geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a star polygon is a type of non-
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 ...
. 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 on regular simple or star polygons.
Branko Grünbaum
Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentJohannes Kepler
Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
, one corresponding to the
regular star 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 ...
s with intersecting edges that do not generate new vertices, and the other one to the
isotoxal
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given tw ...
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 simple polygon that is not convex is called concave, non-convex or ...
simple polygon
In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...
s.Grünbaum & Shephard (1987). Tilings and Patterns. Section 2.5Polygrams include 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 around ...
, but also compound figures like the
hexagram
, can be seen as a compound polygon, 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 l ...
.
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. It is also a simple and didactic way of d ...
, is a polygon having ''q'' ≥ 2 turns (''q'' is called the
turning number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve's number o ...
or
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
), like in
spirolateral
In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,...,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, ...
s.Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p. 24
Names
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:
*triangle, quadrilateral, pentagon, hexagon, octagon ...
, 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:
*triangle, quadrilateral, pentagon, hexagon, octagon ...
'', with the
Greek
Greek may refer to:
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 of all kno ...
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 around ...
''). The prefix is normally a Greek
cardinal
Cardinal or The Cardinal most commonly refers to
* Cardinalidae, a family of North and South American birds
**''Cardinalis'', genus of three species in the family Cardinalidae
***Northern cardinal, ''Cardinalis cardinalis'', the common cardinal of ...
, but synonyms using other prefixes exist. For example, a nine-pointed polygon or ''
enneagram
Enneagram may refer to:
* Enneagram (geometry), a nine-sided star polygon with various configurations
* Enneagram of Personality, a model of human personality illustrated by an enneagram figure
See also
* Enneagon
In geometry, a nonagon () or ...
'' is also known as a ''nonagram'', using the ordinal ''nona'' from
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
. The ''-gram'' suffix derives from '' γραμμή'' (''grammḗ''), meaning a line. The name ''star polygon'' reflects the resemblance of these shapes to the diffraction spikes of real stars.
Regular star polygon
A ''regular star polygon'' is a self-intersecting, equilateral, and equiangular
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
.
A regular star polygon is denoted by its
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
, where ''p'' (the number of vertices) and ''q'' (the
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
) are
relatively prime
In number theory, 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 equiv ...
(they share no factors) and where ''q'' ≥ 2. The density of a polygon can also be called its
turning number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve's number o ...
: 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 the
dihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
D''p'', of order 2''p'', independent of ''q''.
Regular star polygons were first studied systematically by
Thomas Bradwardine
Thomas Bradwardine (c. 1300 – 26 August 1349) was an English cleric, scholar, mathematician, physicist, courtier and, very briefly, Archbishop of Canterbury. As a celebrated scholastic philosopher and doctor of theology, he is often ca ...
, and later
Johannes Kepler
Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
.
Construction via vertex connection
Regular star polygons can be created by connecting one vertex of a regular ''p''-sided simple polygon to another vertex, non-adjacent to the first one, 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 1st to the 3rd vertex, from the 3rd to the 5th vertex, from the 5th to the 2nd vertex, from the 2nd to the 4th vertex, and from the 4th to the 1st vertex.
If ''q'' ≥ ''p''/2, then the construction of will result in the same polygon as ; 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 (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
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
In geometry, the pentagrammic crossed-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 differs from the pentagrammic antiprism by having oppos ...
. 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 diag ...
, which can be seen as a "crossed triangle" cuploid.
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: 1–3 and 4–6. This should be seen not as two overlapping triangles, but as a double-winding single unicursal 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, a 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 ...
s of a convex regular ''core'' polygon. Constructions based on stellation also allow regular polygonal compounds to be obtained in cases where the density ''q'' and amount ''p'' of vertices are not coprime. When constructing star polygons from stellation, however, if ''q'' > ''p''/2, the lines will instead diverge infinitely, and if ''q'' = ''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 (geometry), edge and one Vertex (geometry), vertex. It has Schläfli symbol .Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386 ...
and
digon
In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...
; such polygons do not yet appear to have been studied in detail.
Isotoxal star simple polygons
When the intersecting line segments are removed from a regular star ''n''-gon, the resulting figure is no longer regular, but can be seen as an
isotoxal
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given tw ...
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 simple polygon that is not convex is called concave, non-convex or ...
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 ...
2''n''-gon, alternating vertices at two different radii.
Branko Grünbaum
Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentTilings and patterns
''Tilings and patterns'' is a book by mathematicians Branko Grünbaum and Geoffrey Colin Shephard published in 1987 by W.H. Freeman. The book was 10 years in development, and upon publication it was widely reviewed and highly acclaimed.
Structu ...
'', represents such a star that matches the outline of a regular polygram as , ''n''/''d'', , or more generally with , which denotes an isotoxal concave ''or convex'' simple 2''n''-gon with outer
internal angle
In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...
𝛼.
* For , ''n''/''d'', , the outer internal angle degrees, necessarily, and the inner (new) vertices have an external angle degrees, necessarily.
* For , the outer internal and inner external angles, also denoted by 𝛼 and ''β'', do not have to match those of any regular polygram ; however, degrees and necessarily (here, is concave).
