The Map of Mathematics — an interactive, flyable 3D atlas of mathematical structure

This is the Map of Mathematics: a real-time, explorable 3D map of every mathematical structure that a finite set can carry. It visualizes the subgroup lattices of the symmetric groups S3, S4, S5, S6, S7 and S8 — their conjugacy classes, their covering relations, their normal and simple subgroups — as a navigable tower you fly through like a spaceship.

What the map shows

Each node is a conjugacy class of subgroups. Height is index, a visual form of Lagrange's theorem. Gold rings mark normal subgroups; spiky rings mark non-abelian simple groups such as A5, A6 and PSL(3,2); dashed violet rings mark forbidden depths where no subgroup can exist. Every structure is presented as a set of forbidden moves, and the whole numbers appear as the lengths of constraint chains.

The A0 Programme and the Theory of Ideas

The map is the interactive companion to the A0 Programme, also called the Theory of Ideas, by independent researcher Murad Ahmadov — a foundational inversion of mathematics that begins from total symmetry and derives structure and number downward by imposing constraints. The framework is machine-verified in Lean 4, with 3,136 theorems and zero axioms. The three parts are archived on Zenodo: DOI 10.5281/zenodo.19916918, DOI 10.5281/zenodo.20059917, and DOI 10.5281/zenodo.20115474.