The maths of the spirograph… with the drawings

Well, I sat down and thought this was going to be easy, but it has taken me three hours to work the maths of a smaller inner wheel rolling around inside a large outer wheel: mainly because for the first two of those I neglected the basic insight that the inner wheel rolls in the opposite direction to its direction of travel (think of it this way – as a car wheel moves forward the point at the top of the wheel moves backwards – relative to the centre of the wheel).

And instead of using MetaPost I resorted to a spread sheet – though I might do a MetaPost drawing still.

Anyway – assume you have a big wheel of unit radius and a small wheel inside it of radius \frac{1}{R} .

At any given time the centre of this small wheel will be at cartesian co-ordinates (assuming the big wheel is centred on (0,0):

(cos(\theta)(1 -\frac{1}{R}) , sin(\theta)(1 -\frac{1}{R})) (1)

where \theta is the angle of rotation of the small wheel relative to the centre of the big wheel.

But if the small wheel has moved through angle \theta relative to the centre of the big wheel, then it will have itself rotated through the angle R\theta – in the opposite direction to its rotation around the centre of the big wheel.

This means a fixed point on the surface of the small wheel will now be, compared to the centre of the smaller circle, at cartesian co-ordinates:

(\frac{cos(\theta - R\theta)}{R}, \frac{sin(\theta - R\theta)}{R}) (2)

And we add (1) and (2) together to get the co-ordinates relative to the origin (ie the centre of the bigger circle).

Looking at the above it should be relatively obvious that if R is an integer then the pattern will represent R cusps – and not much less obvious is the fact that if \frac{1}{R} can be expressed as a rational number then the pattern will repeat. But if \frac{1}{R} cannot be expressed as a rational then it turns out there are a countably infinite number i.e., \aleph_0 , number of cusps. In a way this is just a graphical way of representing an irrational number – it is a number that cannot be made to divide up unity (the circle) into equal proportions.

So here are the pretty pictures:

Let R = 2 and we have a degenerate case

2 cusp figure from 'spirograph'Then the 3 cusps of the ‘deltoid’:

Three cusped deltoidThe four cusped astroid:

Four cusped astroidAnd, here is the spreadsheet:
spirograph spreadsheet



  1. Thanks kindly for working this out. I have the dread suspicion that if I’d worked it out on my own I wouldn’t get the interior wheel orientation correct for an embarrassingly long time.

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