Equiangular polygon

An equiangular polygon can be constructed from a regular polygon or regular star polygon where edges are extended as infinite lines. 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 to negative 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 into supplementary 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 convex direct equiangular hexagon, ⟨6⟩, is bounded by 6 lines with 60° angle between. Each line can be moved perpendicular to its direction.

This concave indirect equiangular hexagon, ⟨6-2⟩, is also bounded by 6 lines with 90° angle between, each line moved independently, moving vertices as new intersections.

For a convex equiangular p-gon, each internal angle is 180(1−2/p)°; this is the equiangular polygon theorem.

For a direct equiangular p/q star polygon, density q, each internal angle is 180(1−2q/p)°, with 1 < 2q < p. For w = gcd(p,q) > 1, this represents a w-wound ⁠p/w/q/w⁠ star polygon, which is degenerate for the regular case.

A concave indirect equiangular (p_r+p_l)-gon, with p_r right turn vertices and p_l left turn vertices, will have internal angles of 180(1−2/|p_r−p_l|))°, regardless of their sequence. An indirect star equiangular (p_r+p_l)-gon, with p_r right turn vertices and p_l left turn vertices and q total turns, will have internal angles of 180(1−2q/|p_r−p_l|))°, 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.

Every direct equiangular p-gon can be given a notation ⟨p⟩ or ⟨p/q⟩, like regular polygons {p} and regular star polygons {p/q}, containing p vertices, and stars having density q.

Convex equiangular p-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 equiangular p-gon can be given the notation ⟨p−2c⟩, with c 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 equilateral p-gon can be given the notation ⟨^p−2c/_q⟩ with c counter turn vertices, and q total turns. An equiangular polygon <p−p> is a p-gon with undefined internal angles θ, but can be expressed explicitly as ⟨p−p⟩_θ.

Viviani's theorem holds for equiangular polygons:^[2]

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.

A cyclic 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 if n is odd, a cyclic polygon is equiangular if and only if it is regular.^[3]

For prime p, every integer-sided equiangular p-gon is regular. Moreover, every integer-sided equiangular p^k-gon has p-fold rotational symmetry.^[4]

An ordered set of side lengths ${\displaystyle (a_{1},\dots ,a_{n})}$ gives rise to an equiangular n-gon if and only if either of two equivalent conditions holds for the polynomial ${\displaystyle a_{1}+a_{2}x+\cdots +a_{n-1}x^{n-2}+a_{n}x^{n-1}:}$ it equals zero at the complex value ${\displaystyle e^{2\pi i/n};}$ it is divisible by ${\displaystyle x^{2}-2x\cos(2\pi /n)+1.}$^[5]

Direct equiangular polygons can be regular, isogonal, or lower symmetries. Examples for <p/q> are grouped into sections by p and subgrouped by density q.

Equiangular triangles must be convex and have 60° internal angles. It is an equilateral triangle and a regular triangle, ⟨3⟩={3}. The only degree of freedom is edge-length.

• 
  Regular, {3}, r₆

Direct equiangular quadrilaterals have 90° internal angles. The only equiangular quadrilaterals are rectangles, ⟨4⟩, and squares, {4}.

An equiangular quadrilateral with integer side lengths may be tiled by unit squares.^[6]

• 
  Regular, {4}, r₈
• 
  Spirolateral 2_90°, p₄

Direct equiangular pentagons, ⟨5⟩ and ⟨5/2⟩, have 108° and 36° internal angles respectively.

108° internal angle from an equiangular pentagon, ⟨5⟩

Equiangular pentagons can be regular, have bilateral symmetry, or no symmetry.

• 
  Regular, r₁₀
• 
  Bilateral symmetry, i₂
• 
  No symmetry, a₁

36° internal angles from an equiangular pentagram, ⟨5/2⟩

• 
  Regular pentagram, r₁₀
• 
  Irregular, d₂

Direct equiangular hexagons, ⟨6⟩ and ⟨6/2⟩, have 120° and 60° internal angles respectively.

