Drawing Spirolaterals

This activity can be done with any young child who can count and draw a straight line.  However, the math behind it is quite advanced, for the curious student who wants to dig deeper.

The Idea

Learn to draw spirolaterals, or “square spirals,” to explore the intersection of pattern and numbers.  A spirolateral starts with a segment of length 1, then turns to create a segment of length 2, then a turn and length of 3, etc.     End results vary based on the length and number of segments.

spiro image

Spirolateral of 1,2,3,4,5,6,7,8,9.  The black dot is the starting point for the square spiral.

The Execution


graph paper
pencil & eraser

Read through this tutorial on How to Draw Spirolaterals.  Draw with pencil until you get the hang of it and start seeing the patterns.

Older children can start with regular graph paper.  Younger ones benefit from larger squares, such as this 1/2″ grid paper from Print Free Graph Paper.

The triangular spirolaterals are best for older children, or in pencil … it took me a few tries to get them right!

Some questions to explore:

Why do the spirolaterals based on multiples of 4 not work out? (Because you are making a square with 4 sides, and thus ending back where you started).

What other patterns do you see?

What happens if you skip numbers? (1, 3, 5 instead of 1, 2, 3, for example)

The Extension

Online software that can plot basic spirolaterals

Mathematical Definition:

From Wolfram: Spirolaterals as a subset of Mathematical Images

Inspiration for further exploration (from Robert Krawczyk, the spirolateral expert)

The Art of Spirolateral Reversals (pdf)

Spirolaterals, Complexity From Simplicity (pdf)

Curved Spirolaterals

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