Spiral drawing with gears: how ring, wheel and pen hole shape the pattern

What curve does a gear drawing make?

Wolfram MathWorld defines a hypotrochoid as a curve traced by a point attached to a circle rolling around the inside of a fixed circle, where the point sits some distance from the centre of the rolling circle. That is exactly the set up of a ring, a wheel and a pen hole.

MacTutor, the history of mathematics archive at the University of St Andrews, explains that there are four closely related curves, the epicycloid, epitrochoid, hypocycloid and hypotrochoid, all traced by a point on a circle rolling round a fixed circle. For the hypotrochoid the circle rolls on the inside. MathWorld describes the other three like this:

• Hypocycloid: traced by a fixed point on the edge of a small circle rolling around the inside of a large circle, so it is a hypotrochoid with the point on the rim.
• Epitrochoid: traced by a point attached to a circle rolling around the outside of a fixed circle.
• Epicycloid: traced by a point on the edge of a circle rolling on the outside of another circle, so it is an epitrochoid with the point on the rim.

MacTutor notes that these curves were studied by la Hire, Desargues, Leibniz, Newton and many others. MathWorld lists some special cases of the hypotrochoid, including the ellipse and the rose curve, so a single rolling set up can give shapes that look nothing like each other.

Why does a spiral drawing have that many points?

Each time the wheel rolls along the ring by its own number of teeth, the pen makes one loop or point. The pattern closes when the pen gets back to exactly where it started, which happens when the wheel has rolled a whole number of turns and has also travelled a whole number of laps round the ring.

With gear teeth, you can work this out with simple arithmetic. Take the smallest number that both tooth counts divide into evenly (the lowest common multiple). Divide it by the wheel’s teeth to get the number of points, and by the ring’s teeth to get how many times the wheel goes round the ring.

For example, with a 96 tooth ring and a 36 tooth wheel:

• The lowest common multiple of 96 and 36 is 288.
• 288 divided by 36 is 8, so the pattern has 8 points.
• 288 divided by 96 is 3, so the wheel goes round the ring 3 times before the pattern closes.

A quicker way to see it is to write the wheel over the ring as a fraction and simplify it: 36/96 is 3/8. The bottom number is the number of points and the top is the number of laps. If the fraction cannot be simplified much, the pattern takes many laps and ends up very dense.

How does the pen hole change the shape?

The distance from the pen to the wheel’s centre is the third setting. In MacTutor’s description it is the distance of the point from the centre of the rolling circle. With the pen right on the rim, the curve becomes a hypocycloid, with sharp points where the pen touches the ring. Moving the pen in towards the centre generally gives softer, rounder loops. Same gears, different hole, different drawing.

Tips for new designs

• Change one thing at a time. Keep the ring and wheel, and try every pen hole. Then change only the wheel.
• Layer patterns. Draw one pattern, then another on top with a different wheel or colour.
• Use a fraction that will not simplify for a lacy, dense design, and a simple one, like 1/4 or 1/3, for a bold flower.
• Mix ink colours between layers so overlapping loops stand out.

Questions people ask

Where Gear Drawing helps, and where it does not

Gear Drawing is our Android app for spiral drawing with virtual gears. Pick a ring, a wheel and a pen hole and watch the wheel roll round the ring as the pattern draws itself. It shows how many loops and turns each pattern has, so you can check the fraction rule above. Two layers are free, each with its own colour, and you can save sharp 2048 px pictures. Saved drawings stay editable, and it works offline with no account. A one-time Pro unlock adds up to 12 layers, more palettes, any colour, Gradient and Wobble ink and 4096 px saves. There is no subscription.

What it is good for: trying lots of gear and pen combinations quickly, with no slipping gears or pen running dry.

What it will not do: it draws on a screen, so if you want ink on paper you will need to print your picture or use a physical gear set. It shows loop and turn counts but does not explain the maths as you go; that is what this guide is for.

Questions or bugs: Email us.

Sources

1. University of St Andrews, HypotrochoidMacTutor History of Mathematics · mathshistory.st-andrews.ac.uk
2. HypotrochoidWolfram MathWorld · mathworld.wolfram.com
3. HypocycloidWolfram MathWorld · mathworld.wolfram.com
4. EpitrochoidWolfram MathWorld · mathworld.wolfram.com
5. EpicycloidWolfram MathWorld · mathworld.wolfram.com