What it is and the problem it solves
Fejes Tóth’s work is a set of geometric proof techniques that resolve packing density and volume maximisation for symmetric convex shapes. It solves the problem of proving optimality without exhaustive search.
How it works
Fejes Tóth used geometric reasoning and finite decomposition to reduce infinite packing problems to verifiable cases. He proved lattice arrangements are optimal for centrally symmetric convex sets in the plane. He developed a technique to confirm Steiner’s conjecture for specific solids by bounding volume under surface-area constraints.
What works
Lattice packing is proven optimal for centrally symmetric convex sets in the plane. Maximum density for repeated symmetric convex bodies is achieved by lattices. Regular polytopes maximise volume under surface-area and Platonic equivalence constraints. Steiner’s conjecture holds for cube and dodecahedron via his technique.
What does not
It does not solve sphere packing outright. It does not apply to non-symmetric or non-convex bodies. It does not provide an algorithm—only existence proofs and structural bounds.
What it changes
It changes how we approach infinite geometric optimisation: from seeking global analytic solutions to bounding and enumerating finite configurations. It shifts proof strategy from calculus to combinatorial geometry.
Is it worth your time
Yes—if you work on computational geometry, verification methods, or discrete optimisation. His 1953 finite-reduction insight directly enabled Hales’ computer-assisted proof of Kepler. No implementation cost; it is pure method.