Spirolateral

class=wikitable align=right width=360 |+ Varied cases |120px Simple 690°, 2 cycle, 3 turn |120px Regular unexpected closed spirolateral, 890°1,5 |120px Unexpectedly closed spirolateral 790°4 |120px Crossed rectangle (1,2,-1,-2)60°

120pxCrossed hexagon (1,1,2,-1,-1,-2)90° |120px (-1.2.4.3.2)60° |120px (2...4)90° |120px (2,1,-2,3,-4,3)120°

A simple spirolateral has turns all the same direction.Abelson, Harold, diSessa, Andera, 1980, Turtle Geometry, MIT Press, pp.37-39, 120-122 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. It is denoted by n_θ^a₁,...,a_k, where a_i are indices with negative or concave angles.

{{mathworld|title=Spirolateral|urlname=Spirolateral}} For example, 2_60°² is a crossed rectangle with ±60° internal angles, bending left or right.

An unexpected closed spirolateral returns to the first vertex on a single cycle. Only general spirolaterals may not close. A golygon is a regular unexpected closed spirolateral that closes from the expected direction. An irregular unexpected closed spirolateral 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.

A modern spirolateral, also called a [https://recipesforpi.wordpress.com/2015/09/28/loop-de-loops-and-a-contest/ loop-de-loops]Anna Weltman, This is Not a Math Book A Graphic Activity Book, Kane Miller; Act Csm edition, 2017 by Educator Anna Weltman, is denoted by (i₁,...,i_n)_θ, allowing any sequence of integers as the edge lengths, i₁ to i_n.{{Cite web|url=https://www.whatdowedoallday.com/simple-spirolateral-math-art-for-kids|title = Practice Multiplication with Simple Spirolateral Math Art|date = 23 July 2015}} 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.

: File:Spirolateral 4 90.svg

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The number of cycles it takes to close a spirolateral, n_θ, with k opposite turns can be computed like so. Define p and q such that p/q=360/(180-θ). if the fraction (p-2q)(n-2k)/2p is reduced fully to a/b, then the figure repeats after b cycles, and complete a total turns. If b=1, the figure never closes.

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°

Spirolateral 1 90-fill.svg|1_90°, 4 cycle, 1 turn

Spirolateral 2 90-fill.svg|2_90°, 2 cycle, 1 turn

Spirolateral 3 90-fill.svg|3_90°, 4 cycle, 3 turn

Spirolateral 4 90b.svg|4_90°, never closes

Spirolateral 5 90-fill.svg|5_90°, 4 cycle, 5 turn

Spirolateral 6 90-fill.svg|6_90°, 2 cycle, 3 turn

Spirolateral 7 90.svg|7_90°, 4 cycle, 6 turns

Spirolateral 8 90.svg|8_90°, never closes

Spirolateral 9 90-fill.svg|9_90°, 4 cycle, 9 turn

Spirolateral 10 90-fill.svg|10_90°, 2 cycle, 5 turn

;n_60°:

