144 Middle Harbour Road, Lindfield, NSW 2070, Australia. Corresponding author; email: [email protected] 225 The Grove, Lower Mitcham, SA 5062, Australia.35 Knightsbridge Road, Leabrook, SA 5068, Australia.
Formative Years 1924–39
Eric Barnes was born on 16 January 1924in Cardiff, Wales. He was the only child ofWilliam H. and Dorothy Barnes. His fatherdid not proceed beyond primary schooland spent his life as a manual worker. Hismother attended secondary school andtook various clerical jobs. The family suf-fered during the depression of the 1920s inSouth Wales and this prompted the fatherto migrate to Sydney in 1926, and Eric andhis mother followed in 1927. His mothercould not tolerate the new life and wentback with Eric to Cardiff, but they eventu-ally returned to Sydney in 1929.
Eric proved to be a precocious child,learning to read when very young and ableto tell the time at age 3. He attendedPunchbowl Primary School and acceleratedthrough the grades to the final grade at age9. His results in the Primary Final examina-tion qualified him for admission to a fullhigh school but he was considered tooyoung to proceed. He sat the Primary Finalagain the following year and finished highenough to qualify for a State GovernmentBursary; but alas there was a condition thatone had to be 11 years old by the following1 January, and the award was refused sinceEric was 15 days too young. Nevertheless,with the support of his parents, he enteredCanterbury Boys’ High School in 1935. Atthe end of his first year he took the PrimaryFinal for the third time and was dulyawarded a Government Bursary.
Eric excelled academically at Canter-bury. For the Intermediate Certificate in1937 he gained the maximum of eight As
(and was awarded a pair of gold cuff-links). At the Leaving Examination in 1939he gained first-class honours in the twoMathematics subjects and French, second-class honours in German, and As inEnglish and Latin. He was awarded theBarker Scholarship for Mathematics,shared the Garton Prize for French, andwon a University Exhibition and a StateGovernment Bursary.
University of Sydney 1940–43
Eric was first attracted to a career as anactuary. The arduous seven-year coursewas conducted from Britain. At the end of1939 a torpedo sank the ship carrying theexamination papers and the course wasabandoned for ‘the duration’. Eric optedfor an Honours BA course in both Mathe-matics and French at the University ofSydney. He won prizes in both subjectsevery year, graduating at the beginning of1943 with First-Class Honours in Mathe-matics and French. Although he wasrecommended for University Medals inboth subjects, the recommendation wasrejected because he had taken three and notfour years for the honours courses.
War Service 1943–45
During 1942, Eric had been approached byA.D. Trendall, Professor of Greek at theUniversity of Sydney, as a candidate for aspecial unit of the Australian IntelligenceCorps located at Victoria Barracks in Mel-bourne. Eric duly joined the Citizen Mili-tary Forces in 1943 and served for threeyears in the Intelligence Section, with the
22 Historical Records of Australian Science, Volume 15 Number 1
E. S. Barnes May 1955.
Eric Stephen Barnes 1924–2000 23
particular task of decoding Japanese diplo-matic messages.
It was only in the 1990s that informa-tion concerning the secret work of theAustralian Intelligence Unit was madepublic. In an article ‘Our War of Words’ inthe Sydney Morning Herald (19 September1992), the author David Jenkins profiledthe Australian code-breakers ‘who helpedchange the course of history’. The code-breakers were mainly classicists andmathematicians. To quote from the article:‘There were, however, notable exceptionsto the rule that said that classicists werebetter than mathematicians. One or two ofthe younger mathematicians, Barnes inparticular, proved to be highly skilledcode-breakers’. In conversation in lateryears Eric recalled that he gained his com-mission as a Lieutenant because of hissuccess in cracking a Japanese code thathad baffled the British experts at BletchleyPark.
An interesting account has been givenby some of Eric’s colleagues in the SpecialIntelligence Section of the solving of theso-called Kormoran cryptogram. Althoughthere is some controversy about the matter,HMAS Sydney was apparently sunk, withall crew lost, by the German raider Kormo-ran off the coast of Western Australia on19 November 1941. Captain Detmers andthe crew of the damaged Kormoran aban-doned their ship and were captured andinterned. On 11 January 1945 Detmers,with others, escaped from a POW campand when recaptured had in his possessiona book with coded messages. These weresent to the Section and other Intelligencegroups for decryption. One of Eric’s col-leagues writes: ‘It was not difficult to breakthis cipher, once it had been recognised.I cannot remember the details of thisbreaking, but I am sure the crucial stepswere taken by Barnes, who used to see intwo minutes what would take me twohours.’ The same colleague described Ericas having a ‘laser-sharp mind’.
Although Eric found his war work intel-lectually challenging and satisfying, pro-fessionally the years were a waste. TheMelbourne University Library refused himpermission to borrow books, and therewere so few mathematics books held bythe Public Library that he read them all. Atleast by the end of the war he had made hisdecision to pursue a career in mathematicsand not French.
Sydney 1946–47
Fortunately Eric’s demobilization washastened by T.G. Room, Professor of PureMathematics at the University of Sydney,who appointed Eric a Teaching Fellow inMathematics in 1946. The tutoring andlecturing duties were very demanding,including a term at Armidale teaching fivecourses with a total of twelve lectures perweek. He taught a third-year Honourscourse in Group Theory, a topic which hehad never studied.
Eric applied for and was awarded the‘open’ J.B. Watt scholarship and withencouragement from Professor Trendallapplied for entry to Trinity College, Cam-bridge. At first unsuccessful, he was lateraccepted and departed for Cambridge inAugust 1947.
Cambridge 1947–53
Eric enrolled in the Cambridge Mathe-matical Tripos. Having a Sydney honoursdegree, he was permitted to take the three-year course in two years — a mixedblessing, since the solid two years of workfor the crucial Part II examination had to becrammed into one. In the event, he passedall examinations with flying colours (‘Wran-gler’ Part II, Distinction Part III). The aca-demic atmosphere in Cambridge at this timewas very stimulating for prospectivenumber theorists. Professor L.J. Mordell’sweekly Number Theory Seminar wasattended by some twenty to thirty people,amongst whom there were a number ofbrilliant younger mathematicians. In 1949
24 Historical Records of Australian Science, Volume 15 Number 1
Eric enrolled as a research student in thegeometry of numbers with Mordell assupervisor. A fellow student, John Chalk,had first excited his interest in the subjectby proposing a problem that was the start-ing point for his first paper, published in1950.
What happened in these first years ofresearch is vividly expressed in his ownwords: ‘So in 1949 and 1950 I ate, drankand slept mathematics: reading and writingout notes on or translations of papers in theCambridge Philosophical Society Library,working at problems on binary quadraticand bilinear forms and attending lecturecourses and seminars’. His achievementsover these two years were remarkable. Bymid–1950 he had written seven papers forpublication. These were submitted both forthe Smith’s Prize and for a Trinity CollegeFellowship. He was successful in bothcases, the Smith’s Prize being shared withanother candidate. He was awarded hisPhD (Cantab.) in 1952.
In 1951, Eric was appointed AssistantLecturer in Mathematics, a three-year postin the first instance. In June that year hemarried Stewart Caird, an Australian,daughter of William and Emily Caird ofPreston, Victoria. Their son, Peter, wasborn in Cambridge in 1953. In order tosupplement the income from his lecture-ship, Eric undertook additional tutorial andexamining work. Of his lectures at thistime, Maurice Brearley (now EmeritusProfessor of the University of Melbourne)writes:
I was fortunate to attend a one-term courseof lectures on linear algebra by Eric in Part2 of the Cambridge Mathematical Tripos.His style was lucid and unhurried, hisblackboard work always impeccable. Herarely consulted his notes during a lecture,giving the impression that he was not work-ing from a planned script. Each lecture,however, ended precisely on time at a stagewhere there was a natural break in themathematics; never was he part waythrough a proof when time ran out, which
showed the whole had been meticulouslyplanned. Eric had a dry sense of humour, farremoved from any conscious joke. Afterintroducing the concept of homomorphismhe remarked: ‘One of the Morph brothers’.The characteristic which I most appreciatedwas his ability to make even quite difficultconcepts easy to grasp.
Despite his heavier teaching commit-ments, Eric’s research continued to flour-ish. In particular, he continued a veryfruitful collaboration already begun withPeter Swinnerton-Dyer.
Sydney 1953–58
With increasing family commitments and abarely adequate income from his fellow-ship and assistant lectureship, Eric wasfaced with serious decisions about hisfuture. Cambridge was a great centre ofnumber theory, in which he had establisheda firm reputation and made many friends.But he and Stewart had always intended toreturn to Australia. In 1953, he success-fully applied for a Senior Lectureship atthe University of Sydney. However, onbeing informed that he had been recom-mended for a full Lectureship in Cam-bridge, he turned the Senior Lectureshipdown. Not long after, an offer of a Reader-ship at Sydney arrived, and this heaccepted. Eric, Stewart and their infantson, Peter, left Cambridge in August 1953.
