Sommerville, Duncan MacLaren Young (1879–1934)

British mathematician who made significant contributions to the study of non-Euclidean geometry.

Sommerville was born in Beawar, Rajasthan, India, and educated in Scotland, graduating from St Andrews. In 1915 he emigrated to New Zealand as professor at Victoria University College, Wellington.

Sommerville explained how non-Euclidean geometries arise from the use of alternatives to Euclid's postulate of parallels, and showed that both Euclidean and non-Euclidean geometries – such as hyperbolic and elliptic geometries – can be considered as sub-geometries of projective geometry. He stated that projective geometry is the invariant theory associated with the group of linear fractional transformations. He studied the tessellations of Euclidean and non-Euclidean space and showed that, although there are only three regular tessellations in the Euclidean plane, there are five congruent regular polygons of the same kind in the elliptical plane and an infinite number of such patterns in the hyperbolic plane. The variety is even greater if ‘semi-regular’ networks of regular polygons of different kinds are allowed (because the regular patterns are topologically equivalent to the irregular designs). In his later work on n-dimensional geometry, Sommerville generalized his earlier analysis to include ‘honeycombs’ of polyhedra in three-dimensional spaces and of polytopes in spaces of 4, 5, ..., n dimensions – including both Euclidean and non-Euclidean geometries.

Sommerville's interest in crystallography played a significant part in motivating him to investigate repetitive space-filling geometric patterns.