120° internal angles of an equiangular hexagon, ⟨6⟩

An equiangular hexagon with integer side lengths may be tiled by unit equilateral triangles.^[6]

• 
  Regular, {6}, r₁₂
• 
  Spirolateral (1,2)_120°, p₆
• 
  Spirolateral (1…3)_120°, g₂
• 
  Spirolateral (1,2,2)_120°, i₄
• 
  Spirolateral (1,2,2,2,1,3)_120°, p₂

60° internal angles of an equiangular double-wound triangle, ⟨6/2⟩

• 
  Regular, degenerate, r₆
• 
  Spirolateral (1,3)_60°, p₆
• 
  Spirolateral (1,2)_60°, p₆
• 
  Spirolateral (2,3)_60°, p₆
• 
  Spirolateral (1,2,3,4,3,2)_60°, p₂

Direct equiangular heptagons, ⟨7⟩, ⟨7/2⟩, and ⟨7/3⟩ have 128 4/7°, 77 1/7° and 25 5/7° internal angles respectively.

128.57° internal angles of an equiangular heptagon, ⟨7⟩

• 
  Regular, {7}, r₁₄
• 
  Irregular, i₂

77.14° internal angles of an equiangular heptagram, ⟨7/2⟩

• 
  Regular, r₁₄
• 
  Irregular, i₂

25.71° internal angles of an equiangular heptagram, ⟨7/3⟩

• 
  Regular, r₁₄
• 
  Irregular, i₂

Direct equiangular octagons, ⟨8⟩, ⟨8/2⟩ and ⟨8/3⟩, have 135°, 90° and 45° internal angles respectively.

135° internal angles from an equiangular octagon, ⟨8⟩

• 
  Regular, r₁₆
• 
  Spirolateral (1,2)_135°, p₈
• 
  Spirolateral (1…4)_135°, g₂
• 
  Unequal truncated square, p₂

90° internal angles from an equiangular double-wound square, ⟨8/2⟩

• 
  Regular degenerate, r₈
• 
  Spirolateral (1,2,2,3,3,2,2,1)_90°, d₂
• 
  Spirolateral (2,1,3,2,2,3,1,2)_90°, d₂

45° internal angles from an equiangular octagram, ⟨8/3⟩

• 
  Regular, r₁₆
• 
  Isogonal, p₈
• 
  Isogonal, p₈
• 
  Spirolateral, (1,2)_45°, p₈
• 
  Isogonal, p₈
• 
  Spirolateral (1…4)_45°, g₂

Direct equiangular enneagons, ⟨9⟩, ⟨9/2⟩, ⟨9/3⟩, and ⟨9/4⟩ have 140°, 100°, 60° and 20° internal angles respectively.

140° internal angles from an equiangular enneagon ⟨9⟩

• 
  Regular, r₁₈
• 
  Spirolateral (1,1,3)_140°, i₆

100° internal angles from an equiangular enneagram, ⟨9/2⟩

• 
  Regular {9/2}, p₉
• 
  Spirolateral (1,1,5)_100°, i₆
• 
  Spirolateral 3_100°, g₃

60° internal angles from an equiangular triple-wound triangle, ⟨9/3⟩

• 
  Regular, degenerate, r₆
• 
  Irregular, a₁
• 
  Irregular, a₁
• 
  Irregular, a₁

20° internal angles from an equiangular enneagram, ⟨9/4⟩

• 
  Regular {9/4}, r₁₈
• 
  Spirolateral 3_20°, g₃
• 
  Irregular, i₂

Direct equiangular decagons, ⟨10⟩, ⟨10/2⟩, ⟨10/3⟩, ⟨10/4⟩, have 144°, 108°, 72° and 36° internal angles respectively.