Before 1950, prospective researchmathematicians from Australian univer-sities needed to undertake further studyoverseas. By the early 1950s, however, thePhD degree was becoming established inmathematics at the University of Sydneyand elsewhere. When Eric arrived, theSydney department had a handful ofresearch students who soon included hisfirst PhD student, Jane Pitman. Later, hecollaborated with algebraist Tim (G.E.)Wall on a significant joint paper. However,like most researchers in pure mathematicsin Australia at the time, Eric had to workmainly in isolation, in marked contrastwith his Cambridge experience.
Eric Stephen Barnes 1924–2000 25
Despite these seeming disadvantages,the years he spent in Sydney were some ofthe most fruitful of his career in terms ofresearch. He also proved to be an accom-plished lecturer at all levels, noted for hisbeautifully clear and well organizedpresentation.
On the personal side, Eric and Stewartwere able to settle down as a family withreasonable financial security. Their daugh-ter, Erica, was born in Sydney in 1956.
Eric’s brilliant research in the 1950ssoon gained recognition. In 1954, he wasawarded the Edgeworth David Medal ofthe Royal Society of New South Wales. Hewas elected a Fellow of the newly foundedAustralian Academy of Science in thesame year and was awarded its ThomasRankin Lyle Medal in 1959.
Adelaide 1959–83
When H.W. Sanders retired as Professor ofMathematics at the University of Adelaidein 1958, the Council decided to replace himby two professors, one in Pure Mathematicsand one in Applied Mathematics, whowould alternate as Head of Department forperiods of three years. Eric Barnes wasappointed as Elder Professor of PureMathematics and moved to Adelaide withhis family in January 1959. Ren Potts wasappointed as Professor of Applied Mathe-matics and arrived a few months later.
Eric served initially as Head of theDepartment of Mathematics until the endof 1962. The position entailed a wide rangeof commitments outside the Department,including service on some key Universitycommittees, and his abilities soon gainedrespect. From 1963 onwards, while contin-uing his mathematical commitments, Ericbecame increasingly involved in Universityadministration. It was a loss to mathe-matics when he vacated the Chair of PureMathematics to become one of two DeputyVice-Chancellors in 1975. As Deputy Vice-Chancellor, he had major responsibility inseveral areas, including academic matters
and University entrance, and chaired theUniversity’s Co-ordinating Committee andthe Committee of Deans. In 1980, restruc-turing of the University’s management sawthe two Deputy Vice-Chancellor positionsdiscontinued after the end of their firstterm, and in 1981 the two incumbentsreturned to their respective departments asProfessors. Eric was warmly welcomed bywhat was then the Department of PureMathematics and served as elected Chair-man in 1982. In May 1983, he took up theopportunity of early retirement.
The Mathematics Department
In 1959, despite the presence of some veryable researchers, the Mathematics Depart-ment was relatively inactive. It had pro-duced only one PhD and very few honoursgraduates and its main focus was onservice teaching. On the arrival of RenPotts the two new professors began afriendly co-operation that helped to trans-form the Department into an activemodern department with high standards inboth teaching and research.
The structure of the mathematicscourses was streamlined by introducingGeneral Mathematics, abolishing AppliedMathematics I, and replacing Pure Mathe-matics I by Mathematics I. Priority wasgiven to establishing a comprehensivefourth-year honours course, with flexibleprerequisites. Honours projects were intro-duced with the allocation of staff ashonours supervisors. The results weredramatic. For the first time, HonoursMathematics became an attractive optionfor mathematically inclined students. Thenumber of honours students increasedfrom one in 1959 to twenty in 1964. TheAustralian Mathematical Society coinci-dentally started collecting and publishingfrom 1959 the yearly number of honoursstudents graduating in the mathematicalsciences in the universities in Australia.Adelaide soon topped the list, above muchlarger universities.
26 Historical Records of Australian Science, Volume 15 Number 1
The establishment of a strong post-graduate program was of course more pro-tracted. A staff member graduated with aPhD in 1961, two students in 1964, three in1966, four in 1967 and seven in 1968. Inthe yearly lists mentioned above, Adelaidewas soon near the top.
A separate Department of ComputingScience was already established and in1968 Statistics separated from Mathe-matics to form its own department. In 1970the applied mathematicians proposed afurther split into Pure and Applied Mathe-matics. Although Eric was not in favour, hedid not oppose it and years later he wrotethat the move was ‘completely vindicated’.The separation took effect in 1971, withEric and Ren as the Heads of the two newdepartments which continued to co-operate closely and shared some staff. Thenew structure helped provide a firm foun-dation for the establishment of the Faculty
of Mathematical Sciences for which Ericwas to prove the strongest advocate and itsfirst Dean in 1973. The Faculty comprisedfive departments, Pure, Applied, Statistics,Computing Science and (also in Science)Mathematical Physics. The new Facultywas by student numbers the third largest inthe University (behind Arts and Science).Its new degree provided the flexibilityrequired by the growth of the mathematicalsciences and their links with many otherareas.
Between 1959 and 1974, Eric gaveeffective academic leadership in all aspectsof the work in pure mathematics, includingteaching, research, supervision of researchand honours students, and on-going curric-ulum development.
The University
Eric gave extensive and dedicated serviceto the University in terms of administra-
Figure 1. Professor Barnes lecturing on geometry of numbers in 1982.
Eric Stephen Barnes 1924–2000 27
tion, committee service and service onbehalf of the University to outside bodies.Apart from his terms as Head of Depart-ment, Faculty Dean and Deputy Vice-Chancellor, his service included terms asChief Examiner in Mathematics for thePublic Examinations Board (for manyyears), Chairman of the Board itself,Chairman of the Board of ResearchStudies, Chairman of the Education Com-mittee (now the Academic Board), andelected member of the University Council.
Professional service
During his time in Adelaide, Eric wasactive in the wider mathematical andscientific community.
Through his early work as Chief Exam-iner in Mathematics and the associatedchairmanship of the Mathematics SyllabusCommittee, he soon established friendlyrelations with local school mathematicsteachers and went on to play a leading rolein connection with school mathematics.With the support of A.W. Jones of the StateEducation Department, Eric was respon-sible for establishing the MathematicalAssociation of South Australia (the stateprofessional association of mathematicsteachers) and became its foundation Presi-dent (1959–61). He gave practical help toteachers through presentations at confer-ences of the Association and especially bywriting a useful textbook (see (47) in theBibliography below) jointly with mathe-matics teacher Bruce Robson.
Eric was a foundation member of theAustralian Mathematical Society (from1956) and served the Society in a numberof roles, including President (1962–64),Council Member, Director of the SummerResearch Institute (1962), and member ofthe Editorial Board, and later AssociateEditor (1967–1974), of the Society’sJournal. While in Adelaide he also servedthe Australian Academy of Science asCouncil Member (1962–64) and Secretary,Physical Sciences (1972–76).
Teaching
During the early 1960s, both the new pro-fessors taught mathematics at first- andsecond-year levels. This helped to establishthe curriculum and attract the interest ofstudents. Eric’s meticulous lecture noteswere extremely useful to the lecturers whotook over subjects he had taught and set ahigh standard of preparation.
The outstanding quality of Eric’s teach-ing was immediately recognised. Both inthe early 1960s and in 1982, students par-ticularly appreciated his teaching ofsecond-year Pure Mathematics. EugeneSeneta (now Professor of MathematicalStatistics at the University of Sydney)writes1 (see References below) that ‘heheavily influenced the direction of myfuture work through his lucid teaching ofPure Mathematics 2 in 1961’.
Much of Eric’s teaching was at third-year and honours levels, and his honourscourses attracted graduate students andstaff members as well as honours students.While most of his honours courses were ontopics in number theory, including thegeometry of numbers, there were occa-sional exceptions. A notable example washis 1962 course on Linear Inequalities,mentioned by several of those whoattended, including Seneta. Eric’s interest inthis then new topic had been stimulated byA.W. Tucker on a visit to Sydney, and it washighly relevant to his own research. Thecourse was an eye-opener and formed thebasis of extensive and continuing researchon mathematical programming by some ofthe more applied students and staff.
Research and scholarly work
In the decade 1962–72, Eric supervised fivesuccessful PhD students, Tom Dickson,Paul Scott, Rod Worley, Peter Blanksby (fortwo years) and Dennis Trenerry. Heassisted in supervising the MSc work ofstaff member Marta Sved, who went on tofurther research and a PhD, and also super-vised the MSc work of the late Christopher
28 Historical Records of Australian Science, Volume 15 Number 1
Nelson, who submitted his one paper2 in1972, the year before, sadly, he wasdrowned in floods in New South Wales. Allsix of Eric’s PhD students proceeded tocareers in university mathematics and tofurther research, and three, Pitman, Scottand Blanksby, joined the Adelaide depart-ment. He gave his research students signifi-cant and challenging research problems,and their publications bear witness to hisinfluence. Paul Scott (who recently retiredas Associate Professor at the University ofAdelaide) writes that ‘Eric’s main legacy tome was the ability to write clear andconcise mathematics’ and ‘I feel privilegedto have had Eric first as my mentor, andlater as a colleague’.
