Combinatorics and Graph Theory · Problem 6.2.4
Pólya–Redfield Counting — David Guichard
Mathematics — Combinatorics · burnsides-theorem
Problem
Find the number of different colorings of the edges of a regular tetrahedron with colors, modulo the rigid motions.Intuitive explanation
This is a classic application of Burnside's lemma (also called the Cauchy–Frobenius lemma), which counts distinct objects under group symmetry. The key insight is that instead of counting all edge colorings directly, you count how many colorings stay unchanged under each rigid motion of the tetrahedron, then average over the symmetry group. The rotation group of a regular tetrahedron has 12 elements—identity, rotations around vertex-to-face axes, and rotations around edge-to-edge axes—and each contributes differently to the count of fixed colorings.
Intuitive explanation
This is a classic application of Burnside's lemma (also called the Cauchy–Frobenius lemma), which counts distinct objects under group symmetry. The key insight is that instead of counting all edge colorings directly, you count how many colorings stay unchanged under each rigid motion of the tetrahedron, then average over the symmetry group. The rotation group of a regular tetrahedron has 12 elements—identity, rotations around vertex-to-face axes, and rotations around edge-to-edge axes—and each contributes differently to the count of fixed colorings.