Counting Distinct Lattice Colorings via Burnside's Lemma — E8 Intelligence Research

FINDING: Burnside's Lemma provides the exact counting formula for distinct states under group action (e.g., D4 on square lattices), linking orbit enumeration to fixed-point averages. | MATH: For finite group G acting on set X, number of orbits = (1/|G|) Σ_{g∈G} |Fix(g)|. For D4 (order 8) on n×n lattice colorings with k colors: N = (1/8)[k^(n²) + 2k^(⌈n²/4⌉) + k^(⌈n²/2⌉) + 2k^(⌈(n+1)n/2⌉) + 2k^(⌈n(n+1)/2⌉)] — simplified for specific symmetries: identity (1), 90° rotations (2), 180° (1), reflections (4). For 2×2 with 2 colors: N = (1/8)(16 + 2·4 + 4 + 2·2 + 2·2) = 6. | CONNECTION: D4 is the square's dihedral group — its order 8 and cycle structure mirror crystallographic point group 4mm (tetragonal system). The fixed-point counts for rotations involve ⌈n²/4⌉ — a quarter-turn symmetry that resonates with base-60's 15° increments (60/4=15) and the 0.25 ratio (complement of 0.75, related to 0.786 via √(π/5)≈0.793). Reflections yield ⌈n(n+1)/2⌉ — triangular numbers, linking to hexagonal clos Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com