As might be expected in view of hismany other commitments, Eric’s ownresearch output slowed markedly after hetook up the Chair in Adelaide. However,during the next twelve years or so, his workgained increasing recognition, and papersin the geometry of numbers by his students(some joint with Eric) appeared at a steadyrate. Leading international number theo-rists visited Adelaide, and Eric’s involve-ment ensured that the newly establishedJournal of the Australian MathematicalSociety published papers in number theoryby both international and local authors. Bythe early 1970s, Adelaide was among thehalf dozen internationally recognisedcentres of progress in the geometry ofnumbers, of which the largest was theMoscow/Leningrad school centred on theSteklov Institute in which B.N. Delone andS.S. Ryškov were leading figures.
An Australian Research Grants Commit-tee award enabled Eric to appoint a Post-doctoral Fellow for 1974–76 for a projectrelated to his recent research and to corre-spondence with Ryškov. The appointee wasMichael Cohn (former research student ofPaul Scott and of C.A. Rogers, London),and the project was very successful.
While at Adelaide, Eric had threeperiods of overseas leave. In 1965, he and
his family spent a year of study leave inKuala Lumpur and Bangkok, part of theaim being to further mathematics educa-tion in the region. In 1975 he had a two-month overseas study tour as Deputy Vice-Chancellor, and in 1981 he had a full yearof study leave. Early in 1981 he met NeilSloane (then of Bell Telephone Labora-tories, New Jersey), with whom he beganresearch collaboration. A visit to the USand Canada later in the year providedfurther opportunity for work with Sloane.
Personal
Eric Barnes was an exceptionally quickand incisive thinker with an excellentmemory. He devoted time and hard work toany matter which he took up and had aremarkable ability to master complexdetail and identify the essentials. He wasalso a gifted expositor whose presentationswere clear, logical and appropriate to theiraudiences. These qualities underlay hismathematical research and teaching andcontributed greatly to his work in adminis-tration and academic management.
Eric’s intellectual abilities could attimes be daunting, and by nature he had alow tolerance for inaccuracy. In debate onlarge University committees, these charac-teristics sometimes led to an abrasive styleof argument.
Eric gained and retained the respect andaffection of his colleagues in pure mathe-matics. While he could argue cogently forhis own point of view at departmental meet-ings, he was not dominating and was verymuch the opposite of a ‘God-Professor’. Hehad a genuine concern for students at alllevels and took a special interest in studentsfrom overseas, mathematically gifted stu-dents, and those who did not quite fit thesystem. His colleagues found himapproachable, supportive and encouraging,particularly to new arrivals, those with lessexperience, and new researchers.
Among Eric’s recreational interestswere music, bridge, chess, reading and a
Eric Stephen Barnes 1924–2000 29
love of language and words. He couldlighten the atmosphere with a witty turn ofphrase and he remained fluent in idiomaticFrench. He was a competent pianist andthose who were there remember a happyMathematics Department party at thehome of the Potts family, around 1963,when musical entertainment was providedby a trio (Eric Barnes, piano, GeorgeSzekeres, viola, Ren Potts, clarinet) andnovice pianist Maurice Brearley.
In 1984 the E.S. Barnes Prize (for third-year Pure Mathematics) was established,thanks to former students, friends and col-leagues, in recognition of his contributionto pure mathematics.
Later Years
From the 1960s, Eric had to cope withsome health problems, in particular, achronic respiratory condition. By the early1980s these problems had increased andthis was a factor in his early retirement.After retirement his health gradually dete-riorated, but he maintained contact withthe Pure Mathematics Department as anHonorary Visiting Research Fellow andparticipated actively in the weekly NumberTheory Seminar until 1992.
In his last few years Eric was house-bound and had occasional visits to hospi-tal. He died on 16 October 2000.
Mathematical Work
Apart from his admirable note (41) (seeBibliography below) and his contributionsto the books (47), (48) and (49), Barnes’smathematical publications belong to thegeometry of numbers. We start with somenecessary background and then indicatethe main themes of his work.
Background
Classical problems in number theoryinclude not only Diophantine equations butalso Diophantine inequalities. A typicalhomogeneous Diophantine inequality ofdegree 2 is
|x2 + xy + y2| ≤ 1,
where we seek non-null integral solutions,that is, solutions x = u, y = v such that u andv are integers (0, ± 1, ± 2,…), not bothzero. There are also correspondinginhomogeneous inequalities involving apolynomial in x and y whose terms are notall of the same degree.
Consider a plane with a standard rectan-gular Cartesian co-ordinate system. The‘integral lattice’, which consists of allpoints whose co-ordinates are integers, is aparticular example of the concept of‘lattice’. Investigation of a problem onDiophantine inequalities in two variables isoften helped by consideration of an equiva-lent geometrical problem on lattice pointsin a region of the plane.
More general Diophantine inequalitiesinvolve forms of degree k in n variables.These are homogeneous polynomials ofdegree k with real coefficients, in n realvariables. Such a form is binary if n = 2,ternary if n = 3. A form is quadratic if ithas degree 2, integral if its coefficients areintegers, indefinite if it takes both positiveand negative values, and positive, or posi-tive definite, if its value is always positiveexcept when all variables are zero.
The geometry of numbers is a majorbranch of number theory that was intro-duced by Minkowski in the 1890s. Thesubject deals with n-dimensional latticesand their relationship to bodies inn-dimensional space for all n ≥ 2. Itprovides an effective geometrical frame-work for many problems on Diophantineinequalities and has important applicationsboth within and beyond number theory.
Main themes
Barnes’s research publications can be con-veniently grouped under three headings:
Indefinite forms — Part 1,Indefinite forms — Part 2,Positive quadratic forms and lattices.
30 Historical Records of Australian Science, Volume 15 Number 1
The work on indefinite forms began inCambridge, continued in Australia, mainlyduring the 1950s, and provided topics forthe theses of Pitman, Worley and Blanksby.Part 1 of this work deals with homo-geneous Diophantine inequalities forindefinite forms and uses purely arithmeticmethods. Part 2 deals with inhomogeneousinequalities for indefinite binary quadraticforms and related topics, and uses two-dimensional lattices.
From the mid-1950s onwards, Barnes’smajor research interest was in positivequadratic forms in n variables andn-dimensional lattices. This theme pro-vided topics for the theses of Scott,Dickson, Trenerry and Nelson, and also forthe major research project with Cohn. Theproblems Barnes considered include bothhomogeneous and inhomogeneousinequalities for positive forms. These areequivalent to problems on packing andcovering of n-dimensional space withequal spheres whose centres are at thepoints of a lattice.
Barnes was interested in solvingspecific problems rather than in developingabstract theory, and his preferred mathe-matical tools were those of discrete mathe-matics and geometry. The researchproblems in the geometry of numberswhich he addressed fitted well with thesetastes, gave scope for all his intellectualabilities and mathematical powers, andrequired, in addition, a high degree ofcreative insight. The results he obtained arealmost all in some sense best possible, andall his papers bear the hallmarks of hisstyle: clarity, economy and beautifulorganization.
We shall discuss Barnes’s work on eachof the three main themes further below.The papers considered in more detail havebeen selected to reflect the range of hismain work. We give some background toplace the work in context.
References
References (1), (2), etc., are to the Biblio-graphy of Barnes’s publications at the endof this article. References such as Cassels3
are to the list of References immediatelybefore the Bibliography. Standard refer-ences for the geometry of numbers areCassels3 and Gruber-Lekkerkerker4. Refer-ences such as GL, 17, or GL, xi, are tosections (17 or xi) of Gruber-Lekkerkerker4.
Indefinite Forms — Part 1
Background on indefinite binary quadratic forms
We denote by R2 the space of all realvectors x = (x, y) (viewed as points of theplane) and by Z2 the lattice of all integralpoints u = (u, v). Consider a binary quad-ratic form
f = f(x, y) = ax2 + bxy + cy2
with real coefficients a, b, c and discrimi-nant d(f) = b2 – 4ac 0. We investigate thevalues f(u) for u = (u, v) in Z 2, andparticularly the homogeneous minimum
M(f) = inf |f(u, v)| (u, v integers, not both 0),
where ‘inf’ means infimum, or greatestlower bound.
A linear transformation T on R2 mapsZ2 onto itself if and only if its 2×2 matrixhas integral entries and determinant ± 1.We shall call such a transformation T anautomorph of Z2. A binary quadratic formg is equivalent to f if g(x, y) is identicalwith f[T(x, y)] for some automorph T of Z2.In this case, d(g) = d(f), the set of all valuesg(u) with u in Z2 coincides with the set ofall f(u) with u in Z2, and M(g) = M(f). Thevalue of M(f) / |d(f)|1/2 is unchanged if f isreplaced by a form equivalent to a non-zero multiple of f. The concepts of homo-geneous minimum and equivalence areeasily extended to any form in any numberof variables.
