Thomas Kirkman

Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane as one of ten leading 19th-century British mathematicians. In the 1840s, he obtained an existence theorem for Steiner triple systems that founded the field of combinatorial design theory, while the related Kirkman's schoolgirl problem is named after him.

Early life and education Kirkman was born 31 March 1806 in Bolton, in the north west of England, the son of a local cotton dealer. In his schooling at the Bolton Grammar School, he studied classics, but no mathematics was taught in the school. He was recognised as the best scholar at the school, and the local vicar guaranteed him a scholarship at Cambridge, but his father would not allow him to go. Instead, he left school at age 14 to work in his father's office. Generalizing the quaternions and octonions, Kirkman called a pluquaternion Qa a representative of a system with a imaginary units, a > 3. Kirkman's paper was dedicated to confirming Cayley's assertions concerning two equations among triple-products of units as sufficient to determine the system in case a = 3 but not a = 4. By 1900 these number systems were called hypercomplex numbers, and later treated as part of the theory of associative algebras.

Polyhedral combinatorics Beginning in 1853, Kirkman began working on combinatorial enumeration problems concerning polyhedra, beginning with a proof of Euler's formula and concentrating on simple polyhedra (the polyhedra in which each vertex has three incident edges). He also studied Hamiltonian cycles in polyhedra, and provided an example of a polyhedron with no Hamiltonian cycle, prior to the work of William Rowan Hamilton on the Icosian game. He enumerated cubic Halin graphs, over a century before the work of Halin on these graphs. He showed that every polyhedron can be generated from a pyramid by face-splitting and vertex-splitting operations, and he studied self-dual polyhedra. In 1857, Kirkman was elected as a fellow of the Royal Society for his research on pluquaternions and partitions. He was also an honorary member of the Literary and Philosophical Society of Manchester and the Literary and Philosophical Society of Liverpool, and a foreign member of the Dutch Society of Science.

Since 1994, the Institute of Combinatorics and its Applications has handed out an annual Kirkman medal, named after Kirkman, to recognise outstanding combinatorial research by a mathematician within four years of receiving a doctorate.