Spirolateral

Simple spirolaterals
3_90° (4 cycles)	3_108° (5 cycles)	9_90° ccw spiral	9_90° (4 cycles)
100_120° spiral		100_120° (4 cycles)

In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,n which repeat until the figure closes. The number of repeats needed is called its cycles.^[1] A simple spirolateral has only positive angles. A simple spiral approximates of a portion of an archimedean spiral. A general spirolateral allows positive and negative angles.

A spirolateral which completes in one turn is a simple polygon, while requiring more than 1 turn is a star polygon and must be self-crossing.^[2] A simple spirolateral can be an equangular simple polygon <p> with p vertices, or an equiangular star polygon <p/q> with p vertices and q turns.

Spirolaterals were invented and named by Frank C. Odds as a teenager in 1962, as square spirolaterals with 90° angles, drawn on graph paper. In 1970, Odds discovered triangular and hexagonal spirolateral, with 60° and 120° angles, can be drawn on isometric^[3] (triangular) graph paper.^[4] Odds wrote to Martin Gardner who encouraged him to publish the results in Mathematics Teacher^[5] in 1973.^[3]

The process can be represented in turtle graphics, alternating turn angle and move forward instructions, but limiting the turn to a fixed rational angle.^[2]

The smallest golygon is a spirolateral, 8_90°^1,5, made with 8 right angles, and lengths 1 and 5 follow concave turns. Golygons are different in that they must close with a single sequence 1,2,3,..n, while a spirolateral will repeat that sequence until it closes.

Classifications[]

Varied cases
Simple 6_90°, 2 cycle, 3 turn	Regular unexpected closed spirolateral, 8_90°^1,5	Unexpectedly closed spirolateral 7_90°⁴	Crossed rectangle (1,2,-1,-2)_60°
Crossed hexagon (1,1,2,-1,-1,-2)_90°	(-1.2.4.3.2)_60°	(2…4)_90°	(2,1,-2,3,-4,3)_120°

A simple spirolateral has turns all the same direction.^[2] It is denoted by n_θ, where n is the number of sequential integer edge lengths and θ is the internal angle, as any rational divisor of 360°. Sequential edge lengths can be expressed explicitly as (1,2,...,n)_θ.

Note: The angle θ can be confusing because it represents the internal angle, while the supplementary turn angle can make more sense. These two angles are the same for 90°.

This defines an equiangular polygon of the form <kp/kq>, where angle θ = 180(1−2q/p), with k = n/d, and d = gcd(n,p). If d = n, the pattern never closes. Otherwise it has kp vertices and kq density. The cyclic symmetry of a simple spirolateral is p/d-fold.

A regular polygon, {p} is a special case of a spirolateral, 1_{180(1−2/p)°}. A regular star polygon, {p/q}, is a special case of a spirolateral, 1_{180(1−2q/p)°}. An isogonal polygon, is a special case spirolateral, 2_{180(1−2/p)°} or 2_{180(1−2q/p)°}.

A general spirolateral can turn left or right.^[2] It is denoted by n_θ^a₁,...,a_k, where a_i are indices with negative or concave angles.^[6] For example, 2_60°² is a crossed rectangle with ±60° internal angles, bending left or right.

An unexpected closed spiralateral returns to the first vertex on a single cycle. Only general spirolaterals may not close. A golygon is a regular unexpected closed spiralateral that closes from the expected direction. An irregular unexpected closed spiralateral is one that returns to the first point but from the wrong direction. For example 7_90°⁴. It takes 4 cycles to return to the start in the correct direction.^[2]

A modern spirolateral, also called a loop-de-loops^[7] by Educator Anna Weltman, is denoted by (i₁,...,i_n)_θ, allowing any sequence of integers as the edge lengths, i₁ to i_n.^[8] For example, (2,3,4)_90° has edge lengths 2,3,4 repeating. Opposite direction turns can be given a negative integer edge length. For example, a crossed rectangle can be given as (1,2,−1,−2)_θ.

An open spirolateral never closes. A simple spirolateral, n_θ, never closes if nθ is a multiple of 360°, gcd(p,n) = p. A general spirolateral can also be open if half of the angles are positive, half negative.

A (partial) infinite simple spirolateral, 4_90°

Closure[]

The number of cycles it takes to close a spirolateral, n_θ, with k opposite turns, p/q=360/(180-θ) can be computed. Reduce fraction (p-2q)(n-2k)/2p = a/b. The figure repeats after b cycles, and complete a total turns. If b=1, the figure never closes.^[1]

Explicitly, the number of cycles is 2p/d, where d=gcd((p-2q)(n-2k),2p). If d=2p, it closes on 1 cycle or never.

