Equiangular Polygon

In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths.

For clarity, a planar equiangular polygon can be called ''direct'' or ''indirect''. A direct equiangular polygon has all angles turning in the same direction in a plane and can include multiple turns. Convex equiangular polygons are always direct. An indirect equiangular polygon can include angles turning right or left in any combination. A skew equiangular polygon may be isogonal, but can't be considered direct since it is nonplanar.

A spirolateral ''n''_θ is a special case of an ''equiangular polygon'' with a set of ''n'' integer edge lengths repeating sequence until returning to the start, with vertex internal angles θ.

Construction

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. 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.

Equiangular polygon theorem

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(1-2''q''/''p'')°, with 1<2''q''<''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(1-2''q''/, ''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.

Notation

Every direct equiangular ''p''-gon can be given a notation <''p''> or <''p''/''q''>, like regular polygons and regular star polygons , 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(1-2''q''/''p'')°.

A concave indirect equiangular ''p''-gon can be given the notation <''p''-2''c''>, 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''-2''c''/_''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.

Other properties

Viviani's theorem holds for equiangular polygons:Elias Abboud "On Viviani’s Theorem and its Extensions"
pp. 2, 11
:''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.

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

An ordered set of side lengths $(a_1, \dots , a_n)$ gives rise to an equiangular ''n''-gon if and only if either of two equivalent conditions holds for the polynomial $a_1+a_2x+\cdots + a_x^+a_nx^:$ it equals zero at the complex value $e^;$ it is divisible by $x^2-2x \cos (2\pi /n)+1.$M. Bras-Amorós, M. Pujol: "Side Lengths of Equiangular Polygons (as seen by a coding theorist)", ''The American Mathematical Monthly'', vol. 122, n. 5, pp. 476–478, May 2015. .

Direct equiangular polygons by sides

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

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

Regular polygon 3 annotated.svg, Regular, , r₆

Equiangular quadrilaterals

Direct equiangular quadrilaterals have 90° internal angles. The only equiangular quadrilaterals are rectangles, <4>, and squares, .

An equiangular quadrilateral with integer side lengths may be tiled by unit squares..

Regular polygon 4 annotated.svg, Regular, , r₈
Spirolateral_2_90.svg, Spirolateral 2_90°, p₄

Equiangular pentagons

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

Equiangular pentagon 03.svg, Regular, r₁₀
Equiangular pentagon 02.svg, Bilateral symmetry, i₂
Equiangular pentagon 01.svg, No symmetry, a₁

; 36° internal angles from an equiangular pentagram, <5/2>

File:Regular star polygon 5-2.svg, Regular pentagram, r₁₀
Equiangular_pentagram1.svg, Irregular, d₂

Equiangular hexagons

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.

Regular polygon 6 annotated.svg, Regular, , r₁₂
Spirolateral_2_120.svg, Spirolateral (1,2)_120°, p₆
Spirolateral_3_120.svg, Spirolateral (1…3)_120°, g₂
Spirolateral 1-2-2 120.svg, Spirolateral (1,2,2)_120°, i₄
Spirolateral 1-2-2-2-1-3 120.svg, Spirolateral (1,2,2,2,1,3)_120°, p₂

; 60° internal angles of an equiangular double-wound triangle, <6/2>:

Regular polygon 3 annotated.svg, Regular, degenerate, r₆
Spirolateral 1-3 60.svg, Spirolateral (1,3)_60°, p₆
Spirolateral_2_60.svg, Spirolateral (1,2)_60°, p₆
Spirolateral 2-3 60.svg, Spirolateral (2,3)_60°, p₆
Spirolateral_1-2-3-4-3-2_60.svg, Spirolateral (1,2,3,4,3,2)_60°, p₂

Equiangular heptagons

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 polygon 7 annotated.svg, Regular, , r₁₄
Equiangular heptagon.svg, Irregular, i₂

; 77.14° internal angles of an equiangular heptagram, <7/2>:

Regular star polygon 7-2.svg, Regular, r₁₄
Equiangular heptagram1.svg, Irregular, i₂

; 25.71° internal angles of an equiangular heptagram, <7/3>:

