The math behind string art

Straight lines that draw curves

Put nails in two rows that form a corner. Join the nail farthest from the corner on one row to the nearest one on the other, then repeat, moving one nail along each row. Every string is straight, but the edge they form together is a curve. In mathematics that curve is called an envelope: nobody draws it, it appears where the lines touch one another. Mary Everest Boole used it in the 19th century to teach geometry with thread and card, and it is the basis of all string art.

The skip

On a circle of nails, stringing “by counting” means always going to the nail that is the same number of places away: every 5th, every 9th. That number is the skip. With a fixed skip the string goes around until it is back at the nail it started from, and what it leaves is a star.

Why prime numbers

Whether the string passes through every nail depends on whether the number of nails and the skip share a divisor. With 24 nails and a skip of 6, both divide by 6: the string touches 24 ÷ 6 = 4 nails, draws a square and stops. With 24 nails and a skip of 5 they share no divisor, and the string passes through all 24.

The rule is simple: the string touches as many nails as the number of nails divided by the greatest common divisor of the nails and the skip. A prime number shares no divisors with any skip, so with 23 nails every skip works. That is why the first design of the basic course is the Prime Number Circle.

The circle in the center

The strings of a layer do not reach the center: they go around it and leave an empty circle. The shorter the skip, the larger that circle. You can work it out before stringing:

radius of the inner circle = radius of the nail circle × cosine of (180° × skip ÷ nails)

With 36 nails on a circle of 27 cm radius:

That is how a mandala is built in layers: each color, with a shorter skip, ends up as a ring outside the one before. It is what happens in the Sun Disc.

How far apart the nails go

There are two measurements for spacing nails on a circle. The angle between one nail and the next is 360° divided by the number of nails. The straight-line distance, which is the one you mark with a compass, is 2 × radius × sine of (180° ÷ nails). With 23 nails on a circle of 12 cm radius that gives 15.65° and 3.27 cm. The nail spacing calculator does the math for any size.

How much string a layer takes

The length of one string between two nails is 2 × radius × sine of (180° × skip ÷ nails). Strung by counting, a full layer has as many strings as nails. With 23 nails, a 12 cm radius and a skip of 10, each string is 23.5 cm long and the whole layer 5.4 m. Add 10% for the knots and the turns around each nail. The generator works this out for every color.

The cardioid and the times tables

Number the nails on a circle and join each one to the nail with twice its number: 1 to 2, 2 to 4, 3 to 6. When you go past the last nail, keep counting from the start. A heart-shaped curve appears: the cardioid. If you use three times instead of twice, you get two lobes; with four times, three. The Flower of Venus is made of cardioids.

Try it without hammering anything

In the demo of the string art generator you can open the Prime Number Circle, change the number of nails and the skip of each layer, and see the figure at once. If you would rather do it by hand, the free patterns include ten circles ready to print, and string art ideas lists every design by difficulty.