(Left) In the simplest (“hydrogen-atom”) case, the energy-momentum relationship of the surface states in a topological insulator takes the form of a Dirac cone. The constant energy surfaces are then circles of different radii. (Right) Results of a first-principles computation of the spin texture of the surface states in that we have carried out, showing in quantitative detail how the Dirac cone is warped due to the effect of the crystal potential. The actual shape of the Fermi surface is determined by the natural chemical potential and can be a hexagon or a snowflake. As a result of deviations from the ideal Dirac cone dispersion, a nonzero out-of-the plane spin polarization develops and the conventional nesting channels (red arrows) open up possibilities for magnetic order on the surface.