Eric Stephen Barnes 1924–2000 31
For a form f as above with discriminantd(f)=d and homogeneous minimumM(f)=M, suppose that f is indefinite (d >0).Then
M ≤ 5–1/2 d1/2,
with equality if and only if f is equivalentto a multiple of F0 = x2 – xy – y2. However,if equality does not hold, then
M ≤ 8–1/2 d1/2,
so that the constant 5–1/2 is ‘isolated’. Thisfirst example of isolation was discoveredby Korkine and Zolotareff in 1873, and ledto Markoff’s major study published in1879–80. Markoff found a remarkable infi-nite sequence of forms F0, F1, F2, … (nowthe Markoff forms). The strictly decreasingsequence of values Md–1/2 for these formsstarts with 5–1/2, 8–1/2, 5(221)–1/2,… andconverges to 1/3. Markoff’s main result isthat M is at most d1/2/3 unless f is equiva-lent to a multiple of some Markoff form.
Much more is now known about thestructure of the set of all possible values ofMd–1/2, which provides a standard of com-parison for the ‘spectra’ of constantsarising in other problems. (The reciprocalsd1/2 M–1 form the ‘Markoff spectrum’.)
For the work above, Markoff used anearly version of the now classical contin-ued fraction method. Since much ofBarnes’s work on indefinite forms relies onthis method, we indicate the main ideas.
Suppose again that f = ax2 + bxy + cy2 isindefinite with d(f) = d >0 and M(f) = M.The equation ax2 + bx + c = 0 has realsolutions θ, Φ with |θ| ≤ |Φ|, called theroots of f. We assume that a ≠ 0 and theroots are irrational (otherwise M = 0).Starting from a suitable equivalent form f0with roots θ0, Φ0, we obtain a two-wayinfinite sequence of positive integers
…, a–2, a–1, a0, a1, a2,…
from the regular (or ‘ordinary’) continuedfractions
–Φ0 = (a1; a2 a3,…)
,
1/θ0 = (a0; a–1, a–2, …).
The ai determine a correspondingsequence of equivalent ‘reduced’ forms fi(these are unrelated to the Markoff formsFi).
If u, v are integers, not both zero, suchthat |f(u, v)| < d1/2/2 then f(u, v) is equal tofi(1, 0) for some i. Since there is a simpleformula for fi(1, 0) in terms of continuedfractions, this gives us a powerful tool forstudying these values f(u, v).
Restricted homogeneous minima of indefinite forms
The papers (2) to (7) cover Barnes’s sub-stantial early research in Cambridge [apartfrom (1)]. The interrelated papers (2) to (6)deal with ‘restricted’ minima of certainindefinite forms, and the same circle ofideas includes (7) and the later paper (17).All of this work depends on masterly useof the continued fraction method andsequences of reduced forms, for relevantbinary quadratic forms.
The restricted homogeneous minimumof a form f(x, y, z, t) in four variables is theinfimum of the values of |f(x, y, z, t)| atintegral values of the variables satisfyingxt – yz = ±1. Within this context, we denotethis restricted minimum by M(f).Davenport and Heilbronn had studiedM = M(f) for f = (ax + by)(cz + dt) where ∆= ad – bc ≠ 0. (See last paragraph of GL43.2.) In contrast to Markoff’s results forbinary forms, they found that the thirdlargest value of M ∆–1 is not isolated.
In (2) to (4) Barnes considered the formf(x, y, z, t) given by f = q(x, y)q(z, t) where qis an indefinite binary quadratic form withdiscriminant d=d(q). In (2), he studied thevalues of Md–1, where M = M(f). He founda sequence of quadratic forms Q–1, Q0, Q1,Q3, Q5,… and demonstrated a remarkable
= a1 + 1
a2 + 1/(a3 + …)
32 Historical Records of Australian Science, Volume 15 Number 1
analogue of the Markoff phenomenon forthe forms hi = Qi(x, y)Qi(z, t): The values ofMd–1 for the hi form a strictly decreasingsequence converging to a specified limit Cand Md–1 is at most C except when q isequivalent to a multiple of some Qi. Thepapers (3), (4) deal with restricted minimaassociated with ‘asymmetric’ and ‘one-sided’ inequalities for f as above. Theyinclude further instances of Markoff phe-nomena, some of which contrast withknown results for binary forms.
In (5), Barnes proved the theorem ofDavenport and Heilbronn mentioned aboveby the methods of (2). This led to his studyin (6) of best possible upper bounds for therestricted homogeneous minimum M(f) ofa more general ‘bilinear’ form f(x, y, z, t).
One-sided inequalities for indefinite quadratic forms
Three 1955 papers on indefinite quadraticforms stemmed from a visit to Sydney byA. Oppenheim. One, (17), was mentionedabove. Before discussing the others, weneed some further vocabulary.
The matrix of a ternary quadratic formf(x1, x2, x3) is the 3×3 symmetric matrixA = [aij] such that
f(x) = f(x1, x2, x3) = Σ aij xi xj ,
with summation over all i, j such that 1 ≤ i≤ 3, 1 ≤ j ≤ 3. The determinant of f is detA(the determinant of A). If this is non-zerothen f can be expressed as ± L2
1 ± L22 ± L2
3 ,where the Li are linear forms, and f is saidto be of type (r, s) when there are r plussigns and s (=3–r) minus signs. If f isindefinite, its non-negative minimum M+(f)is the infimum of the non-negative valuestaken by f at integral values of the varia-bles, not all zero, and its positive minimumis defined similarly. (These may differbecause there may be non-null integralsolutions of f(x)=0). These concepts extendto quadratic forms in n variables for any n.
In 1953 Oppenheim had given best pos-sible upper bounds for the positive minima
for n = 3, 4 (see GL, 44.4). In (18), Barnesaddressed the more difficult correspondingproblem for M+(f), and this work wasfurther extended in the joint paper (20)with Oppenheim. For n ≥ 2, r ≥ 1 and s ≥ 1,we consider all indefinite quadratic forms fin n variables of type (r, s) with non-zerodeterminant D. The constant κr, s is the leastpositive constant such that for all suchforms f we have
M+(f) ≤ (κr, s |D|)1/n.
The cases n = 2, 3, 4 are the only onesof interest since, thanks to the 1987 break-through of Margulis (see Dani andMargulis5) we now know that the positiveminimum of f is zero if n ≥ 5.
For n = 2, it was known (see GL, xiv.4)that κ1, 1 is 4 and is not isolated. In (18),Barnes obtained a key lemma on positivevalues of indefinite binary quadratic formsby the continued fraction method. Usingthis, he showed that κ2,1 is 4/3 and isisolated, and derived upper bounds in theother cases with n = 3, 4. The co-operationwith Oppenheim yielded the values of κ1, 2and κ2, 2.
Later, in his thesis, Worley studiedindefinite ternary quadratic forms andachieved major progress on asymmetricand one-sided inequalities (see Worley6, 7).
The work on κr, s was carried further byWorley8, and by Jackson, who, in particu-lar, evaluated κ3, 1 (see GL, xiv, for refer-ences). It seems that κ1, 3 has still not beenprecisely determined.
As well as being of independent inter-est, results on one-sided and asymmetricinequalities often play an important role asauxiliary results in other problems. Forexample, the value of κ2, 1 was essential toBarnes’s later work in (22) and was usedsimilarly by Raka9.
Indefinite Forms — Part 2
Let f = f(x, y) be an indefinite binaryquadratic form with discriminant d(f) >0.For each real αααα = (α, β), let
Eric Stephen Barnes 1924–2000 33
m(f; α, β) = inf |f(u + α, v + β) | (u, v integers),
and write
m(f; α, β) = m(f, αααα).
If (α', β') ≡ (α, β) (mod 1), that is α' – αand β' – β are integers, then m(f; α', β') isequal to m(f; α, β). The inhomogeneousminimum m(f) is the supremum, or leastupper bound, of the values of m(f; α, β) forall real α, β:
m(f) = sup m(f; α, β) (α, β real).
The behaviour of m(f) under equivalenceof forms is exactly similar to that of thehomogeneous minimum M(f).
The inhomogeneous minimum of anyform in any number of variables is definedsimilarly. The concept arises naturally inalgebraic number theory. For quadraticforms it can be interpreted in terms oflattice coverings of Euclidean space.
We now look at the contributions ofBarnes related to inhomogeneous minimaof indefinite binary quadratic forms.
Two-dimensional lattices and Minkowski’s theorem
Let s1x + t1y and s2x + t2y be two real linearforms whose coefficient matrix
has non-zero determinant. The product
f = (s1x + t1y) (s2x + t2y)
is an indefinite binary quadratic form withdiscriminant (detL)2. Every indefinitebinary quadratic form can be factorised inthis way and the linear factors are uniqueup to suitable multiples.
The two-dimensional lattice determinedby the two given linear forms, or equiva-lently by the matrix L, consists of all pointsw = (w1, w2) such that for some integral uand v we have
w1 = s1u + t1v ,
w2 = s2u + t2v .
(The vectors s = (s1, s2), t = (t1, t2) form a‘basis’ of Λ.) If Λ is also determined by amatrix L' then |detL'| = |detL|, and so thedeterminant of Λ is uniquely defined asdet Λ = |detL|.
Minkowski proved the followingseminal theorem geometrically in the1890s. Let Λ be the lattice determined bytwo real linear forms L1(x, y), L2(x, y) withnon-zero determinant. Then for each pointγ = (γ1, γ2) in the plane there is a pointw = (w1, w2) of Λ such that
|(w1 + γ1)( w2 + γ2)| ≤ 1/4det Λ.
