Sperner Triangle Boundary Parity
Boundary color rules force a tiny triangle carrying all three labels, no matter how the allowed colors are arranged.
Free theory articleBoundary color rules force a tiny triangle carrying all three labels, no matter how the allowed colors are arranged.
Concept & interpretation
Sperner's Lemma states that every admissible coloring of a triangulated simplex must contain an odd number of fully labeled simplices. For a 2D triangle, this implies that no matter how we color the internal vertices (subject to boundary rules), there is always at least one small triangle whose three vertices carry three distinct colors (0, 1, and 2).
experiment
In the interactive module at Sperner Triangle Boundary Parity, you can verify the combinatorial certificate on different grids:
- Change the Mesh denominator to control the resolution.
- Adjust the Favored corner to bias colors.
- Set the Tie priority to resolve ties differently.
- Observe that the number of fully labeled triangles is always odd.