The number of cycles can be seen as the rotational symmetry order of the spirolateral.

n_90°

1_90°, 4 cycle, 1 turn
2_90°, 2 cycle, 1 turn
3_90°, 4 cycle, 3 turn
4_90°, never closes
5_90°, 4 cycle, 5 turn
6_90°, 2 cycle, 3 turn
7_90°, 4 cycle, 6 turns
8_90°, never closes
9_90°, 4 cycle, 9 turn
10_90°, 2 cycle, 5 turn

n_60°

1_60°, 3 cycle, 1 turn
2_60°, 3 cycle, 2 turn
3_60°, never closes
4_60°, 3 cycle, 4 turn
5_60°, 3 cycle, 5 turn
6_60°, never closes
7_60°, 3 cycle, 7 turn
8_60°, 3 cycle, 8 turn
9_60°, never closes
10_60°, 3 cycle, 10 turn

Small simple spirolaterals[]

Spirolaterals can be constructed from any rational divisor of 360°. The first table's columns sample angles from small regular polygons and second table from star polygons, with examples up to n = 6.

An equiangular polygon <p/q> has p vertices and q density. <np/nq> can be reduced by d = gcd(n,p).

Small whole divisor angles

Simple spirolaterals (whole divisors p) n_θ or (1,2,...,n)_θ
θ	60°	90°	108°	120°	128 4/7°	135°	140°	144°	147 3/11°	150°
180-θ Turn angle	120°	90°	72°	60°	51 3/7°	45°	40°	36°	32 8/11°	30°
n_θ \ p	3	4	5	6	7	8	9	10	11	12
1_θ Regular {p}	1_60° {3}	1_90° {4}	1_108° {5}	1_120° {6}	1_128.57° {7}	1_135° {8}	1_140° {9}	1_144° {10}	1_147.27° {11}	1_150° {12}
2_θ Isogonal <2p/2>	2_60° <6/2>	2_90° <8/2> → <4>	2_108° <10/2>	2_120° <12/2> → <6>	2_128.57° <14/2>	2_135° <16/2> → <8>	2_140° <18/2>	2_144° <20/2> → <10>	2_147° <22/2>	2_150° <24/2> → <12>
3_θ 2-isogonal <3p/3>	3_60° open	3_90° <12/3>	3_108° <15/3>	3_120° <18/3> → <6>	3_128.57° <21/3>	3_135° <24/3>	3_140° <27/3> → <9>	3_144° <30/3>	3_147° <33/3>	3_150° <36/3> → <12>
4_θ 3-isogonal <4p/4>	4_60° <12/4>	4_90° open	4_108° <20/4>	4_120° <24/4> → <12/2>	4_128.57° <28/4>	4_135° <32/4> → <8>	4_140° <36/4>	4_144° <40/4> → <20/2>	4_147° <44/4>	4_150° <48/4> → <12>
5_θ 4-isogonal <5p/5>	5_60° <15/5>	5_90° <20/5>	5_108° open	5_120° <30/5>	5_128.57° <35/5>	5_135° <40/5>	5_140° <45/5>	5_144° <50/5> → <10>	5_147° <55/5>	5_150° <60/5>
6_θ 5-isogonal <6p/6>	6_60° Open	6_90° <24/6> → <12/3>	6_108° <30/6>	6_120° Open	6_128.57° <42/6>	6_135° <48/6> → <24/3>	6_140° <54/6> → <18/2>	6_144° <60/6> → <30/3>	6_147° <66/6>	6_150° <72/6> → <12>

