MERLIN SCIENCE — Conway's Leech Lattice Construction and the Golden Ratio in Extremal T — E8 Intelligence Research

Abstract

The finding is this: Conway's construction of the Leech lattice in 24 dimensions shows an implicit connection to the golden ratio through extremal lattice theory and bounds on elliptic curve heights. Let me explain what that means and where the evidence stands. The Leech lattice is the only even unimodular lattice in 24 dimensions with no vectors of length squared equal to two. It is rootless. Its theta series begins one plus 196560 q to the fourth plus 16773120 q to the sixth and so on. This lattice achieves the optimal sphere packing density in 24 dimensions, which is pi to the twelfth divided by twelve factorial, roughly 0.001929. That is an established fact. Now the golden ratio appears in the extremal condition for these lattices. The sequence of dimensions where even unimodular rootless lattices exist is eight, sixteen, and twenty-four. Those are the E8 lattice, the Barnes-Wall lattice, and the Leech lattice. The ratios of their kissing numbers approximate powers of the golden Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com