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 ...

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentJohannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being

simple Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

isotoxal

concave polygon A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Pol ...

s. The first usage is included in polygrams which includes polygons like the

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 aro ...

but also compound figures like the

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 ...

. One definition of a ''star polygon'', used in

turtle graphics In computer graphics, turtle graphics are vector graphics using a relative cursor (the " turtle") upon a Cartesian plane (x and y axis). Turtle graphics is a key feature of the Logo programming language. Overview The turtle has three attri ...

, is a polygon having 2 or more turns ( turning number and

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

), like in spirolaterals.Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p.24

Etymology

Star polygon names combine a

numeral prefix 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-cyc ...

, such as '' penta-'', 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 ''

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 aro ...

''). 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, ...

, but synonyms using other prefixes exist. For example, a nine-pointed polygon or ''

enneagram Enneagram is a compound word derived from the Greek neoclassical stems for "nine" (''ennea'') and something "written" or "drawn" (''gramma''). Enneagram may refer to: * Enneagram (geometry), a nine-sided star polygon with various configurations ...

'' is also known as a ''nonagram'', using the ordinal ''nona'' from

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...

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

Regular star polygon

A "regular star polygon" is a self-intersecting, equilateral equiangular

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 t ...

. A regular star polygon 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 mor ...

, where ''p'' (the number of vertices) and ''q'' (the

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

) 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 equival ...

(they share no factors) and ''q'' ≥ 2. The density of a polygon can also be called its turning number, the sum of the turn angles of all the vertices divided by 360°. The

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of is

dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...

''D''_n of order 2''n'', independent of ''k''. Regular star polygons were first studied systematically by Thomas Bradwardine, and later Johannes Kepler.

Construction via vertex connection

Regular star polygons can be created by connecting one

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

of a simple, regular, ''p''-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. Alternatively for integers ''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. For instance, in a regular pentagon, a five-pointed star can be obtained by drawing a line from the first to the third vertex, from the third vertex to the fifth vertex, from the fifth vertex to the second vertex, from the second vertex to the fourth vertex, and from the fourth vertex to the first vertex. If ''q'' is greater than half of ''p'', then the construction will result in the same polygon as ''p''-''q''; connecting every third vertex of the pentagon will yield an identical result to that of connecting every second vertex. However, the vertices will be reached in the opposite direction, which makes a difference when retrograde polygons are incorporated in higher-dimensional polytopes. For example, an

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass ...

formed from a prograde pentagram results in a

pentagrammic antiprism In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. It has 12 faces, 20 edges and 10 vertices. This polyhedron is iden ...

; the analogous construction from a retrograde "crossed pentagram" results in a pentagrammic crossed-antiprism. Another example is the

tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagr ...

, which can be seen as a "crossed triangle"

cuploid In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares ...

.

Degenerate regular star polygons

If ''p'' and ''q'' are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example will appear as a triangle, but can be labeled with two sets of vertices 1-6. This should be seen not as two overlapping triangles, but a double-winding of a single unicursal hexagon.Coxeter, The Densities of the Regular polytopes I, p.43: If d is odd, the truncation of the polygon is naturally . But if not, it consists of two coincident 's; two, because each side arises from an original side and once from an original vertex. Thus the density of a polygon is unaltered by truncation. : Doubly wound hexagon

Construction via stellation

Alternatively, a regular star polygon can also be obtained as a sequence of

stellation 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 ...

s of a convex regular ''core'' polygon. Constructions based on stellation also allow for regular polygonal compounds to be obtained in cases where the density and amount of vertices are not coprime. When constructing star polygons from stellation, however, if ''q'' is greater than ''p''/2, the lines will instead diverge infinitely, and if ''q'' is equal to ''p''/2, the lines will be parallel, with both resulting in no further intersection in Euclidean space. However, it may be possible to construct some such polygons in spherical space, similarly to the

monogon In geometry, a monogon, also known as a henagon, is a polygon with one edge and one vertex. It has Schläfli symbol .Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386-388 In Euclidean geometry In Eucli ...

and

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 visu ...

; such polygons do not yet appear to have been studied in detail.

Simple isotoxal star polygons

When the intersecting lines are removed, the star polygons are no longer regular, but can be seen as

simple Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set In geometry, a subset ...

isotoxal 2''n''-gons, alternating vertices at two different radii, which do not necessarily have to match the regular star polygon angles.

Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentpolygram PolyGram N.V. was a multinational entertainment company and major music record label formerly based in the Netherlands. It was founded in 1962 as the Grammophon-Philips Group by Dutch corporation Philips and German corporation Siemens, to be ...

with a notation more generally, representing an n-sided star with each

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

α<180°(1-2/''n'') degrees. For , ''n''/''d'', , the inner vertices have an exterior angle, β, as 360°(''d''-1)/''n''.

Examples in tilings

These polygons are often seen in tiling patterns. The parametric angle α (degrees or radians) can be chosen to match

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

s of neighboring polygons in a tessellation pattern. Johannes Kepler in his 1619 work '' Harmonices Mundi'', including among other period tilings, nonperiodic tilings like that three regular pentagons, and a regular star pentagon (5.5.5.5/2) can fit around a vertex, and related to modern

penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without ...

s.Branko Grunbaum and Geoffrey C. Shephard, Tilings by Regular Polygons
Mathematics Magazine 50 (1977), 227–247 and 51 (1978), 205–206]

Interiors

The interior of a star polygon may be treated in different ways. Three such treatments are illustrated for a pentagram.

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

These include: * Where a side occurs, one side is treated as outside and the other as inside. This is shown in the left hand illustration and commonly occurs in computer

vector graphics Vector graphics is a form of computer graphics in which visual images are created directly from geometric shapes defined on a Cartesian plane, such as points, lines, curves and polygons. The associated mechanisms may include vector display ...

rendering. *The number of times that the polygonal curve winds around a given region determines its ''

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

''. The exterior is given a density of 0, and any region of density > 0 is treated as internal. This is shown in the central illustration and commonly occurs in the mathematical treatment of

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

. (However, for non-orientable polyhedra density can only be considered modulo 2 and hence the first treatment is sometimes used instead in those cases for consistency.) * Where a line may be drawn between two sides, the region in which the line lies is treated as inside the figure. This is shown in the right hand illustration and commonly occurs when making a physical model. When the area of the polygon is calculated, each of these approaches yields a different answer.

In art and culture

Star polygons feature prominently in art and culture. Such polygons may or may not be

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

but they are always highly symmetrical. Examples include: *The star pentagon (

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 aro ...

) is also known as a pentalpha or pentangle, and historically has been considered by many magical and

religious Religion is usually defined as a social- cultural system of designated behaviors and practices, morals, beliefs, worldviews, texts, sanctified places, prophecies, ethics, or organizations, that generally relates humanity to supernatural, ...

cults to have

occult The occult, in the broadest sense, is a category of esoteric supernatural beliefs and practices which generally fall outside the scope of religion and science, encompassing phenomena involving otherworldly agency, such as magic and mysticism ...

significance. *The and star polygons (

heptagram A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, '' hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ� ...

s) also have occult significance, particularly in the

Kabbalah Kabbalah ( he, קַבָּלָה ''Qabbālā'', literally "reception, tradition") is an esoteric method, discipline and school of thought in Jewish mysticism. A traditional Kabbalist is called a Mekubbal ( ''Məqūbbāl'' "receiver"). The de ...

and in

Wicca Wicca () is a modern Pagan religion. Scholars of religion categorise it as both a new religious movement and as part of the occultist stream of Western esotericism. It was developed in England during the first half of the 20th century and w ...

. *The star polygon (

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 ...

) is a frequent geometrical motif in Mughal

Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslim populations. Referring to characteristic traditions across a wide r ...

and

architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing buildings ...

; the first is on the

emblem of Azerbaijan The state emblem of Azerbaijan ( az, Azərbaycan gerbi) mixes traditional and modern symbols. The focal point of the emblem is a stylized flame. The flame is in the shape of the word "Allah" written in Arabic () to represent the country's majorit ...

. *An eleven pointed star called the

hendecagram In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven vertices. The name ''hendecagram'' combines a Greek numeral prefix, '' hendeca-'', with the Greek suffix ''-gram''. The ''hendeca-'' prefix derives f ...

was used on the tomb of Shah Nemat Ollah Vali.

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

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

Etymology

Regular star polygon

Construction via vertex connection

Degenerate regular star polygons

Construction via stellation

Simple isotoxal star polygons

Examples in tilings

Interiors

In art and culture

See also

References