Australian Mathematical Society Web Site - the Gazette

# Book Reviews

Poetry of the Universe
Reviewer: Marty Ross
ARML-NSML Contests 1984--1994
Reviewer: Peter Taylor
One Dimensional Spline Interpolation Algorithms
Reviewer: Rick Beatson
Two Dimensional Spline Interpolation Algorithms
Reviewer: Rick Beatson

## POETRY OF THE UNIVERSE

Robert Osserman
Publisher: London: Weidenfeld and Nicolson, 1995
Reviewed by: Marty Ross

Dr. Hoenikker used to say that any scientist who couldn't explain to an eight-year old what he was doing was a charlatan.''

Then I'm dumber than an eight-year old,' Miss Pefko mourned. I don't even know what a charlatan is!''

To the general public, most undergraduates included, the notion of research in mathematics and physics is incomprehensible. The common notion is that advanced (and earlier) study in our fields consists of memorizing dry facts from dusty tomes, an improbable task with implausible rewards. There is little sense of research involving exploration and discovery, beauty and fun. This view differs radically from that of the better informed, scientists and journalists of science. There has thus arisen a natural market and an obvious desire for members of the latter group to impart their understanding and enthusiasm to the former, and considerable effort has been expended on popularizations of physics and related disciplines. Sadly but predictably, no comparable effort has been expended in the reading of these works.

The main reason for this tepid response is confusion by the writer over the nature and desires of the reader. For the appreciation of even a gentle popularization, certain minimum levels of knowledge, sophistication and dedication are required. On the other hand, the casual reader, even if of serious intent, is unlikely to be willing to work overly hard. Thus there is presented to the writer a delicate balancing act, one in which few succeed. Overbalancing in one direction, the result is a work which is inaccessible and discouraging, all the more so from being couched in This is easy!' language. (Such works are usually accessible to and occasionally rewarding for the technically proficient, but that is beside the point). In the other direction, one can spend so much time on on character portrayal and and non-technical history that (deliberately?) there is room left for at most a cartoon characterization of the discipline in question; in extreme, this gives rise to coffee table books with glossy pictures of the Mandelbrot Set and its kin, and little else. Such books may be read, or viewed, but the result is not to give the reader any real understanding of our world or the world at large.

A further problem has been the objective quality of the writing, which is seldom more than serviceable and is often very poor. This is a concern with any work, but it is doubly problematic in the exposition of technical material. Here, clumsy phrasing is not just inelegant and annoying, it is a source of possibly critical confusion. The nature of the material together with that of the audience leaves little margin for error.

Poetry of the Universe (POU) is a welcome change. It grew out of the course The Nature of Mathematics, Science and Technology'', a distribution requirement for Humanities students at Stanford University. (In the U.S. now, subjects of this nature are commonly, affectionately or derogatorily, referred to as Physics for Poets'' and the like. The Melbourne University Mathematics Department has just initiated such a course, for which POU is recommended reading.) The students in this class form an excellent trial audience for, reciprocally an excellent market for, popular science writing: they are intelligent and eager, but they tend to be sceptical of and/or fearful of mathematical and scientific thought, acutely aware of their own weaknesses. They can be convinced - the Stanford Course, which was very successful is evidence of that - but they do need convincing, for which is required careful thought and a gentle hand. Osserman (one of three lecturers of the subject) demonstrated his remarkable teaching ability in his lectures and lecture notes, and this has come through in the resulting publication. There is an integrity to the work: the explanations are vast simplifications of technical material, but it is done in a truthful and non-demeaning manner.

As indicated by the subtitle, POU is concerned with the mathematics of cosmology. The approach is historical, beginning with the (successful) attempt of the Greek Eratosthenes to measure the size of the Earth, and ending with contemporary speculation on the geometry and topology of space-time. Osserman links the latter to the former by a chain of historical snapshots and a natural mathematical progression: Columbus and mapping the Earth, and Euler's theorem that any such mapping must introduce distortions; local variation in the shape of the Earth, surveying, triangulation and the notion of Gauss curvature; the power of abstraction, imaginary numbers and Lobachevskian geometry; three-dimensional manifolds as models of the universe and Riemannian geometry (including Dante's vision of S ^3); the discovery of X-rays, radio waves, and mapping the universe; Einstein's theory of relativity and expansion of the universe; Hubble's Law, the Big Bang and dating of the universe; the universe as four-dimensional manifold. Throughout, strong emphasis is placed upon the interplay between practical need and theoretical wonder. The exposition is never rushed, the mathematical explanations blended in with anecdotes and with ponderings on the nature of mathematical and real-world discovery.

