Discrete Math
Sperner’s Lemma

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Discrete Math
Sperner’s Lemma

Sperner’s Lemma

Author

Apurva Nakade

Published

July 13, 2026

Sperner’s lemma is a combinatorial statement about colorings of triangulated triangles, with two very different classic proofs — one combinatorial, one geometric.

Sperner’s Lemma - Geometric Proof

This is my new favorite proof of Sperner’s lemma. In the following app the vertices are slowly morphed to the “cardinal” R, G, B vertices with time while preserving the…

Sperner’s Lemma - Combinatorial Proof

combinatorial proof

This picture is a combinatorial proof of Sperner’s Lemma.

Sperner’s Lemma - Statement

Consider a triangulation of a triangle. (Below, we show a regular triangulation of an equilateral triangle, but the theorem applies to any triangulation of a triangle.) We…

Barycentric Coordinates

barycentric coordinates

Barycentric coordinates are a coordinate system for describing points inside a triangle.

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Barycentric Coordinates