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ObituaryPeter Pleasants
24 August 1939 – 20 April 2008
Peter Pleasants, who made substantial contributions to number theory andquasicrystallography, died on 20 April 2008 from lymphoma in Brisbane, thecity where he also spent his ﬁrst years in Australia in the early 1960s.
Peter Arthur Barry Pleasants was born on 24 August 1939 in Harrow in England, and grew up in the family terrace house. His father Horace was anauditor for London Transport, and this allowed Peter to travel free on the vast
network of buses and underground trains during his school years. His mother,Florence, like many others at the time, stayed at home to look after the family.
Peter was educated at the Harrow County Grammar School, one in the top
tier of state schools in England at that time. He won a place in 1957 to readmathematics at St John’s College, University of Cambridge, where he also
held various scholarships. Apparently he did rather well in his ﬁrst year but
less so in his second. Indeed, he thought that he might not have done enoughto win a postgraduate place, and began telling his friends that he was goingon a long trip after he ﬁnished his third year. His pessimism turned out tobe unfounded, but although Harold Davenport oﬀered to take him on as a
research student, he decided to go on his long trip anyway.
Peter worked for about a year to save up enough money for his trip, and arrived in New Guinea after a long passage on a cargo boat. He was told that
academic positions in Australia were available, made some applications, andtook a junior lecturer position in the Mathematics Department at the University of Queensland in 1961. There he met Ann Johnman, a fellow juniorlecturer, whom he married a few years later. Their devotion to each other has
been the bed-rock of their family life and careers.

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In 1963, Peter returned to St John’s College, University of Cambridge, to embark on his research under Davenport, and Ann came to England soon after.
On completing the work on his PhD, Peter became a Research Fellow at Sidney Sussex College in Cambridge, a post which he held until 1968. However,
his thesis, entitled The representation of primes by quadratic and cubic polynomials, was not submitted until 1967. Peter was oﬀered a lecturer positionat the University of St Andrews, but he declined it in favour of Sidney Sussex
College, at substantially lower remuneration. His pay in Cambridge would
remain tax-free so long as he did not become Dr Pleasants!
In 1968, Peter joined the University College in Cardiﬀ, where he subsequently
became senior lecturer and remained for nearly 20 years. There he was partof a very active and successful research group in number theory. However, inthe mid-1980s, the college refused to implement draconian cutbacks imposedby the then Conservative government. In the ensuing ﬁnancial hardship anduncertainty, the Pleasants family made the decision to return to Australia in
1988. However, before they returned, Peter spent a year at the University ofTexas, where he loved the system under which colleagues were given teaching
duties and then left to get on with them without bureaucratic interference.
Before coming back to Australia, Peter had obtained a Master of Computing
Science. He soon settled into a position in computing at Monash University.
However, mathematics soon took over again, and Peter moved to a lectureshipin mathematics at the University of New England while the family stayed inMelbourne. When Peter moved to a lectureship in mathematics at Macquarie
University in 1992, he famously commented that the bus journeys back toMelbourne became overnight only half as long. Among his many and varied
contributions was the supervision of the honours project of Fran Griﬃn who
is now a continuing member of the academic staﬀ.
Between 1997 and 2001, both Ann and Peter were at the University of theSouth Paciﬁc, until the political uncertainties in Fiji persuaded them that it
was perhaps time they retired. They returned to Brisbane where Peter became
an honorary research consultant in the Department of Mathematics at theUniversity of Queensland.
Outside mathematics, Peter enjoyed bush walking, and had done both theLarapinta Trail and the Carnarvon Gorge. However, perhaps it was cyclingthat was Peter’s greatest sporting passion. He cycled to work in Cardiﬀ before
cycling became fashionable, and he took up long distance cycling the year hewas in Texas. He took part on a number of occasions in the Sydney to theGong cycle ride. In October 2007, he successfully negotiated the Brisbane toGold Coast ride at a very competitive pace. Unfortunately, this was to be hisﬁnal competitive ride.
Throughout his life, Peter was extremely dedicated to mathematics as well as
his family. He continued working while seriously ill in hospital, and would surround himself with books and manuscripts, working on yet another problem.
He was still correcting the proofs for a paper two weeks before he passed away.

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ObituaryPeter is survived by his wife Ann, sister Angela, brother Robert, children
Simon, Robin and Zoë, and grandson Evan, born on 19 May 2008.