Regular star polygon 7-3.svg, Regular, r₁₄
Equiangular heptagram2.svg, Irregular, i₂

Equiangular octagons

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 polygon 8 annotated.svg, Regular, r₁₆
Spirolateral_2_135.svg, Spirolateral (1,2)_135°, p₈
Spirolateral_4_135.svg, Spirolateral (1…4)_135°, g₂
Equiangular_octagon.svg, Unequal truncated square, p₂

; 90° internal angles from an equiangular double-wound square, <8/2>:

Regular polygon 4 annotated.svg, Regular degenerate, r₈
Spirolateral 1-2-2-3-3-2-2-1 90.svg, Spirolateral (1,2,2,3,3,2,2,1)_90°, d₂
Spirolateral 2-1-3-2-2-3-1-2 90.svg, Spirolateral (2,1,3,2,2,3,1,2)_90°, d₂

; 45° internal angles from an equiangular octagram, <8/3>:

Regular star polygon 8-3.svg, Regular, r₁₆
Regular truncation 4 2.svg, Isogonal, p₈
Regular truncation 4 -4.svg, Isogonal, p₈
Spirolateral_2_45.svg, Spirolateral, (1,2)_45°, p₈
Regular truncation 4 -0.2.svg, Isogonal, p₈
Spirolateral_4_45.svg, Spirolateral (1…4)_45°, g₂

Equiangular enneagons

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 polygon 9 annotated.svg, Regular, r₁₈
Spirolateral 1-1-3 140.svg, Spirolateral (1,1,3)_140°, i₆

;100° internal angles from an equiangular enneagram, <9/2>:

Regular star polygon 9-2.svg, Regular , p₉
Spirolateral 1-1-5 140.svg, Spirolateral (1,1,5)_100°, i₆
File:Spirolateral 3 100.svg, Spirolateral 3_100°, g₃

;60° internal angles from an equiangular ''triple-wound triangle'', <9/3>:

Regular polygon 3 annotated.svg, Regular, degenerate, r₆
Equiangular_triple-triangle1.svg, Irregular, a₁
Equiangular_triple-triangle2.svg, Irregular, a₁
Equiangular_triple-triangle3.svg, Irregular, a₁

;20° internal angles from an equiangular enneagram, <9/4>:

Regular star polygon 9-4.svg, Regular , r₁₈
Spirolateral 3 20.svg, Spirolateral 3_20°, g₃
Equiangular enneagram2.svg, Irregular, i₂

Equiangular decagons

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 polygon 10 annotated.svg, Regular, r₂₀
Spirolateral 2 144.svg, Spirolateral (1,2)_144°, p₁₀
Spirolateral_5_144.svg, Spirolateral (1…5)_144°, g₂

;108° internal angles from an equiangular double-wound pentagon <10/2>

Regular polygon 5 annotated.svg, Regular, degenerate
Spirolateral 2 108.svg, Spirolateral (1,2)_108°, p₁₀
Equiangular_double-pentagon1.svg, Irregular, p₂

;72° internal angles from an equiangular decagram <10/3>

Regular star polygon 10-3.svg, Regular , r₂₀
Regular star truncation 5-3 3.svg, Isogonal, p₁₀
Spirolateral_2_72.svg, Spirolateral (1,2)_72°, p₁₀
Equiangular_double-pentagon2.svg, Irregular, i₄
Spirolateral 5 72.svg, Spirolateral (1…5)_72°, g₂

;36° internal angles from an equiangular double-wound pentagram <10/4>

Regular star polygon 5-2.svg, Regular, degenerate, r₁₀
Spirolateral 2 36.svg, Spirolateral (1,2)_36°, p₁₀
Regular polygon truncation 5 3.svg, Isogonal, p₁₀
Regular truncation 5 4.svg, Isogonal, p₁₀
Equiangular double-pentagram3.svg, Irregular, p₂
Equiangular_double-pentagon5.svg, Irregular, p₂
Equiangular_double-pentagram2.svg, Irregular, p₂

Equiangular hendecagons

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 polygon 11 annotated.svg, Regular, , r₂₂

;114° internal angles from an equiangular hendecagram, <11/2>:

Regular star polygon 11-2.svg, Regular , r₂₂

;81° internal angles from an equiangular hendecagram, <11/3>:

Regular star polygon 11-3.svg, Regular , r₂₂

;49° internal angles from an equiangular hendecagram, <11/4>:

Regular star polygon 11-4.svg, Regular , r₂₂

;16° internal angles from an equiangular hendecagram, <11/5>:

Regular star polygon 11-5.svg, Regular , r₂₂

Equiangular dodecagons

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.