Equivalently, in terms of forms: Let f(=L1L2) be an indefinite binary quadraticform with discriminant d(f) = d. Then foreach real αααα = (α, β) there is an integralvector u such that
|f(u + αααα)| ≤ 1/4d1/2.
Thus m(f, αααα) is at most d1/2/4 for all real ααααand the inhomogeneous minimumm = m(f) is at most d1/2/4. Minkowski’sconstant 1/4 is best possible but is notisolated.
Interest in improving Minkowski’stheorem led to detailed study of particularforms f and investigation of alternativeupper bounds for m(f). Heinhold in 1939and others studied the ‘principal normforms’ of real quadratic fields. (See GL,47.5.) These are of special interest becausetheir inhomogeneous minimum is con-nected with the question of whether or notthe field is Euclidean, that is, its domain of‘integers’ has a ‘Euclidean algorithm’ (andhence has the unique factorisationproperty).
In (1), Barnes obtained an upper boundwhich is often sharper than Minkowski’s:If f = ax2 + bxy + cy2 is indefinite withinhomogeneous minimum m = m(f) then
m ≤ 1/4max |a|, |c|, min |a ± b + c|.
Ls1 t1
s2 t2
=
34 Historical Records of Australian Science, Volume 15 Number 1
He gave some significant applications,including simple verification of mostknown examples of Euclidean realquadratic fields. Related work is discussedin GL, 47.4.
The automorph method
While in Cambridge, Barnes pursued thetheme of (1) further in major collaborativeresearch with Swinnerton-Dyer. They usedtwo different approaches, both highly geo-metrical, and from then onwards Barnes’swork was underpinned by geometricalideas, even when the details were arithmet-ical.
In the joint papers (8),(9) they studiedthe inhomogeneous minima of norm formsof real quadratic fields. For square-freeintegral k >1, the field Q( ) is obtainedby adjoining to the field Q of rationalnumbers and its principal norm form is
gk = x2 + xy – 1/4(k – 1)y2
if k – 1 is divisible by 4, and, otherwise,
gk = x2 – ky2.
For this purpose, Barnes andSwinnerton-Dyer developed what we shallcall the automorph method, a general geo-metric method applicable to integral forms(or multiples thereof).
Consider an integral indefinite form f =ax2+bxy+cy2 with a ≠ 0, irrational roots,and inhomogeneous minimum m = m(f).An automorph of f is an automorph T of Z2
such that the form f [T(x, y)] is identicalwith ±f(x, y). The integral form f hasautomorphs T of infinite order, and themethod is based on the fact that, for such Tand real αααα = (α, β), we have
m[f; T(αααα)] = m(f; αααα).
An important role is played by the orbit(mod 1) of αααα under T for certain ‘excep-tional’ points αααα. In (8), Barnes andSwinnerton-Dyer obtained a group oftheorems which together provide a firmtheoretical basis for finding m by this
approach. In (8) and (9) they also gave somegeneral theorems on m(f; αααα) for given inte-gral f. In particular, the set of all values ofm(f; αααα) is closed.
For real f, let m2 = m2(f) be the supre-mum of the values of m(f; αααα) which arestrictly less than m = m(f). Clearly m2 ≤ m.If m2 < m then the minimum m is isolatedand m2 is the second (inhomogeneous)minimum of f. If m2 is isolated there will bea third minimum m3 < m2, and so on. Asconjectured in (9) and later shown byGodwin,10 the second minimum m2 of fneed not be isolated even if f is integral.
Barnes and Swinnerton-Dyer used theautomorph method to study m = m(f) forthe norm form f = gk in many cases,including all square-free k ≤ 101. In (8)they presented theorems covering differentpossibilities and tabulated their results,many of which were new, with referencesfor known results. In all but a handful ofcases, they evaluated m and found thepoints at which m(f; αααα) takes this value,and often they also evaluated m2. Later, bya modification of their method, Godwin11
filled the remaining gaps for k ≤ 101.In (9), Barnes and Swinnerton-Dyer
extended the automorph method to dealwith an infinite sequence of minimam = m(f), m2, m3, … and studied the normforms f = gk for k = 11, 13 in detail. Fork = 11, they found an analogue of theMarkoff phenomenon, and their results aresimilar to those obtained earlier byDavenport for k = 5 and Varnavides fork = 2 (see GL, end of 47.5, for references).The norm form f for k = 13 has unusualproperties, and the results for this case aresurprisingly complicated.
Barnes and Swinnerton-Dyer conjec-tured in (9) that for all integral indefinitebinary quadratic f the inhomogeneousminimum m is always rational, isolated,and taken by m(f; αααα) at some rational pointαααα = (α, β) with α, β both rational. SeeBerend and Moran12 for progress towardsthis.
kk
Eric Stephen Barnes 1924–2000 35
Euclidean real quadratic fields
For square-free k >1, the real quadraticfield Q( ) with norm form f = gk isEuclidean if and only if m(f; αααα) <1 for allrational points αααα = (α, β). A 1951 theoremof Davenport led via this criterion to com-pletion of the proof that there are nofurther real Euclidean quadratic fieldsbeyond those on the ‘classical’ list of 17Euclidean fields, all with k <101. (SeeEnnola13 and GL 48.3.) The results of (8)(9) and Godwin11 confirmed these conclu-sions for all square-free k ≤ 101, with oneexception. Barnes and Swinnerton-Dyershowed in (8) that the last field on the list,Q( ), had been wrongly listed asEuclidean. In 1958 Ennola13 improved theDavenport result and drew on ideas fromseveral articles [in particular (1) and (8)] togive the first unified proof of the correct-ness of the final list of 16 values of k forwhich Q( ) is Euclidean. Barnes madefurther contributions related to this topic inthe later papers (31) and (39).
Background on Delone’s geometric divided cell algorithm
The second method of Barnes andSwinnerton-Dyer is the divided cellmethod, based on a geometric algorithmfor inhomogeneous problems introducedby Delone14 in 1947. (See GL, 48.1 for adetailed account. The automorph methodand the significant further developmentsdiscussed below are not covered in GL.)
Delone’s algorithm is formulated interms of grids. A grid Γ associated with atwo-dimensional lattice Λ is a translate ofΛ, Γ = γ + Λ, for some specified real pointγ = (γ1, γ2). The points of Γ and the linesthey determine are called grid points andgrid lines, and the determinant of Γ isdetΛ. A parallelogram P whose four verti-ces are grid points is a cell of Γ if there areno other grid points inside P or on itsboundary, or, equivalently, if P has areadet Γ. A cell is divided if, further, it has
one vertex in each of the four quadrantsdetermined by the axes x = 0, y = 0.
Suppose a grid Γ has no grid point oneither of the axes and no grid line parallelto an axis. Delone showed that Γ has atleast one divided cell, C0, say, and gave asimple geometric algorithm which startsfrom C0 and yields a two-way infinite chainof divided cells which includes all dividedcells of Γ:
…, C–2, C–1, C0, C1, C2,…
He obtained basic results on divided cellsand gave some significant applications.
Inhomogeneous problems on indefinitebinary quadratic forms are equivalent toproblems on the value of xy at grid points(x, y) of an appropriate grid Γ. The dividedcell algorithm is important because inorder to determine the infimum of |xy| forall grid points, or for all grid points in acertain quadrant, it is sufficient to considergrid points which are vertices of dividedcells.
The divided cell method for asymmetric inequalities
For real τ ≥ 1, consider grids Γ with no gridpoint in the asymmetric region –1 <xy <τ,and let D(τ) be the infimum of their deter-minants. In (15) Barnes and Swinnerton-Dyer extended the divided cell method tostudy D(τ) (see GL, 50.1, regarding relatedwork). They first gave a full presentation ofthe basic theory and obtained formulas forthe vertices of the divided cells in terms oftwo sequences of integers ai and εi whicharise from Delone’s construction. Theresults involve ‘semi-regular’ continuedfractions of a special type, together withseries expansions.
A grid is symmetric if one of its cellshas the origin (0, 0) as its centre. Suchgrids have all εi = 0 and are easier tohandle. Barnes and Swinnerton-Dyerproved that symmetric grids are sufficientfor evaluation of D(τ) and extended thetheory further for this type of grid. They
k
97
k
36 Historical Records of Australian Science, Volume 15 Number 1
obtained lower bounds for D(τ) and illus-trated both the power of the method andthe complex nature of D(τ) by preciseevaluation of D(τ) for τ near 2. In 1991,Grover and Raka15 revisited this work,filled a gap in (15), and used the methodfor further detailed study of D(τ).
Davenport had used results on inhomo-geneous asymmetric inequalities forbinary forms to obtain the analogue ofMinkowski’s constant for an indefiniteternary quadratic form f(x, y, z) with non-zero determinant (see GL, 49.4). In themajor papers (14) and (22), Barnes usedthe divided cell method for asymmetricinequalities to carry this work much fur-ther. In (32), without using (15), he alsogave another proof of a result of Blaney onone-sided inhomogeneous inequalities forindefinite ternary quadratic forms (see GL,50.3). In 1993, Raka9 obtained a majorextension of this result by using some ofthe work of Grover and Raka mentionedabove, together with other auxiliaryresults.
