Star polygon
Non-convex polygon with intersecting edges, often star-shaped.
A star polygon is a non-convex polygon. While regular star polygons have been thoroughly studied, star polygons in general lack a formal definition, though certain notable examples can be created by truncating regular simple or star polygons. Branko Grünbaum noted that Johannes Kepler used the term in two main ways: one for regular star polygons whose intersecting edges don’t create new vertices, and another for isotoxal concave simple polygons. Polygrams cover shapes like the pentagram as well as compound figures such as the hexagram.
- first systematic study
- Thomas Bradwardine
- later studied by
- Johannes Kepler
- Schläfli symbol
- {p/q}
- symmetry group
- dihedral group Dp, of order 2p
- turning number
- q (density)
- prefix example
- penta- with suffix -gram gives pentagram
- alternative name
- polygram
Lore & Background
Regular star polygons were first studied systematically by Thomas Bradwardine, and later Johannes Kepler. A regular star polygon is denoted by its Schläfli symbol {p/q}, where p (the number of vertices) and q (the density) are relatively prime and q ≥ 2. The density can also be called its turning number: the sum of the turn angles of all the vertices, divided by 360°. The symmetry group of {p/q} is the dihedral group Dp, of order 2p, independent of q.
Reader's Guide
Star polygons are significant in geometry as a class of non-convex polygons that extend the concept of regular polygons. Their study, initiated by Thomas Bradwardine and later advanced by Johannes Kepler, has influenced both mathematical theory and artistic design. Regular star polygons are constructed by connecting every qth point out of p points regularly spaced on a circle, producing self-intersecting, equilateral, and equiangular shapes. The notation {p/q} captures both the number of vertices and the density, or turning number. When p and q are not coprime, degenerate polygons with coinciding vertices and edges result, such as {6/2} appearing as a triangle but representing a double-winding unicursal hexagon. Star polygons appear in tilings, as in Kepler's Harmonice Mundi, and in art and culture, including the pentagram, heptagrams, and octagrams. The interior of a star polygon can be treated in multiple ways, leading to different area calculations. Branko Grünbaum and Geoffrey Shephard considered two treatments: as regular star n-gons and as isotoxal concave simple 2n-gons.
Did You Know?
- The name star polygon reflects the resemblance of these shapes to the diffraction spikes of real stars.
- If q ≥ p/2, the construction of {p/q} results in the same polygon as {p/(p − q)}, but vertices are reached in the opposite direction.
- A nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin.
- The tetrahemihexahedron can be seen as a 'crossed triangle' {3/2} cuploid.
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