Regular polygon 12 annotated.svg, Regular, , r₂₄
Regular truncation 6 0.45.svg, Isogonal, p₁₂
Spirolateral_2_150.svg, Spirolateral (1,2)_150°, p₁₂
Spirolateral_3_150.svg, Spirolateral (1…3)_150°, g₄
Spirolateral_4_150.svg, Spirolateral (1…4)_150°, g₃
Spirolateral_6_150.svg, Spirolateral (1…6)_150°, g₂

; 120° internal angles from an equiangular ''double-wound hexagon'', <12/2>

Regular polygon 6 annotated.svg, Regular degenerate, r₁₂
Spirolateral 4 120.svg, Spirolateral, (1…4)_120°, g₃
Equiangular double-hexagon3.svg, Irregular, d₂
Equiangular double-hexagon2.svg, Irregular, d₂

; 90° internal angles from an equiangular ''triple-wound square'', <12/3>

Regular polygon 4 annotated.svg, Regular, degenerate, r₈
Spirolateral_3_90.svg, Spirolateral (1…3)_90°, g₂
Spirolateral 2-3-4-90.svg, Spirolateral (2…4)_90°, g₄
Equiangular triple-square3.svg, Spirolateral (1,1,3)_90°, i₈
Spirolateral 1-2-2 90.svg, Spirolateral (1,2,2)_90°, i₈
Spirolateral_6_90.svg, Spirolateral (1…6)_90°, g₂
Equiangular_triple-square1.svg, Irregular, a₁

; 60° internal angles from an equiangular ''quadruple-wound triangle'', <12/4>

Regular polygon 3 annotated.svg, Regular, degenerate, r₆
Equiangular double-hexagon4.svg, Spirolateral (1,3,5,1)_60°, p₆
Spirolateral 4 60.svg, Spirolateral (1…4)_60°, g₃
Equiangular_quadruple-triangle1.svg, Irregular, a₁

; 30° internal angles from an equiangular '' dodecagram'', <12/5>

Regular star polygon 12-5.svg, Regular , r₂₄
Regular truncation 6 1.5.svg, Isogonal, p₁₂
Spirolateral_2_30.svg, Spirolateral (1,2)_30°, p₁₂
Spirolateral_3_30.svg, Spirolateral (1…3)_30°, g₄
Spirolateral_4_30.svg, Spirolateral (1…4)_30°, g₃
Spirolateral_6_30.svg, Spirolateral (1…6)_30°, g₂

Equiangular tetradecagons

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 polygon 14 annotated.svg, Regular , r₂₈
File:Regular truncation 7 0.1.svg, Isogonal, t, p₁₄

;128.57° internal angles from an equiangular double-wound regular heptagon, <14/2>:

Regular polygon 7 annotated.svg, Regular degenerate, r₁₄
regular_star_truncation_7-5_3.svg, Isogonal, t, p₁₄
Spirolateral_2_129.svg, Spirolateral 2_128.57°

;102.85° internal angles from an equiangular tetradecagram, <14/3>:

Regular star polygon 14-3.svg, Regular , r₂₈
regular_star_truncation_7-3_3.svg, Isogonal t, p₁₄

;77.14° internal angles from an equiangular double-wound heptagram <14/4>:

Regular star polygon 7-2.svg, Regular degenerate, r₁₄
regular_star_truncation_7-3_2.svg, Isogonal, p₁₄
regular_star_truncation_7-3_4.svg, Isogonal, p₁₄
File:Spirolateral 2 77.svg, Spirolateral 2_77.14°

;51.43° internal angles from an equiangular tetradecagram, <14/5>:

Regular star polygon 14-5.svg, Regular , r₂₈
regular_star_truncation_7-5_2.svg, Isogonal, p₁₄
regular_star_truncation_7-5_4.svg, Isogonal, p₁₄

