Math Art

Author

Apurva Nakade

Published

July 13, 2026

Pictures drawn by mathematics. Sperner’s lemma colors the vertices of a triangulated triangle and finds a small triangle showing all three colors, built up here from barycentric coordinates to two very different classic proofs — one combinatorial, one geometric. Spirographs trace the path of a pen on one circle rolling around another, a curve whose whole shape is decided by the ratio of the two radii.

Barycentric Coordinates

Introduces barycentric coordinates on a triangle as the setting for Sperner’s Lemma.

Sperner’s Lemma - Statement

States Sperner’s Lemma on a triangulated triangle and lets you color vertices to test the condition interactively.

Sperner’s Lemma - Combinatorial Proof

Proves Sperner’s Lemma with the ‘doors’ argument: a path that must exit through an odd number of bicolored edges.

Sperner’s Lemma - Geometric Proof

Proves Sperner’s Lemma with a discrete degree argument that counts oriented RGB triangles.

Spirographs

Rolls one circle inside or outside another and traces the curve drawn by a pen fixed to the rolling circle.