Examples in tilings
These polygons are often seen in tiling patterns. The parametric angle 𝛼 (in degrees or radians) can be chosen to match
internal angle
In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...
s of neighboring polygons in a tessellation pattern. In his 1619 work ''
Harmonice Mundi
''Harmonice Mundi'' (Latin: ''The Harmony of the World'', 1619) is a book by Johannes Kepler. In the work, written entirely in Latin, Kepler discusses harmony and congruence in geometrical forms and physical phenomena. The final section of t ...
'', among periodic tilings,
Johannes Kepler
Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
includes nonperiodic tilings, like that with three regular pentagons and one regular star pentagon fitting around certain vertices, 5.5.5.5/2, and related to modern
Penrose tiling
A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and a tiling is ''aperiodic'' if it does not contain arbitrarily large Perio ...
s.Branko Grunbaum and Geoffrey C. Shephard, Tilings by Regular Polygons, Mathematics Magazine #50 (1977), pp. 227–247, and #51 (1978), pp. 205–206 /ref>
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 (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent
These three treatments are:
* Where a line segment 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 are 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 displ ...
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 ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
''. 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 (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
. (However, for non-orientable polyhedra, density can only be considered modulo 2 and hence, in those cases, for consistency, the first treatment is sometimes used instead.)
* Wherever a line segment may be drawn between two sides, the region in which the line segment 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 result.
In art and culture
Star polygons feature prominently in art and culture. Such polygons may or may not be regular, but they are always highly
symmetrical
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. 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 around ...
) is also known as a pentalpha or pentangle, and historically has been considered by many magical and
religious
Religion is a range of social- cultural systems, including designated behaviors and practices, morals, beliefs, worldviews, texts, sanctified places, prophecies, ethics, or organizations, that generally relate humanity to supernatural ...
cults to have
occult
The occult () is a category of esoteric or supernatural beliefs and practices which generally fall outside the scope of organized religion and science, encompassing phenomena involving a 'hidden' or 'secret' agency, such as magic and mysti ...
significance.
* The and star polygons (
heptagram
A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
s) also have occult significance, particularly in the
Kabbalah
Kabbalah or Qabalah ( ; , ; ) is an esoteric method, discipline and school of thought in Jewish mysticism. It forms the foundation of Mysticism, mystical religious interpretations within Judaism. A traditional Kabbalist is called a Mekubbal ...
and in
Wicca
Wicca (), also known as "The Craft", is a Modern paganism, modern pagan, syncretic, Earth religion, Earth-centred religion. Considered a new religious movement by Religious studies, scholars of religion, the path evolved from Western esote ...
.
* The star polygon (
octagram
In geometry, an octagram is an eight-angled star polygon.
The name ''octagram'' combine a Greek numeral prefix, ''wikt:octa-, octa-'', with the Greek language, Greek suffix ''wikt:-gram, -gram''. The ''-gram'' suffix derives from γραμμή ...
) is a frequent geometrical motif in
Mughal
Mughal or Moghul may refer to:
Related to the Mughal Empire
* Mughal Empire of South Asia between the 16th and 19th centuries
* Mughal dynasty
* Mughal emperors
* Mughal people, a social group of Central and South Asia
* Mughal architecture
* Mug ...
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 Muslims, Muslim populations. Referring to characteristic traditions across ...
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 construction, constructi ...
hendecagram
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven Vertex (geometry), vertices.
The name ''hendecagram'' combines a Greek numeral prefix, ''wikt:hendeca-, hendeca-'', with the Greek language, Greek suffix ...
Five-pointed star
A five-pointed star (☆), geometrically an equilateral concave decagon, is a common ideogram in modern culture.
Comparatively rare in classical heraldry, it was notably introduced for the flag of the United States in the Flag Act of 1777 and si ...
*
Magic star
An ''n''-pointed magic star is a star polygon with Schläfli symbol in which numbers are placed at each of the ''n'' vertices and ''n'' intersections, such that the four numbers on each line sum to the same magic constant. A normal magic star c ...
*
Moravian star
A Moravian star () is an illuminated decoration used during the Christian liturgical seasons of Advent, Christmas, and Epiphanytide, Epiphany representing the Star of Bethlehem pointing towards the infant Jesus. The Moravian Church teaches:
Th ...
*
Pentagramma mirificum
''Pentagramma mirificum'' (Latin for "miraculous pentagram") is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book '' Mirif ...
Star (glyph)
In typography, a star is any of several glyphs with a number of points arrayed within an imaginary circle. A commonly used star symbol is the asterisk.
Four points
Five points
See also
* Mullet (heraldry)
* Pentagram
Six points
See ...
*
Star polyhedron
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon, nonconvexity giving it a star-like visual quality.
There are two general kinds of star polyhedron:
*Polyhedra which self-intersect in a repetit ...
,
Kepler–Poinsot polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four Regular polyhedron, regular Star polyhedron, star polyhedra.
They may be obtained by stellation, stellating the regular Convex polyhedron, convex dodecahedron and icosahedron, and differ f ...
Starfish
Starfish or sea stars are Star polygon, star-shaped echinoderms belonging to the class (biology), class Asteroidea (). Common usage frequently finds these names being also applied to brittle star, ophiuroids, which are correctly referred to ...
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-Strauss, ''The Symmetries of Things'', 2008, (Chapter 26, p. 404: Regular star-polytopes Dimension 2)
*
Branko Grünbaum
Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentStar symbols