;25.71° internal angles from an equiangular double-wound heptagram, <14/6>:

Regular star polygon 7-3.svg, Regular degenerate, r₁₄
Regular truncation 7 1000.svg, Isogonal, p₁₄
File:Regular truncation 7 4.svg, Isogonal, p₁₄
File:Regular truncation 7 -0.5.svg, Isogonal, p₁₄
Equiangular_star-14-6.svg, Irregular, d₂

Equiangular pentadecagons

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 polygon 15 annotated.svg, Regular, , r₃₀

;132° internal angles from an equiangular pentadecagram, <15/2>:

Regular star polygon 15-2.svg, Regular, , r₃₀

;108° internal angles from an equiangular triple-wound pentagon, <15/3>:

Regular polygon 5 annotated.svg, Regular, degenerate, r₁₀
Equiangular_triple-pentagon1.svg, spirolateral (1…3)_108°, g₅

;84° internal angles from an equiangular pentadecagram, <15/4>:

Regular star polygon 15-4.svg, Regular, , r₃₀

;60° internal angles from an equiangular 5-wound triangle, <15/5>:

Regular polygon 3 annotated.svg, Regular, degenerate, r₆
Equiangular 5-wound triangle1.svg, Irregular, a₁

;36° internal angles from an equiangular triple-wound pentagram, <15/6>:

Regular star polygon 5-2.svg, Regular, degenerate, r₁₀
Equiangular triple-pentagram2.svg, Irregular, a₁
File:Spirolateral 4 36.svg, Spirolateral (1…4)_36°, g₅

;12° internal angles from an equiangular pentadecagram, <15/7>:

Regular star polygon 15-7.svg, Regular, , r₃₀

Equiangular hexadecagons

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 polygon 16 annotated.svg, Regular, , r₃₂
File:Regular truncation 8 0.45.svg, Isogonal, t, p₁₆
File:Spirolateral 4 1575.svg, Spirolateral (1…4)_157.5°, g₄

;135° internal angles from an equiangular double-wound octagon, <16/2>:

Regular polygon 8 annotated.svg, Regular, degenerate, r₁₆
Equiangular double octagon1.svg, Irregular, p₁₆

;112.5° internal angles from an equiangular hexadecagram, <16/3>:

Regular star polygon 16-3.svg, Regular, , r₃₂

;90° internal angles from an equiangular 4-wound square, <16/4>:

Regular polygon 4 annotated.svg, Regular, degenerate, r₈
Equiangular 4-wound square1.svg, Irregular, a₁

;67.5° internal angles from an equiangular hexadecagram, <16/5>:

Regular star polygon 16-5.svg, Regular, , r₃₂

;45° internal angles from an equiangular double-wound regular octagram, <16/6>:

Regular star polygon 8-3.svg, Regular, degenerate, r₁₆
Spirolateral 3 45.svg, spirolateral (1…3)_45°, g₈

;22.5° internal angles from an equiangular hexadecagram, <16/7>:

Regular star polygon 16-7.svg, Regular, , r₃₂
File:Regular truncation 8 -10.svg, Isogonal, p₁₆

Equiangular octadecagons

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 polygon 18 annotated.svg, Regular, , r₃₆
File:Regular truncation 9 0.1.svg, Isogonal, t, p₁₈

;140° internal angles from an equiangular double-wound enneagon, <18/2>:

Regular polygon 9.svg, Regular, degenerate
Spirolateral 2 140.svg, Spirolateral 2_140°, p₁₈

; 120° internal angles of an equiangular 3-wound hexagon <18/3>:

Regular polygon 6.svg, Regular, degenerate, r₁₈
Equilateral triple-wound-hexagon1.svg, irregular, a₁

; 100° internal angles of an equiangular double-wound enneagram <18/4>:

Regular star polygon 9-2.svg, Regular, degenerate, r₁₈
File:Spirolateral_6_100.svg, Spirolateral 2_100°, g₃

; 80° internal angles of an equiangular octadecagram :

Regular star polygon 18-5.svg, Regular, , r₃₆

; 60° internal angles of an equiangular 6-wound triangle <18/6>:

