Mathematical String Art
Place N equally-spaced points around a circle and connect each point i to point (i × multiplier) mod N. Only straight lines are drawn — but curves emerge. A cardioid from ×2, a nephroid from ×3, higher epicycloids beyond. The envelope of tangent lines, from pure modular arithmetic.
String art begins as a craft tradition: hammer nails into a board in a pattern, then wind thread between them. The result is surprising — straight threads somehow trace curves. What you see is an envelope: the family of straight lines that are each tangent to an invisible curve. No one draws the curve itself. It emerges from the density of lines that almost touch it.
The mathematical version places numbered points around a circle and connects each point i to point (i × m) mod N, where m is the multiplier and N is the number of points. This rule is modular multiplication — clock arithmetic applied to geometry. The envelope of these chords is an algebraic curve whose shape depends entirely on m.
When the multiplier is 2, each chord connects a point at angle θ on the circle to the point at angle 2θ. One endpoint moves around the circle twice as fast as the other. The envelope of this family of chords is a cardioid — a heart-shaped curve.
The cardioid is also an epicycloid: the curve traced by a point on a circle of radius r rolling around the outside of a fixed circle of the same radius r. It appears in the Mandelbrot set as the main bulb boundary, in microphone polar patterns, and in the caustic of light reflected inside a coffee cup. The connection to string art is that the chords of a circle, parameterized so one endpoint moves twice as fast, are exactly the tangent lines of this rolling curve.
The multiplier m creates an epicycloid with exactly (m − 1) cusps. With ×2 you get one cusp (the cardioid). With ×3 you get two cusps (the nephroid, kidney-shaped). With ×4, three cusps. The pattern is clean: the number of cusps equals the number of times a small rolling circle fits around the fixed circle.
The cusps are the points where the envelope curve has a sharp turn — where the tangent direction reverses. They sit at equally spaced angles around the circle, giving each curve its characteristic rotational symmetry. Higher multipliers produce more intricate patterns, and non-integer multipliers create beautiful transitional forms that belong to no named family.
The rule i → (i × m) mod N is modular multiplication — the same operation that underpins cryptography, hash functions, and clock arithmetic. What makes it visual here is that modular multiplication has structure: it maps evenly-spaced inputs to a pattern that wraps around the circle, and the geometry of that wrapping produces the envelope.
This connection was popularized by the Mathologer video on times tables on a circle. The visual beauty comes directly from number theory. When m and N share common factors, some points map to themselves and the pattern decomposes into separate components. When m is coprime to N, every point connects to a distinct target and the full curve appears. The interplay between multiplication and modular reduction is what makes these images rich.
The circle is the starting shape, but the idea extends: place points on an ellipse, a polygon, a spiral, or any closed curve and apply the same modular connection rule. The envelopes change — sometimes dramatically — because the geometry of the base curve interacts with the arithmetic of the connection.
Physical string art installations take this further. Artists like Gabriel Dawe create room-sized thread sculptures that use hundreds of threads to form luminous curved surfaces in three dimensions. The mathematics is the same: each thread is straight, but the collective envelope forms a ruled surface — a surface composed entirely of straight lines. Hyperboloids, helicoids, and more complex forms all emerge from thread and geometry.