Spirolateral 1 60-fill.svg|1_60°, 3 cycle, 1 turn

Spirolateral 2 60-fill.svg|2_60°, 3 cycle, 2 turn

Spirolateral 3 60.svg|3_60°, never closes

Spirolateral 4 60-fill.svg|4_60°, 3 cycle, 4 turn

Spirolateral 5 60-fill.svg|5_60°, 3 cycle, 5 turn

Spirolateral 6 60.svg|6_60°, never closes

Spirolateral 7 60-fill.svg|7_60°, 3 cycle, 7 turn

Spirolateral 8 60-fill.svg|8_60°, 3 cycle, 8 turn

Spirolateral 9 60.svg|9_60°, never closes

Spirolateral 10 60-fill.svg|10_60°, 3 cycle, 10 turn

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

class=wikitable |+ Simple spirolaterals (whole divisors p) nθ 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°
nθ \ p||3||4||5||6||7||8||9||10||11||12
align=center valign=bottom !valign=top|1θ Regular {p} |100px 160° {3} |100px 190° {4} |100px 1108° {5} |100px 1120° {6} |100px 1128.57° {7} |100px 1135° {8} |100px 1140° {9} |100px 1144° {10} |100px 1147.27° {11} |100px 1150° {12}
align=center valign=bottom !valign=top|2θ Isogonal <2p/2> |100px 260° <6/2> |BGCOLOR="#fff0f0"|100px 290° <8/2> → <4> |100px 2108° <10/2> |BGCOLOR="#fff0f0"|100px 2120° <12/2> → <6> |100px 2128.57° <14/2> |BGCOLOR="#fff0f0"|100px 2135° <16/2> → <8> |100px 2140° <18/2> |BGCOLOR="#fff0f0"|100px 2144° <20/2> → <10> |100px 2147° <22/2> |BGCOLOR="#fff0f0"|100px 2150° <24/2> → <12>
align=center valign=bottom !valign=top|3θ 2-isogonal <3p/3> |BGCOLOR="#c0c0c0"|100px 360° open |100px 390° <12/3> |100px 3108° <15/3> |BGCOLOR="#f0fff0"|80px 3120° <18/3> → <6> |100px 3128.57° <21/3> |100px 3135° <24/3> |BGCOLOR="#f0fff0"|100px 3140° <27/3> → <9> |100px 3144° <30/3> |100px 3147° <33/3> |BGCOLOR="#f0fff0"|100px 3150° <36/3> → <12>
align=center valign=bottom !valign=top|4θ 3-isogonal <4p/4> |100px 460° <12/4> |BGCOLOR="#c0c0c0"|80px 490° open |100px 4108° <20/4> |BGCOLOR="#fff0f0"|100px 4120° <24/4> → <12/2> |100px 4128.57° <28/4> |BGCOLOR="#ffc0c0"|100px 4135° <32/4> → <8> |100px 4140° <36/4> |BGCOLOR="#fff0f0"|100px 4144° <40/4> → <20/2> |100px 4147° <44/4> |BGCOLOR="#ffc0c0"|100px 4150° <48/4> → <12>
align=center valign=bottom !valign=top|5θ 4-isogonal <5p/5> |100px 560° <15/5> |100px 590° <20/5> |BGCOLOR="#c0c0c0"|80px 5108° open |100px 5120° <30/5> |100px 5128.57° <35/5> |100px 5135° <40/5> |100px 5140° <45/5> |BGCOLOR="#e0e0ff"|80px 5144° <50/5> → <10> |100px 5147° <55/5> |100px 5150° <60/5>
align=center valign=bottom !valign=top|6θ 5-isogonal <6p/6> |BGCOLOR="#c0c0c0"|100px 660° Open |BGCOLOR="#fff0f0"|80px 690° <24/6> → <12/3> |100px 6108° <30/6> |BGCOLOR="#c0c0c0"|70px 6120° Open |100px 6128.57° <42/6> |BGCOLOR="#fff0f0"|100px 6135° <48/6> → <24/3> |BGCOLOR="#f0fff0"|100px 6140° <54/6> → <18/2> |BGCOLOR="#fff0f0"|80px 6144° <60/6> → <30/3> |100px 6147° <66/6> |BGCOLOR="#ffffc0"|80px 6150° <72/6> → <12>