The divided cell method for inhomogeneous minima of forms
Let f(x, y) be a real indefinite binary quad-ratic form with irrational roots andinhomogeneous minimum m = m(f). Theevaluation of m is equivalent to a problemon vertices of divided cells in terms ofassociated sequences of integers ai and εi.However, this is more difficult than theproblem in (15) because infinitely manydifferent sequences of ai must be consid-ered and the εi are in general non-zero. Inthe important paper (16), Barnes overcamethese difficulties by deriving further theo-retical results on the divided cell method.He thus developed the method as an arith-metical tool for the study of m(f) for realforms f as above. He illustrated its advan-tages over the automorph method byapplying it to the relatively difficult normforms f = gk with k = 19 and k = 46(correcting (8) for k = 19).
In (21) Barnes modified the divided cellmethod to deal with the inhomogeneousDiophantine approximation constantsk(θ, β), k+(θ, β), for positive irrational θ andsuitable real β. These constants are relatedto Diophantine inequalities of the type
|x(θx – y – β)| <C
with the conditions x ≠ 0 or x >0. As easyapplications, Barnes gave short proofs oftwo parallel theorems on these constants.The first strengthened a much earlier resultof Morimoto (Fukasawa)16 (whose otherwork seems to have been neglected in theliterature).
In their theses Pitman and Blanksbygave further major applications of thedivided cell method and supplemented thetheory (see Pitman17, 18, Blanksby19, 20).
Recent developments
In the decade from 1973, research oninequalities for indefinite binary quadraticforms concentrated mainly on homo-geneous problems involving the Markoffspectrum and related topics. Since about1990 there has been renewed interest in theinhomogeneous problems investigated byBarnes, with emphasis on the whole spec-trum of values of m(f; αααα) or k(θ, β) or k+
(θ, β) for fixed f or θ. In particular, WilliamMoran (who succeeded Barnes in the Chairof Pure Mathematics at the University ofAdelaide) and his collaborators have con-tributed in this area. The divided cellmethod remains a powerful tool, andinvestigation of new approaches to theautomorph method and the ideas of (8), (9)and (15) has begun. Berend and Moran12
used the methods of topological dynamicsto study the values of m(f; αααα) for indefinitebinary quadratic f. Recent developmentsare further illustrated by the work of Groverand Raka15 and the paper of Raka9 men-tioned earlier, and by the papers of Cusick,Moran and Pollington21 and of Pinner22 oninhomogeneous approximation.
Eric Stephen Barnes 1924–2000 37
Positive Quadratic Forms and Lattices
From the mid-fifties on, Barnes’s majorresearch interest was in positive quadraticforms in n variables and their associatedlattices of points in n-dimensional space.The standard reference work in the area isConway-Sloane.23
Mathematical background
For n ≥ 3, n-dimensional lattices are anatural extension of the case n = 2 dis-cussed earlier. We give an informalaccount of the geometry of 3-dimensionallattices. In the process, some of the basicideas of the general n-dimensional theorywill be introduced.
Consider 3-dimensional space as inEuclidean geometry with one point O per-manently selected as origin, and let A,B,Cbe points such that O,A,B,C are not copla-nar. The (3-dimensional) lattice Λ withbasis , , consists of all pointsP such that
for some integral u, v, w. The parallel-epiped Π determined by O and the basisvectors is the solid body made up of allpoints Q such that
for real numbers x, y, z in the interval [0, 1].A parallelepiped is a cell of Λ if its 8
vertices are all lattice points (i.e. points ofΛ) and it has the same volume as Π. Inparticular, Π itself is a cell, and every cellwith O as a vertex is obtained similarlyfrom some basis , , .
A lattice is unimodular if its cells havevolume 1. Since every lattice is just anexpanded or contracted version of a uni-modular one, it is often appropriate toconfine attention to unimodular lattices.Two lattices are said to belong to the samecongruence class if they are congruent inthe usual sense of Euclidean geometry and,if so, their cells have the same volume.
We now look more closely at the geo-metry of an individual lattice Λ. Expandeach lattice point to a sphere of fixedradius r, assuming for the moment that notwo spheres overlap. Then the proportionof space covered by the spheres is just thevolume of a sphere divided by the volumeof a cell. Even when the spheres overlap,this number still makes sense as a ‘cover-ing ratio’.
Two cases are of particular interest.They are related to two lattice constantsM = M(Λ) and m = m(Λ). To simplify thepresent discussion, we assume that Λ isunimodular, so that the covering ratioabove becomes simply 4πr3/3.
The more obvious constant, M, is theminimum squared distance between dis-tinct lattice points. When r = M1/2/2,certain of the spheres will touch but no twowill actually overlap. In other words, thisvalue of r provides the closest packing ofequal spheres with centres at the latticepoints. The corresponding packing densityis 4π(M1/2/2)3/3 = πM3/2/6.
The mathematical problem that ariseshere is to determine the congruence class(or classes) for which the packing densityis greatest. Gauss solved this problem in1831, confirming the conjecture that themaximum packing density is π /6 =0.74….
In defining the other constant, m, weborrow an image from Conway-Sloane: thepoints of space are where children live,schools are at the lattice points and a childattends the nearest school; m1/2 is then thegreatest distance any child has to go toschool. A little thought shows that r = m1/2
is the smallest radius for which the spherescover the whole of space. In other words,this value of r yields the thinnest coveringof space by equal spheres with centres atthe lattice points. The covering density is4πm3/2/3. The corresponding mathematicalproblem turns out to be harder than thepacking problem. Only in 1954 was it
OA OB OC
OP uOA vOB wOC+ +=
OQ xOA yOB zOC+ +=
OD OE OF
2
38 Historical Records of Australian Science, Volume 15 Number 1
shown by Bambah that the minimumcovering density is 5 π/34 = 1.46 … .
The above ideas can be translated intothe language of coordinate geometry. Thisprovides the link to positive quadraticforms and is an essential step in generaliz-ing the theory to higher dimensions.
Let Λ be a lattice with basis , , as described earlier. Real coordinates
x = (x, y, z) are then assigned to each pointP of space via the vector equation
,
the points of Λ itself being those withintegral coordinates. If the basis vectorsare mutually orthogonal vectors oflength 1, we have an ordinary rectangularCartesian coordinate system and the dis-tance OP is given by the familiar formulaOP2 = x2 + y2 + z2. In general, OP2 is amore complicated quadratic form f(x) in x,y, z. By its meaning, f is positive. It is astandard result of linear algebra that everypositive quadratic form arises in this wayfrom a suitable lattice Λ.
We say that the form f above is adistance function of Λ and that Λ is alattice associated with f. It can be shownthat the determinant of f is the square of thevolume of a cell of Λ. Accordingly, f iscalled unimodular if its determinant is 1.
Consider another basis , , of Λ and let the new coordinates of P andcorresponding distance formula be X=(X, Y, Z) and OP2 = F(X). The generalrelation between X and x is that each of X,Y, Z is an integral linear combination of x,y, z and vice versa. The forms f and F areaccordingly equivalent in the sense used inearlier sections. The forms equivalent to fmake up its equivalence class.
These considerations suffice to translateproblems about lattices into problemsabout positive quadratic forms. It can beseen that, if f is a distance function of aunimodular lattice Λ with lattice constantsM(Λ) and m(Λ), then M(Λ) = M(f) andm(Λ) = m(f), where M(f) and m(f) are the
homogeneous and inhomogeneous minimaof f as defined in earlier sections. Thereforethe packing problem for lattices is equiva-lent to determining those equivalenceclasses of unimodular positive quadraticforms for which the homogeneousminimum is greatest. The coveringproblem translates similarly.
We conclude with a brief indication ofhow the 3-dimensional theory is general-ized to higher dimensions. By definition,the points P of n-dimensional Euclideanspace, Rn, are the n-tuples x = (x1, …, xn)of real numbers and distances from theorigin O are given by OP2 = x1
2 + … + xn2.
Now let a1, … , an be an algebraic basis ofRn, so that each x = X1a1 + … + Xnan withunique coefficients X1,…Xn. Then X =(X1, … , Xn) is the coordinate vector of Pwith respect to this basis and OP2 is apositive quadratic form F(X) in the newcoordinates. Those points P whose coordi-nates X1,…,Xn are integers form the latticewith the above basis.
All that has been said about the3-dimensional case carries over in areasonably straightforward way to then-dimensional case. In particular, thecounterparts for positive quadratic formsof the covering and packing problems forlattices are classical problems of numbertheory.
Barnes’s papers on positive quadraticforms and lattices will now be discussedunder the appropriate subject headings.
Lattice packings of spheres
All but one of Barnes’s nine papers in thisarea were written in the period 1955–58during his time at the University ofSydney. The ninth, a joint paper withSloane, appeared in 1983, the year ofBarnes’s retirement. Although related tothe last of the earlier papers, its back-ground is in the theory of error-correctingcodes, a subject that came to maturity onlyin the intervening decades. Three of thepapers are discussed at some length below.
