Spirograph

In a spirograph, a wheel with $r$ teeth rolls tooth into tooth round a fixed ring with $R$ teeth, inside or outside it, and a pen in the wheel at distance $d = \rho r$ from its centre draws. The teeth on both gears are spaced alike, so in one unit their radii are $R$ and $r$. At $\rho = 1$ the pen is on the rim, and beyond 1 it sits on an arm reaching out of the wheel. Here you can count the teeth that decide a pattern, move the pen to see where sharp points and loops come from, and keep patterns as layers.

The faint line is the whole pattern the pen will draw. Colour runs through the pattern’s period, so a stretch drawn again keeps its colour.

The gears are back where they started when the same two teeth meet again. That takes a number of teeth that is a whole number of turns for both gears, and the first such number is their least common multiple. With $g = \gcd(R, r)$,

Each $r$ teeth rolled trace one petal, a stretch that need not be a closed loop, and each $R$ teeth make one lap round the fixed ring. With the pen at the centre the pattern is just a circle, complete after one lap. So the number of petals depends on the common factor, not on the wheel’s size: 96 with 36 teeth gives 8 petals, and 96 with 35 gives 96. The inset follows the meeting teeth, ring’s across and wheel’s up, and its ticks, $g$ apart, mark where petals start.

The wheel cannot slip, so at each instant the point touching the ring is at rest and the wheel turns about it. The pen therefore moves at right angles to the line from the contact point to the pen, at a speed proportional to that line’s length. Arms draws the line and the pen’s velocity. With the pen inside the rim the line never shrinks to nothing, so the pen never stops and the pattern has no sharp points. With the pen on the rim, the pen reaches the contact point and stops once a petal, which makes a sharp point, except in the straight-line case below. Beyond the rim, once a petal, the pen moves against the wheel’s travel for a stretch, and this usually opens each point into a loop.

Let θ be the angle the wheel’s centre has travelled round the ring. As a complex number, the pen is at

That is two arms end to end, each turning at a steady rate: one from the ring’s centre to the wheel’s, one of length $d$ from the wheel’s centre to the pen.

Some settings give named curves. With the pen on the rim, rolling inside draws a hypocycloid, the deltoid at a ratio of 3 : 1 and the astroid at 4 : 1, and rolling outside draws an epicycloid, the cardioid when the gears are equal. Rolling inside, a wheel half the ring turns its two arms equally fast in opposite directions, so a pen inside the rim draws an ellipse and a pen on the rim runs along a diameter, the Tusi couple.

The curves are called hypotrochoids and epitrochoids. The British engineer Denys Fisher developed the Spirograph toy in the 1960s.