Sperner’s Lemma

Author

Apurva Nakade

Published

October 16, 2026

Sperner’s lemma colors the vertices of a triangulated triangle and finds a small triangle showing all three colors. These pages build it up in order, from barycentric coordinates to two very different classic proofs, one combinatorial and one geometric.

Barycentric Coordinates

Explore barycentric coordinates on a triangle, the setting for Sperner’s Lemma.

Sperner’s Lemma - Statement

Color the vertices of a triangulated triangle and test Sperner’s Lemma.

Sperner’s Lemma - Combinatorial Proof

Prove Sperner’s Lemma by walking through doors.

Sperner’s Lemma - Geometric Proof

Prove Sperner’s Lemma by counting oriented triangles.