(for gcd(''p'',''q'') = 1, ''q'' > 1) or in a set of regular polygon compounds (if gcd(''p'',''q'') > 1).

Etymology

The polygram names combine a

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 ...

, such as ''

penta- Numeral or number prefixes are prefixes derived from 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-cycle, 2-cycle, 3-cy ...

'', 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'' (in this case generating the word ''

''). The prefix is normally a Greek

cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of cardinal in the family Cardinalidae **''Cardinalis cardinalis'', or northern cardinal, the ...

, but synonyms using other prefixes exist. The ''-gram'' suffix derives from ''γραμμῆς'' (''grammos'') meaning a line.

Generalized regular polygons

A regular polygram, as a general

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

, is denoted by its

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 ...

, where ''p'' and ''q'' are

relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

(they share no factors) and ''q'' ≥ 2. For

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s ''p'' and ''q'', it can be considered as being constructed by connecting every ''q''th point out of ''p'' points regularly spaced in a circular placement.

Regular compound polygons

In other cases where ''n'' and ''m'' have a common factor, a ''polygram'' is interpreted as a lower polygon, , with ''k'' = gcd(''n'',''m''), and rotated copies are combined as a compound polygon. These figures are called regular compound polygons.

References

*Cromwell, P.; ''Polyhedra'', CUP, Hbk. 1997, . Pbk. (1999), . p. 175 * 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) * Robert Lachlan, ''An Elementary Treatise on Modern Pure Geometry''. London: Macmillan, 1893, p. 83 polygrams

*

Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentTypes of polygons Star symbols

Etymology

Generalized regular polygons

Regular compound polygons

See also

References