144° internal angles from an equiangular decagon ⟨10⟩

• 
  Regular, r₂₀
• 
  Spirolateral (1,2)_144°, p₁₀
• 
  Spirolateral (1…5)_144°, g₂

108° internal angles from an equiangular double-wound pentagon ⟨10/2⟩

• 
  Regular, degenerate
• 
  Spirolateral (1,2)_108°, p₁₀
• 
  Irregular, p₂

72° internal angles from an equiangular decagram ⟨10/3⟩

• 
  Regular {10/3}, r₂₀
• 
  Isogonal, p₁₀
• 
  Spirolateral (1,2)_72°, p₁₀
• 
  Irregular, i₄
• 
  Spirolateral (1…5)_72°, g₂

36° internal angles from an equiangular double-wound pentagram ⟨10/4⟩

• 
  Regular, degenerate, r₁₀
• 
  Spirolateral (1,2)_36°, p₁₀
• 
  Isogonal, p₁₀
• 
  Isogonal, p₁₀
• 
  Irregular, p₂
• 
  Irregular, p₂
• 
  Irregular, p₂

Direct equiangular hendecagons, ⟨11⟩, ⟨11/2⟩, ⟨11/3⟩, ⟨11/4⟩, and ⟨11/5⟩ have 147 3/11°, 114 6/11°, 81 9/11°, 49 1/11°, and 16 4/11° internal angles respectively.

147° internal angles from an equiangular hendecagon, ⟨11⟩

• 
  Regular, {11}, r₂₂

114° internal angles from an equiangular hendecagram, ⟨11/2⟩

• 
  Regular {11/2}, r₂₂

81° internal angles from an equiangular hendecagram, ⟨11/3⟩

• 
  Regular {11/3}, r₂₂

49° internal angles from an equiangular hendecagram, ⟨11/4⟩

• 
  Regular {11/4}, r₂₂

16° internal angles from an equiangular hendecagram, ⟨11/5⟩

• 
  Regular {11/5}, r₂₂

Direct equiangular dodecagons, ⟨12⟩, ⟨12/2⟩, ⟨12/3⟩, ⟨12/4⟩, and ⟨12/5⟩ have 150°, 120°, 90°, 60°, and 30° internal angles respectively.

150° internal angles from an equiangular dodecagon, ⟨12⟩

Convex solutions with integer edge lengths may be tiled by pattern blocks, squares, equilateral triangles, and 30° rhombi.^[6]

• 
  Regular, {12}, r₂₄
• 
  Isogonal, p₁₂
• 
  Spirolateral (1,2)_150°, p₁₂
• 
  Spirolateral (1…3)_150°, g₄
• 
  Spirolateral (1…4)_150°, g₃
• 
  Spirolateral (1…6)_150°, g₂

120° internal angles from an equiangular double-wound hexagon, ⟨12/2⟩

• 
  Regular degenerate, r₁₂
• 
  Spirolateral, (1…4)_120°, g₃
• 
  Irregular, d₂
• 
  Irregular, d₂

90° internal angles from an equiangular triple-wound square, ⟨12/3⟩

• 
  Regular, degenerate, r₈
• 
  Spirolateral (1…3)_90°, g₂
• 
  Spirolateral (2…4)_90°, g₄
• 
  Spirolateral (1,1,3)_90°, i₈
• 
  Spirolateral (1,2,2)_90°, i₈
• 
  Spirolateral (1…6)_90°, g₂
• 
  Irregular, a₁

60° internal angles from an equiangular quadruple-wound triangle, ⟨12/4⟩

• 
  Regular, degenerate, r₆
• 
  Spirolateral (1,3,5,1)_60°, p₆
• 
  Spirolateral (1…4)_60°, g₃
• 
  Irregular, a₁

30° internal angles from an equiangular dodecagram, ⟨12/5⟩

• 
  Regular {12/5}, r₂₄
• 
  Isogonal, p₁₂
• 
  Spirolateral (1,2)_30°, p₁₂
• 
  Spirolateral (1…3)_30°, g₄
• 
  Spirolateral (1…4)_30°, g₃
• 
  Spirolateral (1…6)_30°, g₂

Direct equiangular tetradecagons, ⟨14⟩, ⟨14/2⟩, ⟨14/3⟩, ⟨14/4⟩, and ⟨14/5⟩, ⟨14/6⟩, have 154 2/7°, 128 4/7°, 102 6/7°, 77 1/7°, 51 3/7° and 25 5/7° internal angles respectively.

