Initial research by Wittke and Garrido (1959) and by Niggli and Wondratschek (1960) identified the relation between the colour groups of an object and the subgroups of the object's geometric symmetry group. In 1961 van der Waerden and Burckhardt built on the earlier work by showing that colour groups can be defined as follows: in a colour group of a pattern (or object) each of its geometric symmetry operations s is associated with a permutation σ of the k colours in such a way that all the pairs (s, σ) form a group. Senechal showed that the permutations are determined by the subgroups of the geometric symmetry group G of the uncoloured pattern. When each symmetry operation in G is associated with a unique colour permutation the pattern is said to be perfectly coloured.