Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, a spirolateral is a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

created by a sequence of fixed vertex

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

s and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M. ''Worm Paths'' Ch. 17 ''Knotted Doughnuts and Other Mathematical Entertainments'' New York: W. H. Freeman, pp. 205-221, 1986

/ref> A ''simple spirolateral'' has only positive angles. A simple spiral approximates of a portion of an

archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a ...

. 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 In geometry, a star polygon is a type of non- convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operatio ...

and must be self-crossing. 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 Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves ...

. In 1970, Odds discovered ''triangular and hexagonal spirolateral'', with 60° and 120° angles, can be drawn on isometric (triangular) graph paper. Odds wrote to

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...

who encouraged him to publish the results in '' Mathematics Teacher'' in 1973.Focus on...Spirolaterals
Secondary Magazine Issue 78 The process can be represented in

turtle graphics In computer graphics, turtle graphics are vector graphics using a relative cursor (the "turtle") upon a Cartesian plane (x and y axis). Turtle graphics is a key feature of the Logo programming language. Overview The turtle has three attribut ...

, alternating turn angle and move forward instructions, but limiting the turn to a fixed rational angle. The smallest

golygon A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a ...

is a spirolateral, 7_90°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

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

, as any

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

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 In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

can make more sense. These two angles are the same for 90°. This defines an

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 w ...

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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

. The cyclic symmetry of a simple spirolateral is ''p''/''d''-fold. A

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

, is a

special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case ...

of a spirolateral, 1_{180(1−2/''p'')°}. A

regular star polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

, , is a

of a spirolateral, 1_{180(1−2''q''/''p'')°}. An

isogonal polygon In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fac ...

, is a special case spirolateral, 2_{180(1−2/''p'')°} or 2_{180(1−2''q''/''p'')°}. A general spirolateral can turn left or right. It is denoted by ''n''_θ^{''a''₁,...,''a''_''k''}, where ''a''_''i'' are indices with negative or concave angles. For example, 2_60°² is a

crossed rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...

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

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. A modern spirolateral, also called
loop-de-loops
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''. 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. : 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''-2''q'')(''n''-2''k'')/2''p'' = ''a''/''b''. The figure repeats after ''b'' cycles, and complete ''a'' total turns. If ''b''=1, the figure never closes. 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° 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

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

<''p''/''q''> has ''p'' vertices and ''q'' density. <''np''/''nq''> can be reduced by ''d'' = gcd(''n'',''p''). ; Small whole divisor angles ; Small rational divisor angles

References

{{reflist * Alice Kaseberg Schwandt ''Spirolaterals: An advanced Investignation from an Elementary Standpoint'', Mathematical Teacher, Vol 72, 1979, 166-16

* Margaret Kenney and Stanley Bezuszka, ''Square Spirolaterals'' Mathematics Teaching, Vol 95, 1981, pp.22-2

* Gascoigne, Serafi
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

* Krawczyk, Robert J. ''The Art of Spirolateral reversals'

External links

Spirolaterals
Javascript App Types of polygons

Classifications

Closure

Small simple spirolaterals

See also

References

External links