154.28° internal angles from an equiangular tetradecagon, ⟨14⟩

• 
  Regular {14}, r₂₈
• 
  Isogonal, t{7}, p₁₄

128.57° internal angles from an equiangular double-wound regular heptagon, ⟨14/2⟩

• 
  Regular degenerate, r₁₄
• 
  Isogonal, t{7/2}, p₁₄
• 
  Spirolateral 2_128.57°

102.85° internal angles from an equiangular tetradecagram, ⟨14/3⟩

• 
  Regular {14/3}, r₂₈
• 
  Isogonal t{7/3}, p₁₄

77.14° internal angles from an equiangular double-wound heptagram ⟨14/4⟩

• 
  Regular degenerate, r₁₄
• 
  Isogonal, p₁₄
• 
  Isogonal, p₁₄
• 
  Spirolateral 2_77.14°

51.43° internal angles from an equiangular tetradecagram, ⟨14/5⟩

• 
  Regular {14/5}, r₂₈
• 
  Isogonal, p₁₄
• 
  Isogonal, p₁₄

25.71° internal angles from an equiangular double-wound heptagram, ⟨14/6⟩

• 
  Regular degenerate, r₁₄
• 
  Isogonal, p₁₄
• 
  Isogonal, p₁₄
• 
  Isogonal, p₁₄
• 
  Irregular, d₂

Direct equiangular pentadecagons, ⟨15⟩, ⟨15/2⟩, ⟨15/3⟩, ⟨15/4⟩, ⟨15/5⟩, ⟨15/6⟩, and ⟨15/7⟩, have 156°, 132°, 108°, 84°, 60° and 12° internal angles respectively.

156° internal angles from an equiangular pentadecagon, ⟨15⟩

• 
  Regular, {15}, r₃₀

132° internal angles from an equiangular pentadecagram, ⟨15/2⟩

• 
  Regular, {15/2}, r₃₀

108° internal angles from an equiangular triple-wound pentagon, ⟨15/3⟩

• 
  Regular, degenerate, r₁₀
• 
  spirolateral (1…3)_108°, g₅

84° internal angles from an equiangular pentadecagram, ⟨15/4⟩

• 
  Regular, {15/4}, r₃₀

60° internal angles from an equiangular 5-wound triangle, ⟨15/5⟩

• 
  Regular, degenerate, r₆
• 
  Irregular, a₁

36° internal angles from an equiangular triple-wound pentagram, ⟨15/6⟩

• 
  Regular, degenerate, r₁₀
• 
  Irregular, a₁
• 
  Spirolateral (1…4)_36°, g₅

12° internal angles from an equiangular pentadecagram, ⟨15/7⟩

• 
  Regular, {15/7}, r₃₀

Direct equiangular hexadecagons, ⟨16⟩, ⟨16/2⟩, ⟨16/3⟩, ⟨16/4⟩, ⟨16/5⟩, ⟨16/6⟩, and ⟨16/7⟩, have 157.5°, 135°, 112.5°, 90°, 67.5° 45° and 22.5° internal angles respectively.

157.5° internal angles from an equiangular hexadecagon, ⟨16⟩

• 
  Regular, {16}, r₃₂
• 
  Isogonal, t{8}, p₁₆
• 
  Spirolateral (1…4)_157.5°, g₄

135° internal angles from an equiangular double-wound octagon, ⟨16/2⟩

• 
  Regular, degenerate, r₁₆
• 
  Irregular, p₁₆

112.5° internal angles from an equiangular hexadecagram, ⟨16/3⟩

• 
  Regular, {16/3}, r₃₂

90° internal angles from an equiangular 4-wound square, ⟨16/4⟩

• 
  Regular, degenerate, r₈
• 
  Irregular, a₁

67.5° internal angles from an equiangular hexadecagram, ⟨16/5⟩

• 
  Regular, {16/5}, r₃₂

45° internal angles from an equiangular double-wound regular octagram, ⟨16/6⟩

• 
  Regular, degenerate, r₁₆
• 
  spirolateral (1…3)_45°, g₈

22.5° internal angles from an equiangular hexadecagram, ⟨16/7⟩

• 
  Regular, {16/7}, r₃₂
• 
  Isogonal, p₁₆

Direct equiangular octadecagons, <18}, ⟨18/2⟩, ⟨18/3⟩, ⟨18/4⟩, ⟨18/5⟩, ⟨18/6⟩, ⟨18/7⟩, and ⟨18/8⟩, have 160°, 140°, 120°, 100°, 80°, 60°, 40° and 20° internal angles respectively.