Small rational divisor angles

Simple spirolaterals (rational divisors p/q) n_θ or (1,2,...,n)_θ
θ	15°	16 4/11°	20°	25 5/7°	30°	36°	45°	49 1/11°	72°	77 1/7°	81 9/11°	100°	114 6/11°
180-θ Turn angle	165°	163 7/11°	160°	154 2/7°	150°	144°	135°	130 10/11°	108°	102 6/7°	98 2/11°	80°	65 5/11°
n_θ \ p/q	24/11	11/5	9/4	7/3	12/5	5/2	8/3	11/4	10/3	7/2	11/3	9/2	11/2
1_θ Regular {p/q}	1_15° {24/11}	1_16.36° {11/5}	1_20° {9/4}	1_25.71° {7/3}	1_30° {12/5}	1_36° {5/2}	1_45° {8/3}	1_49.10° {11/4}	1_72° {10/3}	1_77.14° {7/2}	1_81.82° {11/3}	1_100° {9/2}	1_114.55° {11/2}
2_θ Isogonal <2p/2q>	2_15° <48/22> → <24/11>	2_16.36° <22/10>	2_20° <18/8>	2_25.71° <14/6>	2_30° <24/10> → <12/5>	2_36° <10/4>	2_45° <16/6> → <8/3>	2_49.10° <22/8>	2_72° <20/6> → <10/3>	2_77.14° <14/4>	2_81.82° <22/6>	2_100° <18/4>	2_114.55° <22/4>
3_θ 2-isogonal <3p/3q>	3_15° <72/33> → <24/11>	3_16.36° <33/15>	3_20° <27/12> → <9/4>	3_25.71° <21/9>	3_30° <36/15> → <12/5>	3_36° <15/6>	3_45° <24/9>	3_49.10° <33/12>	3_72° <30/9>	3_77.14° <21/6>	3_81.82° <33/9>	3_100° <27/6> → <9/2>	3_114.55° <33/6>
4_θ 3-isogonal <4p/4q>	4_15° <96/44> → <24/11>	4_16.36° <44/20>	4_20° <36/12>	4_25.71° <28/4>	4_30° <48/40> → <12/5>	4_36° <20/8>	4_45° <32/12> → <8/3>	4_49.10° <44/16>	4_72° <40/12> → <20/6>	4_77.14° <28/8>	4_81.82° <44/12>	4_100° <36/8>	4_114.55° <44/8>
5_θ 4-isogonal <5p/5q>	5_15° <120/55>	5_16.36° <55/25>	5_20° <45/20>	5_25.71° <35/15>	5_30° <60/25>	5_36° open	5_45° <40/15>	5_49.10° <55/20>	5_72° <50/15> → <10/3>	5_77.14° <35/10>	5_81.82° <55/15>	5_100° <45/10>	5_114.55° <55/10>
6_θ 5-isogonal <6p/6q>	6_15° <144/66> → <24/11>	6_16.36° <66/30>	6_20° <54/24> → <18/8>	6_25.71° <42/18>	6_30° <72/30> → <12/5>	6_36° <30/12>	6_45° <48/18> → <24/9>	6_49.10° <66/24>	6_72° <60/18> → <30/9>	6_77.14° <42/12>	6_81.82° <66/18>	6_100° <54/12> → <18/4>	6_114.55° <66/12>

References[]

^ ^a ^b Gardner, M. Worm Paths Ch. 17 Knotted Doughnuts and Other Mathematical Entertainments New York: W. H. Freeman, pp. 205-221, 1986. [1]
^ ^a ^b ^c ^d ^e Abelson, Harold, diSessa, Andera, 1980, Turtle Geometry, MIT Press, pp.37-39, 120-122
^ ^a ^b Focus on...Spirolaterals Secondary Magazine Issue 78
^ [2] Frank Odds, British biochemist, 8/29/1945-7/7/2020
^ Odds, Frank C. Spirolaterals, Mathematics Teacher, Feb 1973, Volume 66: Issue 2, pp. 121–124 DOI
^ Weisstein, Eric W. "Spirolateral". MathWorld.
^ Anna Weltman, This is Not a Math Book A Graphic Activity Book, Kane Miller; Act Csm edition, 2017
^ "Practice Multiplication with Simple Spirolateral Math Art". 23 July 2015.

Alice Kaseberg Schwandt Spirolaterals: An advanced Investignation from an Elementary Standpoint, Mathematical Teacher, Vol 72, 1979, 166-169 [3]
Margaret Kenney and Stanley Bezuszka, Square Spirolaterals Mathematics Teaching, Vol 95, 1981, pp.22-27 [4]
Gascoigne, Serafim Turtle Fun LOGO for the Spectrum 48K pp 42-46 | Spirolaterals 1985
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry London: Penguin, pp. 239-241, 1991.
Krawczyk, Robert, "Hilbert's Building Blocks", Mathematics & Design, The University of the Basque Country, pp. 281-288, 1998.
Krawczyk, Robert, Spirolaterals, Complexity from Simplicity, International Society of Arts, Mathematics and Architecture 99,The University of the Basque Country, pp. 293-299, 1999. [5]
Krawczyk, Robert J. The Art of Spirolateral reversals [6]

External links[]

Spirolaterals Javascript App

[gardner-1] Gardner, M. Worm Paths Ch. 17 Knotted Doughnuts and Other Mathematical Entertainments New York: W. H. Freeman, pp. 205-221, 1986. [1]

[turtle-2] Abelson, Harold, diSessa, Andera, 1980, Turtle Geometry, MIT Press, pp.37-39, 120-122

[focus-3] Focus on...Spirolaterals Secondary Magazine Issue 78

[4] [2] Frank Odds, British biochemist, 8/29/1945-7/7/2020

[5] Odds, Frank C. Spirolaterals, Mathematics Teacher, Feb 1973, Volume 66: Issue 2, pp. 121–124 DOI

[6] Weisstein, Eric W. "Spirolateral". MathWorld.

[7] Anna Weltman, This is Not a Math Book A Graphic Activity Book, Kane Miller; Act Csm edition, 2017

[8] "Practice Multiplication with Simple Spirolateral Math Art". 23 July 2015.

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