Spirolateral

{{Short description|Geometric polygon}} {| class=wikitable align=right width=360 |+ Simple spirolaterals |- align=center |120px 390° (4 cycles) |120px 3108° (5 cycles) |120px 990° ccw spiral |120px 990° (4 cycles) |- align=center |colspan=2|240px 100120° spiral |colspan=2|240px 100120° (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.<ref name="gardner">Gardner, M. ''Worm Paths'' Ch. 17 ''Knotted Doughnuts and Other Mathematical Entertainments'' New York: W. H. Freeman, pp. 205-221, 1986. [https://books.google.com/books?id=jU4FEAAAQBAJ&dq=spirolateral+Worm+Paths&pg=PA221]</ref> 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.<ref name="turtle"/> 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<ref name=focus/> (triangular) graph paper.<ref>[http://www.bsmm.org/2020/07/24/obituary-for-professor-frank-odds] Frank Odds, British biochemist, 8/29/1945-7/7/2020</ref> Odds wrote to Martin Gardner who encouraged him to publish the results in '' Mathematics Teacher''<ref>Odds, Frank C. ''Spirolaterals'', Mathematics Teacher, Feb 1973, Volume 66: Issue 2, pp. 121–124 [https://doi.org/10.5951/MT.66.2.0121 DOI]</ref> in 1973.<ref name=focus>[https://web.archive.org/web/20110310190929/https://www.ncetm.org.uk/resources/30192 Focus on...Spirolaterals] Secondary Magazine Issue 78</ref>

The process can be represented in turtle graphics, alternating turn angle and move forward instructions, but limiting the turn to a fixed rational angle.<ref name="turtle"/>

The smallest golygon is a spirolateral, 790°4, made with 7 right angles, and length 4 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== {| 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.<ref name="turtle">Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, pp.37-39, 120-122</ref> 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(1−2''q''/''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, 1180(1−2/''p'')°. A regular star polygon, {''p''/''q''}, is a special case of a spirolateral, 1180(1−2''q''/''p'')°. An isogonal polygon, is a special case spirolateral, 2180(1−2/''p'')° or 2180(1−2''q''/''p'')°.

A '''general spirolateral''' can turn left or right.<ref name="turtle"/> It is denoted by '''''n''θ''a''1,...,''a''''k''''', where ''a''''i'' are indices with negative or concave angles.<ref> {{mathworld|title=Spirolateral|urlname=Spirolateral}}</ref> For example, 260°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 790°4. It takes 4 cycles to return to the start in the correct direction.<ref name="turtle"/>

A '''modern spirolateral''', also called a [https://recipesforpi.wordpress.com/2015/09/28/loop-de-loops-and-a-contest/ loop-de-loops]<ref>Anna Weltman, ''This is Not a Math Book A Graphic Activity Book'', Kane Miller; Act Csm edition, 2017</ref> by Educator Anna Weltman, is denoted by '''(''i''1,...,''i''''n'')θ''', allowing any sequence of integers as the edge lengths, ''i''1 to ''i''''n''.<ref>{{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}}</ref> 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 ''n''θ 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. : 320px|thumb|left|A (partial) infinite simple spirolateral, 490° {{clear}}

== Closure == 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''-2''q'')(''n''-2''k'')/2''p'' 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.<ref name=gardner/>

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

The number of cycles can be seen as the rotational symmetry order of the spirolateral. ;''n''90° <gallery> Spirolateral 1 90-fill.svg|190°, 4 cycle, 1 turn Spirolateral 2 90-fill.svg|290°, 2 cycle, 1 turn Spirolateral 3 90-fill.svg|390°, 4 cycle, 3 turn Spirolateral 4 90b.svg|490°, never closes Spirolateral 5 90-fill.svg|590°, 4 cycle, 5 turn Spirolateral 6 90-fill.svg|690°, 2 cycle, 3 turn Spirolateral 7 90.svg|790°, 4 cycle, 6 turns Spirolateral 8 90.svg|890°, never closes Spirolateral 9 90-fill.svg|990°, 4 cycle, 9 turn Spirolateral 10 90-fill.svg|1090°, 2 cycle, 5 turn </gallery> ;''n''60°: <gallery> Spirolateral 1 60-fill.svg|160°, 3 cycle, 1 turn Spirolateral 2 60-fill.svg|260°, 3 cycle, 2 turn Spirolateral 3 60.svg|360°, never closes Spirolateral 4 60-fill.svg|460°, 3 cycle, 4 turn Spirolateral 5 60-fill.svg|560°, 3 cycle, 5 turn Spirolateral 6 60.svg|660°, never closes Spirolateral 7 60-fill.svg|760°, 3 cycle, 7 turn Spirolateral 8 60-fill.svg|860°, 3 cycle, 8 turn Spirolateral 9 60.svg|960°, never closes Spirolateral 10 60-fill.svg|1060°, 3 cycle, 10 turn </gallery>

== 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 {| class=wikitable |+ 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 |- 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 <2''p''/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 <3''p''/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 <4''p''/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 <5''p''/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 <6''p''/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'') ''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 |- 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 <2''p''/2''q''> |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 <3''p''/3''q''> |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 <4''p''/4''q''> |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 <5''p''/5''q''> |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 <6''p''/6''q''> |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> |}

== See also == {{Commons category}} * Turtle graphics represent a computer language that defines an open or close path as move lengths and turn angles.

==References== {{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]

== External links == * [http://thewessens.net/ClassroomApps/Main/spirolaterals.html Spirolaterals] JavaScript App

Category:Types of polygons