For clarity, aplanar equiangular polygon can be calleddirect orindirect. Adirect equiangular polygon has all angles turning in the same direction in a plane and can include multipleturns.Convex equiangular polygons are always direct. Anindirect equiangular polygon can include angles turning right or left in any combination. Askew equiangular polygon may beisogonal, but can't be considered direct since it is nonplanar.
Aspirolateralnθ is a special case of anequiangular polygon with a set ofn integer edge lengths repeating sequence until returning to the start, with vertex internal angles θ.
Anequiangular polygon can be constructed from aregular polygon orregular star polygon where edges are extended as infinitelines. Each edges can be independently moved perpendicular to the line's direction. Vertices represent the intersection point between pairs of neighboring line. Each moved line adjusts its edge-length and the lengths of its two neighboring edges.[1] If edges are reduced to zero length, the polygon becomes degenerate, or if reduced tonegative lengths, this will reverse the internal and external angles.
For an even-sided direct equiangular polygon, with internal angles θ°, moving alternate edges can invert all vertices intosupplementary angles, 180-θ°. Odd-sided direct equiangular polygons can only be partially inverted, leaving a mixture of supplementary angles.
Every equiangular polygon can be adjusted in proportions by this construction and still preserve equiangular status.
This convexdirect equiangular hexagon,⟨6⟩, is bounded by 6lines with 60° angle between. Each line can be moved perpendicular to its direction.
This concaveindirect equiangular hexagon,⟨6-2⟩, is also bounded by 6 lines with 90° angle between, each line moved independently, moving vertices as new intersections.
For aconvex equiangularp-gon, eachinternal angle is 180(1−2/p)°; this is theequiangular polygon theorem.
For a direct equiangularp/qstar polygon,densityq, each internal angle is 180(1−2q/p)°, with1 < 2q <p. Forw = gcd(p,q) > 1, this represents aw-woundp/w/q/w star polygon, which is degenerate for the regular case.
A concaveindirect equiangular(pr+pl)-gon, withpr right turn vertices andpl left turn vertices, will have internal angles of180(1−2/|pr−pl|))°, regardless of their sequence. Anindirect star equiangular(pr+pl)-gon, withpr right turn vertices andpl left turn vertices andq totalturns, will have internal angles of180(1−2q/|pr−pl|))°, regardless of their sequence. An equiangular polygon with the same number of right and left turns has zero total turns, and has no constraints on its angles.
Convex equiangularp-gons⟨p⟩ have internal angles 180(1−2/p)°, while direct star equiangular polygons,⟨p/q⟩, have internal angles 180(1−2q/p)°.
A concave indirect equiangularp-gon can be given the notation⟨p−2c⟩, withc counter-turn vertices. For example,⟨6−2⟩ is a hexagon with 90° internal angles of the difference,⟨4⟩, 1 counter-turned vertex. A multiturn indirect equilateralp-gon can be given the notation⟨p−2c/q⟩ withc counter turn vertices, andq totalturns. An equiangular polygon <p−p> is ap-gon with undefined internal anglesθ, but can be expressed explicitly as⟨p−p⟩θ.
The sum of distances from an interior point to the sides of an equiangular polygon does not depend on the location of the point, and is that polygon's invariant.
Acyclic polygon is equiangular if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal and sides 2, 4, ... are equal). Thus ifn is odd, a cyclic polygon is equiangular if and only if it is regular.[3]
For primep, every integer-sided equiangularp-gon is regular. Moreover, every integer-sided equiangularpk-gon hasp-foldrotational symmetry.[4]
An ordered set of side lengths gives rise to an equiangularn-gon if and only if either of two equivalent conditions holds for the polynomial it equals zero at the complex value it is divisible by[5]
Direct equiangular polygons can be regular, isogonal, or lower symmetries. Examples for <p/q> are grouped into sections byp and subgrouped by densityq.
Equiangular triangles must be convex and have 60° internal angles. It is anequilateral triangle and aregular triangle,⟨3⟩={3}. The only degree of freedom is edge-length.
^Abboud, Elias (2010). "Viviani's Theorem and Its Extension".The College Mathematics Journal.41 (3):203–211.doi:10.4169/074683410x488683.
^De Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons",Mathematical Gazette 95, March 2011, 102-107.
^McLean, K. Robin (2004). "A powerful algebraic tool for equiangular polygons".Mathematical Gazette (88):513–514.doi:10.1017/S002555720017617X.
^Bras-Amorós, M.; Pujol, M. (2015). "Side Lengths of Equiangular Polygons (as seen by a coding theorist)".The American Mathematical Monthly.122 (5):476–478.doi:10.4169/amer.math.monthly.122.5.476.