5
OA OBOC
OP xOA yOB zOC+ +=
OD OE OF
Eric Stephen Barnes 1924–2000 39
The perfect forms in 6 variables. Let fbe a positive quadratic form in n variablesand Λ an associated n-dimensional lattice.Those f for which the packing density of Λis either absolutely or locally maximal areof particular interest. Thus, f is calledabsolutely extreme if Λ provides thedensest possible lattice packing of spheresin Rn, extreme if no lattice obtained byslightly deforming Λ provides a denserlattice packing. The perfect forms referredto above are a somewhat wider class thanthe extreme forms, and their precise defini-tion need not be given here.
We briefly explain what the above defi-nitions mean in purely algebraic terms.With f as above, write γ(f) = MD–1/n, whereM = M(f) and D = D(f) are the homo-geneous minimum and determinant of f.Then γ(f) is a function of the coefficients off: it achieves its absolute maximum at theabsolutely extreme forms and its localmaxima at the extreme forms.
The absolute maximum just referred tois Hermite’s constant γn. Its precise value isknown only for n ≤ 8. By the 1950s therehad been extensive research by manywriters on γn and on perfect, extreme andabsolutely extreme forms. The determi-nation of the perfect forms in three vari-ables goes back to Gauss in 1831 and in 4, 5variables to Korkine and Zolotareff24 in1877. Hofreiter25 in 1933 claimed to havedetermined the extreme forms in 6 variablesbut his list turned out to be both erroneousand incomplete. Barnes [(25), (26)] closeda chapter in this history by determining theperfect forms in 6 variables.
Twenty years later, Barnes’s resultswere confirmed, by a different method andwith the aid of a computer, by Stacey26,who herself26,27 made considerable pro-gress in classifying the perfect forms in 7variables. The latter were completely class-ified only in 1991. The problem for 8variables remains unsolved. The numbersof essentially different perfect forms in 6and 7 variables are 7 and 31 respectively.
For 8 variables, the number is known to beat least 10,916 (Martinet,28 p.218).
The method that Barnes uses, due toVoronoi,29 is essentially geometrical.Although it applies to n-variable forms ingeneral, we confine attention here to thecase n = 6. A positive quadratic form in 6variables has 21 coefficients and is there-fore representable by a point in 21-dimen-sional space R21. Each unimodular perfectform f determines (in a certain way thatwill not be explained here) a 21-dimen-sional cone V(f) in R21. Each such Voronoicone is bounded by a finite number of ‘flat’20-dimensional faces, called its ‘facets’. Agiven facet of V(f) is also a facet of exactlyone other Voronoi cone V(f ') and thefinitely many unimodular perfect forms f 'obtained in this way are the neighbours off.
We are now in a position to describeVoronoi’s method of determining theperfect forms in 6 variables. A list of formsf1,…,fk is constructed step by step. The firstentry f1 may be any unimodular perfectform (examples are known). The neigh-bours of f1 are then determined one by oneand, if not equivalent to a form alreadythere, entered onto the list. The procedureis then repeated with the neighbours of f2,and so on. After a finite number of steps,no further new forms are produced and thelist is complete.
Voronoi’s method, although straight-forward in principle, presents considerabledifficulties in practice. Each perfect formhas at least 21 neighbours and in determin-ing each of them it is necessary to solvecertain systems of linear inequalities in 21variables. Barnes modified Voronoi’s pro-cedure in a simple but ingenious way thattakes advantage of the symmetry of theassociated lattice. Even so, determiningthe 36 neighbours of the absolutelyextreme form proved very difficult andBarnes’s treatment of this case is a tour deforce of combinatorial algebra.
40 Historical Records of Australian Science, Volume 15 Number 1
Dense lattice packings in large dimen-sions. In contrast to (25), the Barnes-Wallpaper (30) is essentially concerned withforms in a large number of variables. Inwhat follows, fn stands for a positive quad-ratic form in n variables and γ(fn), asearlier, for its ‘scaled homogeneous mini-mum’ MD–1/n, where M = M(fn) and D =D(fn).
We recall that the greatest possiblevalue of γ(fn), assumed when fn is abso-lutely extreme, is Hermite’s constant γn.Although its precise value is known onlyfor small n, there are quite good estimatesof its ultimate size: it lies between n/2πeand n/πe for sufficiently large n. In thissense, ‘γn has order n as n → ∞’.
Let n1, n2, … be a strictly increasingsequence of positive integers. Considernow a corresponding infinite sequence offorms obtained by taking a positive form fnin n variables for n = n1, n2, …. The aboveconsiderations show that the correspond-ing numerical values γ(fn) could be oforder n as n → ∞. However, in all suchsequences of forms explicitly constructedup to 1959, the values γ(fn) turned out to bebounded, that is, all were less than somefixed value C. The achievement of (30) wasto construct for the first time a sequence offorms for which the numbers γ(fn) areunbounded — in fact, of order n1/2 forlarge n.
The values ni chosen in (30) are thepowers 2i. For each such value, a numberof forms are constructed, many of themextreme. For n = 2i, we now understand fnto mean the ‘best’ of the forms constructedin 2i variables. For these forms, the valueof γ(fn) is in fact (n/2)1/2.
The forms fn with n = 2i are of interestfor small, as well as large, i. Thus, f4 and f8
are known to be absolutely extreme, whilef16 is conjectured to be so. The forms f32
and f64 also provide dense lattice pack-ings, although these have now beensurpassed.
The papers (27), (28) should be men-tioned here. Various infinite sequences ofextreme forms had long been known.Barnes used uniform methods to constructfurther such sequences. Of particular inter-est is his method of constructing forms inn + 1 variables from known forms in nvariables. A comprehensive generalizationof Barnes’s work was made by formerresearch student Scott.30, 31
Lattice packings constructed fromcodes. Over 20 years elapsed between thepaper (30) discussed above and Barnes’sfinal paper on sphere packing (45).Although this was not realized at the time,the former may be viewed as an applica-tion of the Reed-Muller codes to packingtheory. The latter, written jointly withSloane, is concerned with the systematicapplication of coding theory to the con-struction of lattice packings of spheres.
Coding theory originated in the late1940s and by the 1960s was widely recog-nised as an independent subject. Leech32, 33
in 1964 and 1967 made explicit use ofcodes to construct dense packings indimensions 2n and 24, respectively. Gener-alizing Leech’s work, Leech and Sloane in1971 gave general methods for manu-facturing packings of equal spheres out ofcodes. (This paper is reprinted asChapter 5 of Conway-Sloane.23) Many ofthe resulting packings were non-lattice(where the centres of the spheres do notform a lattice) and they included onesdenser than any previously known.
The paper (45) is similar in purpose andgeneral structure to the Leech-Sloanepaper, except that it is concerned entirelywith lattice packings. There are spectac-ular applications. For example, starting outfrom the famous Leech lattice in R24,lattices are constructed in Rn for everymultiple n of 24 up to 98,328. Apart from afew lower-dimensional exceptions, thecorresponding lattice packings are thedensest known in their dimensions.
Eric Stephen Barnes 1924–2000 41
Lattice covering by spheres
Of Barnes’s three papers in this area, thefirst, which appeared in 1956, was his firstsubstantial work on positive forms. Theother two, written some time later, werejoint papers with former research studentsDickson (1967) and Trenerry (1972).
The absolutely extreme and extremelattices and forms of packing theory havenatural counterparts in covering theory.Barnes calls them by the same names withthe qualification ‘in the sense of coveringtheory’. To avoid confusion here, we callthem optimal and locally optimal,respectively.
The locally optimal ternary forms.Barnes’s paper (23) is concerned with lat-tices in R3 or equivalently with ternarypositive forms. At the time of writing ofthe paper, the ternary extreme forms hadlong been known but a ternary optimalform had been discovered for the first timeby Bambah34 in 1954. Barnes sharpensBambah’s result by proving that everylocally optimal ternary form is equivalentto a multiple of Bambah’s form.
Barnes uses a second geometricalmethod due to Voronoi.35 Consider a3-dimensional lattice Λ and a point P of Λ.Those points of space for which P is theclosest lattice point form a certain region Ω.By its meaning, the covering radiusr = r(Λ) is the greatest distance of any pointof Ω from P. The region Ω is a polytope andconsequently the points of Ω farthest fromP are among its finitely many vertices.
In the light of these considerations, thepaper proceeds as follows. The geometry isused first to express r, and thence thecovering density δ = δ(Λ), as functions ofthe coefficients of a suitable distance func-tion f for Λ. Algebra is then used to deter-mine those values of the coefficients atwhich the function δ has a local minimum.As usual in Barnes’s work, this programmeis carried out with exceptional skill andelegance.
The locally optimal forms in 4 variableswere subsequently determined byDickson36 in 1967 and the optimal formsin 5 variables by Ryškov and Bar-anovskii 37 in 1975.
Covering theory is more difficult thanpacking theory and less is known about it.The papers of Barnes and Dickson (33) andBarnes and Trenerry (35) make significantcontributions to general covering theory.
A basic theorem of Voronoi29 clarifiesthe relation between perfect and extremeforms. An analogous theorem in coveringtheory is proved in (33). Essential use of(33) is made by Dickson in his determi-nation of the locally optimal forms in 4variables.
