  We explore an optimal partition problem on surfaces using a computational
approach. The problem is to minimise the sum of the first Dirichlet
Laplace--Beltrami operator eigenvalues over a given number of partitions of a
surface. We consider a method based on eigenfunction segregation and perform
calculations using modern high performance computing techniques. We first test
the accuracy of the method in the case of three partitions on the sphere then
explore the problem for higher numbers of partitions and on other surfaces.
