In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech lattice. Description of the algorithm Let be a hyperbolic reflection group. Choose any point ; we shall call it the basic (or initial) point. The fundamental domain of its stabilizer is a polyhedral cone in . Let be the faces of this cone, and let be outer normal vectors to it. Consider the half-spaces There exists a unique fundamental polyhedron of contained in and containing the point . Its faces containing are formed by faces of the cone . The other faces and the corresponding outward normals are constructed by induction. Namely, for we take a mirror such that the root orthogonal to it satisfies the conditions (1) ; (2) for all ; (3) the distance is minimum subject to constraints (1) and (2). References Hyperbolic geometry Reflection groups