Some of Peter Pleasant’s mathematical work
The work of Peter Pleasants is characterised by its depth, clarity of exposition
and meticulous attention to detail. His early work is in mainstream number
theory, analytic in nature but on occasions in somewhat algebraic settings.
Subsequently he applied his number theoretic techniques to study problems inquasicrystallography which form the bulk of his most recent work.
During his time at Cambridge, Peter was heavily inﬂuenced by the work ofH. Davenport and D.J. Lewis, and also by the work of G.L. Watson. His
early papers were on the representation of primes by quadratic and cubic
polynomials and on the representation of integers by cubic forms, utilising thewell-known Hardy–Littlewood method and its variants. He later extended thisstudy to settings over algebraic number ﬁelds.
In the 1970s, Peter expanded his interests in a number of ways. His work on
word problems was inﬂuenced by M. Morse and G.A. Hedlund, while his workon forms over p-adic ﬁelds was inﬂuenced by R.R. Laxton and Lewis. Indeed,in the latter, Peter was able to replace the algebraic geometry by elementary
methods and unify several known low-dimensional cases of the Artin conjecture regarding non-trivial zeros of forms over local ﬁelds. On another front, hiswork on a sum related to distribution modulo one gave rise to generalisationsof some sieve inequalities due to Davenport and H. Halberstam. In the early
1980s, he added factorisations in an algebraic number ﬁeld and prime factorsof binomial coeﬃcients to his expanding repertoire.
Peter became interested in quasicrystallography in the early 1980s. His ﬁrst
paper on the subject was, according to the reviewer, written two years before
the discovery of quasicrystals and contributed in various ways to the theoryof non-periodic patterns. In a second paper, written with W.F. Lunnon1, ageneral method for constructing ﬁnite sets of aperiodic tiles was found.
In the 1990s, Peter began a very successful collaboration with M. Baake. WithJ. Roth, they applied elementary number theory to the theory of coincidencesof Z-modules generated by the vectors of a regular n-star, gave a complete
description of the coincidence rotations when the cyclotomic ﬁeld of n-th rootsof unity has class number 1, and described in detail applications to two quasiperiodic tilings. With J. Hermisson, they computed the number of tilings in

1Ann Pleasants recalls the occasion when Peter was working on these problems on the beachin France during a family holiday. Peter had just made a discovery and wanted very badly
to write to Lunnon as soon as possible. Unfortunately, they were rather short of money andthe banks were closed over the weekend. Eventually, some money intended for ice creamwas sacriﬁced in favour of postage stamps.

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each local isomorphism class of a tiling which are invariant under various subgroups of the full symmetry group. With R.V. Moody, they studied the set
VL of all points of a lattice L in Rn that are visible from the origin and the set
Fk of all k-th power-free integers, and showed that their diﬀraction measures
are pure point, supported respectively on dense sets Q[L∗ ] and Q, where L∗
is the dual lattice of L. Now both VL and Fk contain large holes and so are
not Meyer sets. They thus gave the ﬁrst examples of pure point diﬀractive
discrete sets that are not Meyer.
In the last few years, Peter further expanded his list of collaborators as wellas interests.
With J.C. Lagarias, he considered Delone sets, subsets of Rd that are uniformly discrete and relatively dense, studied their complexity via patch counting and repetitivity, and established their relationships to crystallinity andquasicrystallinity.
With A. Granville, he used Aurifeuillian factorisation to study factorisationsof numbers of the form bn ±1, where b = 2, 3, . . . are small. This is part of what
is known as the Cunningham project, and in 1962, A. Schinzel gave a list ofidentities useful in this project. They showed that Schinzel’s list is exhaustivein some sense, and also succeeded in factorising terms of the form f (bn ) for any
given polynomial f , using deep results of G. Faltings from algebraic geometryand M.D. Fried from the classiﬁcation of ﬁnite simple groups.
With Baake and U. Rehmann, he showed that various results and conjectures
related to the coincidence site lattice problem and its generalisation of Zmodules in R3 can be proved in a uniﬁed way, making use of maximal orders
in quaternion algebras of class number 1 over real algebraic number ﬁelds.
In his ﬁnal paper, Peter succeeded in generalising these results by removing
many of the assumptions. In particular his methods work for quaternion algebras over arbitrary number ﬁelds, although his most precise result requires
the quaternion algebra to satisfy the Eichler condition. He introduced thenotion of a primefree ideal; these correspond bijectively to pairs of maximalorders. Peter showed that an ideal over a maximal order factors uniquely as
the product of a 2-sided ideal and a primefree ideal.
Peter enjoyed combinatorics too. He also loved puzzles, and was the proud
winner very recently of a $50 book voucher from the Gazette of the AustralianMathematical Society for one of his solutions!
William Chen1 , Keith Matthews2 and Ross Street3
1 Department of Mathematics, Macquarie University, Sydney, NSW 2109.

E-mail: wchen@maths.mq.edu.au
2 Department of Mathematics, University of Queensland, Brisbane, QLD 4072.

E-mail: keithmatt@gmail.com
3 Department of Mathematics, Macquarie University, Sydney, NSW 2109.

E-mail: street@maths.mq.edu.au