Regular polygon 3.svg, Regular, degenerate, r₆
Equilateral 6-wound-triangle1.svg, irregular, a₁

; 40° internal angles of an equiangular octadecagram <18/7>:

Regular star polygon 18-7.svg, Regular, , r₃₆
regular_star_truncation_9-7_2.svg, Isogonal, p₁₈
regular_star_truncation_9-7_4.svg, Isogonal, p₁₈
regular_star_truncation_9-7_5.svg, Isogonal, p₁₈

; 20° internal angles of an equiangular double-wound enneagram <18/8>:

Regular star polygon 9-4.svg, Regular, degenerate, r₁₈
regular_polygon_truncation_9_5.svg, Isogonal, p₁₈
regular_polygon_truncation_9_4.svg, Isogonal, p₁₈
regular_polygon_truncation_9_3.svg, Isogonal, p₁₈
regular_polygon_truncation_9_2.svg, Isogonal, p₁₈
Spirolateral 2 20.svg, Spirolateral 2_20°, p₁₈
File:Spirolateral 6 20.svg, Spirolateral 6_20°, g₃

Equiangular icosagons

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 polygon 20 annotated.svg, Regular, , r₄₀
Spirolateral 1-3 162.svg, Spirolateral (1,3)_162°, p₂₀

;144° internal angles from an equiangular double-wound decagon, <20/2>:

Regular polygon 10 annotated.svg, Regular, degenerate, r₂₀
Spirolateral_4_144.svg, Spirolateral (1…4)_144°, g₅

;126° internal angles from an equiangular icosagram, <20/3>:

Regular star polygon 20-3.svg, Regular , p₄₀
Spirolateral_1-3_126.svg, Spirolateral (1,3)_126°, p₂₀

;108° internal angles from an equiangular 4-wound pentagon, <20/4>:

Regular polygon 5 annotated.svg, Regular degenerate, r₁₀
Spirolateral 4 108.svg, Spirolateral (1…4)_108°, g₅
Equiangular quadruple-pentagon1.svg, Irregular, a₁

;90° internal angles from an equiangular 5-wound square, <20/5>:

Regular polygon 4 annotated.svg, Regular degenerate, r₈
Spirolateral 5 90.svg, Spirolateral (1…5)_90°, g₄
Spirolateral 1-2-3-2-1 90.svg, Spirolateral (1,2,3,2,1)_90°, i₈

;72° internal angles from an equiangular double-wound decagram, <20/6>:

Regular star polygon 10-3.svg, Regular degenerate, r₂₀
Spirolateral 2 72.svg, Spirolateral (1,2)_72°, p₁₀
Spirolateral 4 72.svg, Spirolateral (1…4)_72°, g₅

;54° internal angles from an equiangular icosagram, <20/7>:

Regular star polygon 20-7.svg, Regular , r₄₀
regular_star_truncation_10-3_2.svg, Isogonal, p₂₀
regular_star_truncation_10-3_3.svg, Isogonal, p₂₀
regular_star_truncation_10-3_5.svg, Isogonal, p₂₀

;36° internal angles from an equiangular quadruple-wound pentagram, <20/8>:

Regular star polygon 5-2.svg, Regular degenerate, r₁₀
Spirolateral_4_36.svg, Spirolateral (1…4)_36°, g₅
Equiangular quadruple-pentagram1.svg, irregular, a₁

;18° internal angles from an equiangular icosagram, <20/9>:

Regular star polygon 20-9.svg, Regular , r₄₀
regular_polygon_truncation_10_5.svg, Isogonal, p₂₀
regular_polygon_truncation_10_4.svg, Isogonal, p₂₀
regular_polygon_truncation_10_3.svg, Isogonal, p₂₀
regular_polygon_truncation_10_2.svg, Isogonal, p₂₀

See also

* Spirolateral

References

*Williams, R. ''The Geometrical Foundation of Natural Structure: A Source Book of Design''. New York: Dover Publications, 1979. p. 32

External links

A Property of Equiangular Polygons: What Is It About?
a discussion of Viviani's theorem at Cut-the-knot.
*

{{polygons

Types of polygons