**Thomas Gerald Room** FRS FAA (10 November 1902 – 2 April 1986) was an Australian mathematician who is best known for Room squares. He was a Foundation Fellow of the Australian Academy of Science.^{[1]}^{[2]}

## Biography

Thomas Room was born on 10 November 1902, near London, England. He studied mathematics in St John's College, Cambridge, and was a wrangler in 1923. He continued at Cambridge as a graduate student, and was elected as a fellow in 1925, but instead took a position at the University of Liverpool.^{[1]} He returned to Cambridge in 1927, at which time he completed his PhD, with a thesis supervised by H. F. Baker.^{[1]}^{[3]} Room remained at Cambridge until 1935, when he moved to the University of Sydney, where he accepted the position of Chair of the Mathematics Department, a position he held until his retirement in 1968.^{[1]}^{[4]}

During World War II he worked for the Australian government, helping to decrypt Japanese communications.^{[1]} He was one of four in a "cypher section" at the University of Sydney who were recruited by Eric Nave and moved to FRUMEL in June 1941.^{[5]} He moved to the Army Central Bureau with Nave in October 1942.

After the war, Room returned to the University of Sydney, where he was dean of the faculty of science from 1952 to 1956 and again from 1960 to 1965.^{[1]} He also held visiting positions at the University of Washington in 1948, and the Institute for Advanced Study and Princeton University in 1957.^{[1]}^{[6]}^{[7]} He retired from Sydney in 1968 but took short-term positions afterwards at Westfield College in London and the Open University before returning to Australia in 1974. He died on 2 April 1986.

Room married Jessica Bannerman, whom he met in Sydney, in 1937; they had one son and two daughters.^{[1]}^{[8]}

## Research

Room's PhD work concerned generalizations of the Schläfli double six, a configuration formed by the 27 lines on a cubic algebraic surface.^{[1]}^{[3]}

In 1938 he published the book *The geometry of determinantal loci* through the Cambridge University Press.^{[1]} Nearly 500 pages long, the book combines methods of synthetic geometry and algebraic geometry to study higher-dimensional generalizations of quartic surfaces and cubic surfaces. It describes many infinite families of algebraic varieties, and individual varieties in these families, following a unifying principle that nearly all loci arising in algebraic geometry can be expressed as the solution to an equation involving the determinant of an appropriate matrix.^{[1]}^{[9]}

In the postwar period, Room shifted the focus of his work to Clifford algebra and spinor groups.^{[1]} Later, in the 1960s, he also began investigating finite geometry, and wrote a textbook on the foundations of geometry.^{[1]}

Room invented Room squares in a brief note published in 1955.^{[10]} A Room square is an *n* × *n* grid in which some of the cells are filled by sets of two of the numbers from 0 to *n* in such a way that each number appears once in each row or column and each two-element set occupies exactly one cell of the grid. Although Room squares had previously been studied by Robert Richard Anstice,^{[11]} Anstice's work had become forgotten and Room squares were named after Room. In his initial work on the subject, Room showed that, for a Room square to exist, *n* must be odd and cannot equal 3 or 5. It was later shown by W. D. Wallis in 1973 that these are necessary and sufficient conditions: every other odd value of *n* has an associated Room square. The nonexistence of a Room square for *n* = 5 and its existence for *n* = 7 can both be explained in terms of configurations in projective geometry.^{[1]}

Despite retiring in 1968, Room remained active mathematically for several more years, and published the book *Miniquaternion geometry: An introduction to the study of projective planes* in 1971 with his student Philip B. Kirkpatrick.^{[1]}

## Awards and honours

In 1941, Room won the Thomas Ranken Lyle Medal of the Australian National Research Council and was elected as a Fellow of the Royal Society.^{[1]}^{[12]}^{[13]} He was one of the Foundation Fellows of the Australian Academy of Science, chartered in 1954.^{[1]}^{[2]} From 1960 to 1962, he served as president of the Australian Mathematical Society and he later became the first editor of its journal.^{[1]}

The T. G. Room award of the Mathematical Association of New South Wales, awarded to the student with the best score in the NSW Higher School Certificate Mathematics Extension 2 examination, is named in Room's honour.^{[1]}^{[14]}

## References

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}^{l}^{m}^{n}^{o}^{p}^{q}^{r}^{s}Hirschfeld, J. W. P.; Wall, G. E. (1987). "Thomas Gerald Room. 10 November 1902 – 2 April 1986".*Biographical Memoirs of Fellows of the Royal Society*.**33**: 574. doi:10.1098/rsbm.1987.0020. JSTOR 769963.. Also published in*Historical Records of Australian Science***7**(1): 109–122, doi:10.1071/HR9870710109. An abridged version is online at the web site of the Australian Academy of Science. - ^
^{a}^{b}John Mack. "Room, Thomas Gerald (1902–1986)".*Australian Dictionary of Biography*. National Centre of Biography, Australian National University. First published in*Australian Dictionary of Biography*, Volume 18, (MUP), 2012. - ^
^{a}^{b}Thomas Gerald Room at the Mathematics Genealogy Project **^**"The University. Chair of Mathematics. Professor T. G. Room",*The Sydney Morning Herald*, 21 December 1934.**^**Smith, Michael (2000).*The Emperor's Codes: Bletchley Park and the breaking of Japan's secret ciphers*. London: Bantam Press. p. 81. ISBN 0593 046412.**^**"Princeton Appoints 17 Guest Professors",*The New York Times*, 4 September 1957.**^**"Institute Names 128 For Research; Scholars Will Do Advanced Study on Historical Topics And in Mathematics",*The New York Times*, 15 September 1957.**^**"Professor and Bride Dodge Rice",*The Sydney Morning Herald*, 8 November 1937.**^**Review of*The geometry of determinantal loci*by Virgil Snyder (1939),*Bulletin of the AMS***45**: 499–501, doi:10.1090/S0002-9904-1939-07011-0.**^**Room, T. G. (1955), "A new type of magic square",*Mathematical Gazette*,**39**: 307.**^**O'Connor, John J.; Robertson, Edmund F., "Robert Anstice",*MacTutor History of Mathematics archive*, University of St Andrews..**^**"Lyle Medals Awarded",*The Sydney Morning Herald*, 10 July 1941.**^**Thomas Ranken Lyle Medal Archived 28 November 2010 at the Wayback Machine, Australian Academy of Science. Retrieved 6 June 2010.**^**The T G Room Award Archived 4 August 2012 at archive.today, Mathematical Association of New South Wales. Retrieved 1 June 2010.