; Small rational divisor angles

class=wikitable |+ Simple spirolaterals (rational divisors p/q) nθ 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°
nθ \ 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
align=center valign=bottom !valign=top|1θ Regular {p/q} |100px 115° {24/11} |100px 116.36° {11/5} |100px 120° {9/4} |100px 125.71° {7/3} |100px 130° {12/5} |100px 136° {5/2} |100px 145° {8/3} |100px 149.10° {11/4} |100px 172° {10/3} |100px 177.14° {7/2} |100px 181.82° {11/3} |100px 1100° {9/2} |100px 1114.55° {11/2}
align=center valign=bottom !valign=top|2θ Isogonal <2p/2q> |BGCOLOR="#fff0f0"|100px 215° <48/22> → <24/11> |100px 216.36° <22/10> |100px 220° <18/8> |100px 225.71° <14/6> |BGCOLOR="#fff0f0"|100px 230° <24/10> → <12/5> |100px 236° <10/4> |BGCOLOR="#fff0f0"|100px 245° <16/6> → <8/3> |100px 249.10° <22/8> |BGCOLOR="#fff0f0"|100px 272° <20/6> → <10/3> |100px 277.14° <14/4> |100px 281.82° <22/6> |100px 2100° <18/4> |100px 2114.55° <22/4>
align=center valign=bottom !valign=top|3θ 2-isogonal <3p/3q> |BGCOLOR="#f0fff0"|100px 315° <72/33> → <24/11> |100px 316.36° <33/15> |BGCOLOR="#f0fff0"|100px 320° <27/12> → <9/4> |100px 325.71° <21/9> |BGCOLOR="#f0fff0"|100px 330° <36/15> → <12/5> |100px 336° <15/6> |100px 345° <24/9> |100px 349.10° <33/12> |100px 372° <30/9> |100px 377.14° <21/6> |100px 381.82° <33/9> |BGCOLOR="#f0fff0"|100px 3100° <27/6> → <9/2> |100px 3114.55° <33/6>
align=center valign=bottom !valign=top|4θ 3-isogonal <4p/4q> |BGCOLOR="#ffc0c0"|100px 415° <96/44> → <24/11> |100px 416.36° <44/20> |100px 420° <36/12> |100px 425.71° <28/4> |BGCOLOR="#ffc0c0"|100px 430° <48/40> → <12/5> |100px 436° <20/8> |BGCOLOR="#ffc0c0"|75px 445° <32/12> → <8/3> |100px 449.10° <44/16> |BGCOLOR="#fff0f0"|100px 472° <40/12> → <20/6> |100px 477.14° <28/8> |100px 481.82° <44/12> |100px 4100° <36/8> |100px 4114.55° <44/8>
align=center valign=bottom !valign=top|5θ 4-isogonal <5p/5q> |100px 515° <120/55> |100px 516.36° <55/25> |100px 520° <45/20> |100px 525.71° <35/15> |100px 530° <60/25> |BGCOLOR="#c0c0c0"|100px 536° open |100px 545° <40/15> |100px 549.10° <55/20> |BGCOLOR="#e0e0ff"|100px 572° <50/15> → <10/3> |100px 577.14° <35/10> |100px 581.82° <55/15> |100px 5100° <45/10> |100px 5114.55° <55/10>
align=center valign=bottom !valign=top|6θ 5-isogonal <6p/6q> |BGCOLOR="#ffffc0"|100px 615° <144/66> → <24/11> |100px 616.36° <66/30> |BGCOLOR="#f0fff0"|100px 620° <54/24> → <18/8> |100px 625.71° <42/18> |BGCOLOR="#ffffc0"|110px 630° <72/30> → <12/5> |100px 636° <30/12> |BGCOLOR="#fff0f0"|100px 645° <48/18> → <24/9> |100px 649.10° <66/24> |BGCOLOR="#fff0f0"|100px 672° <60/18> → <30/9> |100px 677.14° <42/12> |100px 681.82° <66/18> |BGCOLOR="#f0fff0"|100px 6100° <54/12> → <18/4> |100px 6114.55° <66/12>

{{Commons category}}

• Turtle graphics represent a computer language that defines an open or close path as move lengths and turn angles.

{{reflist}}

• Alice Kaseberg Schwandt Spirolaterals: An advanced Investignation from an Elementary Standpoint, Mathematical Teacher, Vol 72, 1979, 166-169 [https://www.jstor.org/stable/27961580?seq=1]
• Margaret Kenney and Stanley Bezuszka, Square Spirolaterals Mathematics Teaching, Vol 95, 1981, pp. 22–27 [https://www.atm.org.uk/Mathematics-Teaching-Journal-Archive/28719]
• Gascoigne, Serafim [https://link.springer.com/chapter/10.1007%2F978-1-349-08240-7_8 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. [https://mypages.iit.edu/~krawczyk/isama99.pdf]
• Krawczyk, Robert J. The Art of Spirolateral reversals [https://www.researchgate.net/publication/2588913_The_Art_Of_Spirolateral_Reversals/link/552f0c040cf2d495071aa80b/download]

• [http://thewessens.net/ClassroomApps/Main/spirolaterals.html Spirolaterals] Javascript App

Category:Types of polygons