Sperner Fixed-Point Certificate
Turn a boundary labeling rule into a combinatorial fixed-point certificate.
Free theory articleTurn a boundary labeling rule into a combinatorial fixed-point certificate.
Concept & interpretation
Sperner's Lemma is a combinatorial counterpart to Brouwer's Fixed-Point Theorem. By dividing a simplex (a triangle in 2D) into smaller triangles and labeling the vertices according to a specific boundary rule, Sperner's lemma guarantees that there will always be an odd number of "fully labeled" sub-triangles (triangles with all distinct labels). Each of these fully labeled cells traps a fixed point of an associated continuous mapping.
Experiment
To explore the relationship between the subdivision size and the localization accuracy, interact with the Sperner Fixed-Point Certificate module:
- Set the Subdivision to
3and Map toCyclic. Observe that there are 10 vertices and 1 fully labeled triangle. - Change the Subdivision to
4. Notice the number of fully labeled triangles remains 1 (Odd parity), and the witness barycenter lands at (1/3, 1/3, 1/3) with residual0. - Switch the Map to
Swap first twoand observe how the coloring and labels change along the edges, but Sperner's parity remains Odd.