160° internal angles from an equiangular octadecagon, ⟨18⟩

• 
  Regular, {18}, r₃₆
• 
  Isogonal, t{9}, p₁₈

140° internal angles from an equiangular double-wound enneagon, ⟨18/2⟩

• 
  Regular, degenerate
• 
  Spirolateral 2_140°, p₁₈

120° internal angles of an equiangular 3-wound hexagon ⟨18/3⟩

• 
  Regular, degenerate, r₁₈
• 
  irregular, a₁

100° internal angles of an equiangular double-wound enneagram ⟨18/4⟩

• 
  Regular, degenerate, r₁₈
• 
  Spirolateral 2_100°, g₃

80° internal angles of an equiangular octadecagram {18/5}

• 
  Regular, {18/5}, r₃₆

60° internal angles of an equiangular 6-wound triangle ⟨18/6⟩

• 
  Regular, degenerate, r₆
• 
  irregular, a₁

40° internal angles of an equiangular octadecagram ⟨18/7⟩

• 
  Regular, {18/7}, r₃₆
• 
  Isogonal, p₁₈
• 
  Isogonal, p₁₈
• 
  Isogonal, p₁₈

20° internal angles of an equiangular double-wound enneagram ⟨18/8⟩

• 
  Regular, degenerate, r₁₈
• 
  Isogonal, p₁₈
• 
  Isogonal, p₁₈
• 
  Isogonal, p₁₈
• 
  Isogonal, p₁₈
• 
  Spirolateral 2_20°, p₁₈
• 
  Spirolateral 6_20°, g₃

Direct equiangular icosagon, ⟨20⟩, ⟨20/3⟩, ⟨20/4⟩, ⟨20/5⟩, ⟨20/6⟩, ⟨20/7⟩, and ⟨20/9⟩, have 162°, 126°, 108°, 90°, 72°, 54° and 18° internal angles respectively.

162° internal angles from an equiangular icosagon, ⟨20⟩

• 
  Regular, {20}, r₄₀
• 
  Spirolateral (1,3)_162°, p₂₀

144° internal angles from an equiangular double-wound decagon, ⟨20/2⟩

• 
  Regular, degenerate, r₂₀
• 
  Spirolateral (1…4)_144°, g₅

126° internal angles from an equiangular icosagram, ⟨20/3⟩

• 
  Regular {20/3}, p₄₀
• 
  Spirolateral (1,3)_126°, p₂₀

108° internal angles from an equiangular 4-wound pentagon, ⟨20/4⟩

• 
  Regular degenerate, r₁₀
• 
  Spirolateral (1…4)_108°, g₅
• 
  Irregular, a₁

90° internal angles from an equiangular 5-wound square, ⟨20/5⟩

• 
  Regular degenerate, r₈
• 
  Spirolateral (1…5)_90°, g₄
• 
  Spirolateral (1,2,3,2,1)_90°, i₈

72° internal angles from an equiangular double-wound decagram, ⟨20/6⟩

• 
  Regular degenerate, r₂₀
• 
  Spirolateral (1,2)_72°, p₁₀
• 
  Spirolateral (1…4)_72°, g₅

54° internal angles from an equiangular icosagram, ⟨20/7⟩

• 
  Regular {20/7}, r₄₀
• 
  Isogonal, p₂₀
• 
  Isogonal, p₂₀
• 
  Isogonal, p₂₀

36° internal angles from an equiangular quadruple-wound pentagram, ⟨20/8⟩

• 
  Regular degenerate, r₁₀
• 
  Spirolateral (1…4)_36°, g₅
• 
  irregular, a₁

18° internal angles from an equiangular icosagram, ⟨20/9⟩

• 
  Regular {20/9}, r₄₀
• 
  Isogonal, p₂₀
• 
  Isogonal, p₂₀
• 
  Isogonal, p₂₀
• 
  Isogonal, p₂₀

• Spirolateral