Star polygon

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Set of regular star polygons
Star polygon 5-2.svg
{5/2}
Star polygon 7-2.svg
{7/2}
Star polygon 7-3.svg
{7/3}
Star polygon 8-3.svg
{8/3}
Star polygon 9-2.svg
{9/2}
Star polygon 9-4.svg
{9/4}
Star polygon 10-3.svg
{10/3}
...
Schläfli symbol
2<2q<p
gcd(p,q)=1
{p/q}
Vertices and Edges p
Density q
Coxeter–Dynkin diagram CDel node 1.pngCDel p.pngCDel rat.pngCDel dq.pngCDel node.png
Symmetry group Dihedral (Dp)
Dual polygon Self-dual
Internal angle
(degrees)
\frac{180(p-2q)}{p}[1]

A regular star polygon (not to be confused with a star-shaped polygon or a star domain) is a regular non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined.

Etymology[edit]

Modern star polygon names are created by combining a numeral prefix, such as penta-, with the Greek suffix -gram (in this case creating pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. For example, a nine-pointed polygon is called an enneagram, but is also known as a nonagram, using the ordinal nona from Latin.

Although this prefix+suffix formula can be used to create or find star polygon names, it does not necessarily reflect the word's history. For example, pentagram derives from pentagrammos / pentegrammos ("five lines") whose -grammos derives from grammē meaning "line". The -gram suffix, however, derives from gramma meaning "to write". Gramma and grammē are however very similar in sound, writing (γράμμα, γραμμή) and meaning ("written character, letter, that which is drawn", "stroke or line of a pen[2]"), and are possibly cognates.

Regular star polygons[edit]

In geometry, a "regular star polygon" is a self-intersecting, equilateral equiangular polygon, created by connecting one vertex of a simple, regular, p-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again.[3] Alternatively for integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement.[4] 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. The notation for such a polygon is {p/q} (see Schläfli symbol), which is equal to {p/p-q}. Regular star polygons will be produced when p and q are relatively prime (they share no factors). A regular star polygon can also be represented as a sequence of stellations of a convex regular core polygon. Regular star polygons were first studied systematically by Thomas Bradwardine.

Examples[edit]

Regular convex and star polygons from 3 to 12 vertices labelled with their Schläfli symbols

Star figures[edit]

Star figure
hexagram
2{3} or {6/2}
Star figure
enneagram
3{3} or {9/3}

If the number of sides n is divisible by m, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a "degenerate star polygon".

In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called "star figures" or "improper star polygons" or "compound polygons". The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m.

A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form k{n/m}, as 2{5/2}, rather than the commonly used {10/4}.

A six-pointed star, like a hexagon, can be created using a compass and a straight edge:

  • Make a circle of any size with the compass.
  • Without changing the radius of the compass, set its pivot on the circle's circumference, and find one of the two points where a new circle would intersect the first circle.
  • With the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked.
  • With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles.

Symmetry[edit]

Regular star polygons and star figures can be thought of as diagramming cosets of the subgroups x\mathbb{Z}_n of the finite group \mathbb{Z}_n.

The symmetry group of {n/k} is dihedral group Dn of order 2n, independent of k.

Irregular star polygons[edit]

The white line in this graph is an irregular pentagonal cyclic polygon, defining the a vertex figure for the great retrosnub icosidodecahedron. The edge lengths are defined by the distance between alternate vertices in the faces of the uniform polyhedron.

A star polygon need not be regular. Irregular cyclic star polygons occur as vertex figures for the uniform polyhedra, defined by the sequence of regular polygon faces around each vertex, allowing for both multiple turns, and retrograde directions. (See vertex figures at List of uniform polyhedra)[5]

The final stellation of the icosahedron can be seen as a polyhedron with irregular {9/4} star polygon faces with Dih3 dihedral symmetry.

Enneagram 9-4 icosahedral.svg

The unicursal hexagram is another example of a cyclic irregular star polygon, containing Dih2 dihedral symmetry.

Solid unicursal hexagram.svg

Interiors of star polygons[edit]

Star polygons leave an ambiguity of interpretation for interiors. This diagram demonstrates three interpretations of a pentagram.

Pentagram interpretations.svg

  • The left-hand interpretation has the 5 vertices of a regular pentagon connected alternately on a cyclic path, skipping alternate vertices. The interior is everything immediately left (or right) from each edge (until the next intersection). This makes the core convex pentagonal region actually "outside", and in general you can determine inside by a binary even-odd rule of counting how many edges are intersected from a point along a ray to infinity.
  • The middle interpretation also has the 5 vertices of a regular pentagon connected alternately on a cyclic path. The interior may be treated either:
    • as the inside of a simple 10-sided polygon perimeter boundary, as below.
    • with the central convex pentagonal region surrounded twice, because the starry perimeter winds around it twice.
  • The right-hand interpretation creates new vertices at the intersections of the edges (5 in this case) and defines a new concave decagon (10-pointed polygon) formed by perimeter path of the middle interpretation; it is in fact no longer a pentagram.

What is the area inside the pentagram? Each interpretation leads to a different answer.

Example interpretations of a star prism[edit]

{7/2} heptagrammic prism:

Septagram prism-2-7.png
Heptagrams with
2-sided interior
Heptagrammic prism 7-2.png
Heptagrams with
a simple perimeter interior

The heptagrammic prism above shows different interpretations can create very different appearances.

Builders of polyhedron models, like Magnus Wenninger, usually represent star polygon faces in the concave form, without internal edges shown.

Star polygons in art and culture[edit]

Star polygons feature prominently in art and culture. Such polygons may or may not be regular but they are always highly symmetrical. Examples include:

An {8/3} star polygon (octagram) constructed in an octagon
Seal of Solomon (interlaced hexagram, with circle and dots)

See also[edit]

References[edit]

  1. ^ Kappraff, Jay (2002). Beyond measure: a guided tour through nature, myth, and number. World Scientific. p. 258. ISBN 978-981-02-4702-7. 
  2. ^ γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  3. ^ Coxeter, Harold Scott Macdonald (1973). Regular polytopes. Courier Dover Publications. ISBN 978-0-486-61480-9. 
  4. ^ Weisstein, Eric W., "Star Polygon", MathWorld.
  5. ^ H. S. M. Coxeter, M. S. Longuet-Higgins, J. C. P. Miller, Uniform polyhedra, Phil. Trans. 1954 (Tables 6-8)
  • Cromwell, P.; Polyhedra, CUP, Hbk. 1997, ISBN 0-521-66432-2. Pbk. (1999), ISBN 0-521-66405-5. p.175
  • Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
  • 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, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

External links[edit]