Onsager famously announced the following expression for the spontaneous magnetization M of a two-dimensional Ising ferromagnet on the square lattice at two different conferences in 1948, though without proof M = ( 1 − − [ sinh ⁡ ⁡ 2 β β J 1 sinh ⁡ ⁡ 2 β β J 2 ] − − 2 ) 1 8 where J 1 and J 2 are horizontal and vertical interaction energies.