The Leech lattice has dimension 24.
It is an even unimodular lattice in 24-dimensional Euclidean space, commonly denoted Λ24. In simple terms, it is a highly structured infinite arrangement of points whose geometry supports exceptionally dense sphere packing and deep connections to coding theory, group theory, and string theory.
The lattice is notable because its shortest nonzero vectors have squared length 4, so it has no roots—vectors of squared length 2. That absence distinguishes it from the other positive-definite even unimodular lattices in the same dimension. Its 196,560 shortest vectors also form the famous kissing configuration for 24-dimensional spheres.
The number 24 appears repeatedly in the surrounding mathematics. The Leech lattice is related to the binary Golay code, the Mathieu group M24, and the 24-dimensional Niemeier lattices. It is sometimes confused with the E8 lattice, which has dimension 8, or with the 26-dimensional Lorentzian lattice used in one construction of it; those are related structures, not the Leech lattice’s dimension.