String Art

Threads between pins on a circle: pin i connects to pin i·k mod n, and the chords envelope a curve the thread never draws.

What it is

The multiplication table on a circle. k = 2 envelopes the cardioid, k = 3 the nephroid, and each higher multiplier adds a cusp; layered multipliers (distinguished by line weight, the print-safe stand-in for thread colour) build the familiar layered nail-and-thread look.

History

Curve stitching was invented by Mary Everest Boole in the late 19th century as a way to teach children the geometry of envelopes; the modular multiplication form became a mid-20th-century string-art and classroom staple, and the cardioid-in-a-mug connection keeps it alive in mathematical folklore.

The implementation’s guarantees

• The chord count is measured, not pins × layers: i·k ≡ i (mod n) has gcd(k−1, n) fixed points and those threads do not exist — a test pins the arithmetic (100 pins, k = 51: fifty chords, not ninety-eight).
• Ink is the negotiation: the escalation ladder thins the thread to the 0.75 pt validator floor and then halves the chord density. Every stroke weight stays at or above the floor — the first version thinned past it and the validator said so.
• Rating basis: none — a design. Honesty fields: pins, multipliers, the drawn chord count, and the attempt the page settled at.