Various infinite sequences of extremeforms arise naturally and have been knownat least since Korkine and Zolotareff’swork in the 1870s. The situation for locallyoptimal forms is, however, quite different.An infinite sequence of such forms wasfirst constructed by Bleicher38 in 1962.Barnes and Trenerry (35) construct asecond such sequence of surprisingly com-plicated form.
Lattice quantizers
The one paper on this subject was the jointpaper (46) with Sloane, published in 1983.
The simple process of rounding offnumbers to the nearest integer can beformulated in lattice terms. The integers 0,± 1, ± 2, … form the lattice Z1 on the realline R1. Rounding off a number r in R1 tothe nearest integer means replacing it bythe nearest lattice point. It can be shownthat the average squared error incurred inthis process is 1/12.
Now let Λ be a lattice in Rn. Theassociated process of quantization replaceseach point of Rn by the nearest point of Λ.Let (Λ) denote the resulting averagesquared distance error. Making technicaladjustments for dimension and scale, onearrives at an ‘absolute’ measure G(Λ). The
G
42 Historical Records of Australian Science, Volume 15 Number 1
smaller the value of G(Λ), the more effi-cient Λ is said to be as a quantizer.
The value of G(Λ) for the plane latticeZ2 (and indeed for Zn in general) is1/12 = 0.08333…. However, the mostefficient plane lattice is actually the hexag-onal lattice, for which G(Λ) = 5/(36√3) =0.08018….
In general terms, the larger the dimen-sion the more efficient lattices can be. LetGn denote the smallest possible value ofG(Λ) for lattices Λ in Rn. Zador39 in 1963proved the remarkable result that Gn tendsto the value 1/2πe = 0.05855… as n tendsto infinity.
Despite Zador’s result, actually con-structing highly efficient lattices in Rn
remains a difficult problem. For example,the q-optimal lattices in Rn for which G(Λ)attains the minimum possible value Gn
have been determined only for n = 1, 2, 3.The case n = 3 is due to Barnes and Sloane(46). Their work yields the valueG3 = 19/(192.21/3) = 0.07854…, which isroughly 2% smaller than G2.
In general outline, the proof by Barnesand Sloane follows Barnes’s proof in (23)rather closely. Indeed, in the quantizationprocess associated with a lattice Λ in R3,the polytope Ω about a lattice point Pconsists precisely of the points rounded offto P. Moreover, the average, (Λ), takenover R3 is the same as the average takenover Ω. This is calculated by an ingeniousargument, which involves dissecting Ωinto 60 tetrahedra. Both hand and com-puter calculations are involved.
Reduction of positive forms
The five papers briefly reviewed here werepublished in the period 1975–82.
We are concerned with positive quad-ratic forms in n variables. Insight into thetotality of such forms may be gained bypicking out one or more representativeforms from each equivalence class.‘Hermite reduction’ and ‘Minkowskireduction’ are two methods for doing this.
Reduction theory, which deals with suchmethods in general, is one of the mostimportant (and oldest) branches of thearithmetical theory of positive forms.
A reduced set of forms is one such thatevery form is equivalent to at least one,and at most finitely many, of its members.The region in RN [N = n(n + 1)/2] formedby the coefficient vectors of the membersof such a set is called a reduction domain.A fundamental domain is a reductiondomain with the stronger property that notwo of its interior points represent equiva-lent forms.
Minkowski defined certain reductionand fundamental domains Φ and Φ+, bothof which are convex polyhedral cones.Barnes and Cohn (37) explicitly deter-mined their edges (i.e. 1-dimensionalfaces) when n ≤ 4 : Φ has 323 edges andΦ+ 109 when n = 4. Heavy calculations(partly by computer) are involved.
It is natural to ask whether alternativemethods of reduction might lead to simplerdomains. Again for n = 4, Barnes andCohn (38) construct a fundamental domainwith just 12 edges. It is constructed in aningenious way from Voronoi cones.
It was proved by Minkowski himselfthat there is a constant λn such that theinequality a11a22 … ann ≤ λn holds for everyunimodular Minkowski-reduced form
f(x1, …, xn) = Σaij xi xj .
Several mathematicians observed that, inthe cases n = 2, 3, this can be sharpened byreplacing the left hand side of the inequal-ity by a more complicated function of a11,… ann.. In (42), Barnes extends theseresults to the case n = 4, giving a clearexplanation of how such inequalities comeabout. The results of Barnes and Cohn (37)are required in the calculations.
What is involved in refining Minkow-ski’s inequality for general n was furtherclarified by Barnes in (43). Finally, Barnesand Trenerry (44) showed that no suchrefinement exists when n = 5. There is,
G
Eric Stephen Barnes 1924–2000 43
however, a refinement that holds when a55
is large enough compared with the otherdiagonal elements. Heavy algebra isrequired in the analysis and several naturalquestions are left open.
Concluding Remarks
Eric Barnes was internationally recognisedas a leading contributor to the growth ofknowledge in the geometry of numbers.The continued relevance of his outstandingwork can be seen from recent publicationsin the area, as well as from the two maingeneral reference works4, 23 and the recentmore specialized book of Martinet.28
Through his research, teaching, scholarlywork and professional service, he made amajor contribution to the development ofresearch and education in mathematics inAustralia.
Acknowledgments
We thank the Barnes family for their helpwith this project and especially for the loanof an autobiographical note by EricBarnes, which was most useful eventhough not completed. We are very grate-ful to the many other people who haveprovided information, recollections orhelpful comments. These include ProfessorJ.W.S. Cassels, Professor J.C. Davies, MrD.C.S. Sissons, and the late ProfessorI.H. Smith, as well as former students andmathematical colleagues in Australia.
The photographs have been kindlymade available by the Australian Academyof Science (portrait) and the University ofAdelaide Archives (Fig. 1).
References
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2. C.E. Nelson. The reduction of positive defi-nite quinary quadratic forms. AequationesMath. 11, 163–168 (1974).
3. J.W.S. Cassels. An introduction to the geo-metry of numbers. Springer, Berlin, 1959.
4. P.M. Gruber and C.G. Lekkerkerker. Geo-metry of numbers. 2nd edition. North Hol-land, Amsterdam, 1987.
5. S.G. Dani and G.A. Margulis. Values ofquadratic forms at integral points: an elemen-tary approach. Enseign. Math. (2) 36,143–174 (1990).
6. R.T. Worley. Asymmetric minima of indefi-nite ternary quadratic forms. J. Austral. Math.Soc. 7, 191–238 (1967).
7. R.T. Worley. Minimum determinant of asym-metric quadratic forms. J. Austral. Math. Soc.7, 177–190 (1967).
8. R.T. Worley. Non-negative values of quad-ratic forms. J. Austral. Math. Soc. 12,224–238 (1971).
9. M. Raka. Inhomogeneous minima of a classof ternary quadratic forms. J. Austral. Math.Soc. Ser. A 55, 334–354 (1993).
10. H.J. Godwin. On a conjecture of Barnes andSwinnerton-Dyer. Proc. Cambridge Philos.Soc. 59, 519–522 (1963).
11. H.J. Godwin. On the inhomogeneous minimaof certain norm forms. J. London Math. Soc.30, 114–119 (1955).
12. D. Berend and W. Moran. The inhomo-geneous minimum of binary quadratic forms.Math. Proc. Cambridge Philos. Soc. 112,7–19 (1992).
13. V. Ennola. On the first inhomogeneous mini-mum of indefinite binary quadratic forms andEuclid’s algorithm in real quadratic fields.Ann. Univ. Turku Ser. A I 28 (1958). Separatemonograph, 58 pp.
14. B.N. Delone. An algorithm for the ‘dividedcells’ of a lattice (Russian). Izv. Akad. NaukSSSR Ser. Mat. 11, 505–538 (1947).
15. V.K. Grover and M. Raka. On inhomo-geneous minima of indefinite binary quad-ratic forms. Acta Math. 167, 287–298 (1991).
16. S. Morimoto.* Über die Grössenordnung desabsoluten Betrages von einer lineareninhomogenen Form II. Japanese J. Math. 3,91–106 (1926). *Author also known asS. Fukasawa.
17. J. Pitman. The inhomogeneous minima of asequence of symmetric Markov forms. ActaArith. 5, 81–116 (1959).
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44 Historical Records of Australian Science, Volume 15 Number 1
20. P.E. Blanksby. A restricted inhomogeneousminimum for forms. J. Austral. Math. Soc. 9,363–386 (1969).
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23. J.H. Conway and N.J.A. Sloane. Sphere pack-ings, lattices and groups. 3rd edition.Springer, Berlin, 1998.
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26. K.C. Stacey. The enumeration of perfectseptenary forms. J. London Math. Soc. (2) 10,97–104 (1975).
27. K.C. Stacey. The perfect septenary formswith ∆4 = 2. J. Austral. Math. Soc. Ser. A 22,144–164 (1976).
28. J. Martinet. Perfect lattices in Euclideanspaces. Springer, Berlin, 2003.
29. G.F. Voronoi. Sur quelques propriétés desformes quadratiques positives parfaites.J. Reine Angew. Math. 133, 97–178 (1908).
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