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A non-convex regular polygon is a regular star polygon. The most common example is the pentagram , which has the same vertices as a pentagon , but connects alternating vertices. For an n -sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as { n / m }.
Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well.
A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if =, where r, s, k ≥ 0 and where the p i are distinct Pierpont primes greater than 3 (primes of the form +). [8]: Thm. 2 These polygons are exactly the regular polygons that can be constructed with Conic section, and the regular polygons ...
Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, with the pentagram, a non-convex regular polygon (star polygon), appearing as early as the 7th century B.C. on a krater by Aristophanes, found at Caere and now in the Capitoline Museum. [40] [41]
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons.
The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but ...
Any straight-sided digon is regular even though it is degenerate, because its two edges are the same length and its two angles are equal (both being zero degrees). As such, the regular digon is a constructible polygon. [3] Some definitions of a polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean ...
Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges. There are seven families of isogonal figures , each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the ...