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.

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

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

Small rational divisor angles

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

• 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]

• Spirolaterals Javascript App