Introduction to Spirolaterals

Spirolaterals were first encountered while investigating space curves and fractals in Abelson.  What was intriguing about them was the simple procedure to generate them and the great variety that could result from modifying a small set of parameters.  As with researching space curves and fractals the focus was to develop computer-based methods to investigate these two-dimensional forms and begin to suggest three-dimensional versions of them.  Quickly discovering that spirolaterals can generate an infinite number of variations, the focus changed to investigate the rules of generating them that would result in visually interesting designs. 

Further research uncovered what seems to be the first description of this geometrical form by Frank C. Odds, a British biochemist.  A spirolateral is created by drawing a set of lines; the first at a unit length, then each additional line increasing by one unit length while turning a constant direction.  Figure 1 shows the systematic generation of an order 3 spirolateral; one that consists of 3 segments at turns of 90 degrees

Figure 1. Generation of an order 3 spirolateral

Step 1: turn 90 degrees, draw one unit segment, turn 90 degrees, draw two unit segment, turn 90 degrees, draw three unit segment Step 2, 3 and 4: repeat Step 1.

In this example by repeating the initial three-segment construction four times, the spirolateral closes on itself. 

Figure 2. A “square spiral”

Odds writes that name spirolateral is derived from two roots: lateral, referring to a flat surface, and spiro, since the original series of spirolaterals was generated from the “square spiral” as shown in Figure 2.  In this drawing, each segment is one unit longer than the one drawn before it, and each segment turns a constant 90 degrees from its predecessor.  To complete a spirolateral, it is necessary to only to repeat the “square spiral” design, until the starting point is reached.


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