It is axiomatic that popularizations of mathematics should not employ calculus as an explanatory tool, but this still leaves ample scope for drowning the reader in a sea of calculation and technical description. A remarkable feature of POU is that it uses only one mathematical formula: in a Euclidean world the circumference of a circle of radius r is 2\pi r. The notion of curvature is then explained by describing how this quantity is altered in a world of positive or negative or variable curvature, and by the use of clear, simple diagrams: POU is an essay to be read rather than a textbook to be worked through.

The non-formulaic approach obviously precludes a lot of desirable detail: precise definitions of curvature, specific formulas for map projections, spherical and hyperbolic formulae, Maxwell's equations, etc. Osserman makes up for this shortcoming in the main text by including very extensive endnotes, providing detail on technical points, as well as information on side issues, good stories which don't quite fit in the main text. The effect is to make POU two-levelled: the main body, which is self-contained, is accessible to any thoughtful reader; the endnotes, to varying degrees, are accessible to a first or second year science undergraduate.

There are a number of quibbles one can make with POU: even for an essay there is room for more diagrams; some of the poetical quotations, Euclid alone has looked upon Beauty bare'' and the like are as annoyingly grandiose here as always; some of the analogies between musical and mathematical history seem forced; the description of the use of Hubble's law to date the universe is a bit dodgy; it is not made clear that the given circle-characterization of curvature is for sectional curvature and thus could depend upon the selected orientation of the circle; the discussion on fractal dimension in the last chapter appears gratuitous, and seems to confuse the notion of whether dimension'' refers to degrees of freedom or to a scaling exponent. However, all of these complaints are minor in extent. None is a serious distraction.

With the weakness of other popularizations in mind, it is easy to exaggerate the merits of POU and the size of its potential readership. In truth, Osserman's work is not going to outsell Anne Rice's latest vampire epic, and cosmology is not going to displace the state of Warne's finger as the hot topic for discussion at the local pub. Nonetheless POU is unarguably a wonderful book, clear and honest in its aims and beautifully written. For whatever small audience exists, that semi-mythic public willing and keen to learn about mathematics and its application to understanding the world, Poetry of the Universe is a delightful offering.
University of Melbourne

D. Fomin and A. Kirichenko
Publisher: Westford, MD: MathPro Press, 1994

## ARML-NSML CONTESTS 1984-1994

L. Zimmerman and G. Kessler
Publisher: Westford, MD: MathPro Press, 1995
Reviewed by: Peter Taylor

There is a small, but growing market in the publication of books on contests and enrichment mathematics. This is because of the increased interest by talented students in extending their mathematical skills and knowledge. This usually results from a love of the subject, and the challenge of problem solving, most of the time, but it has an added value of providing a wider knowledge base and enhancing future university study.

Often such students study in what the Russians call circles'', namely a club of students often drawn from a number of schools in a particular locality, studying under the encouragement and support of a local school or university teacher.

There are now a number of these circles'' in Australia, as developed over a thirty year period. All have their own character, and some enter contests, such as those run by the Australian Mathematics Olympiad Committee, International Mathematics Tournament of Towns or local Olympiad Contests, which serve as a focal point for discussion in the meetings. In fact I would argue that contests are irrelevant as stand-alone items, with little value. The value is in the discussions which take place beforehand or afterwards.

Books on enrichment mathematics, enabling students to extend their syllabus knowledge, are vital to these activities. They generally fall into two classes, one being collections of problems and solutions from national or international competitions, while the other is the collection of books which teach certain topics. These topics form the unwritten syllabi of competitions, generally algebra, number theory, discrete mathematics and geometry, but rarely calculus.

Enrichment books in Russian and published in Eastern Europe are wide-spread. Contests and enrichment mathematics groups or circles'' certainly developed in Eastern European countries such as Hungary, Russia and Bulgaria to a greater extent than in English speaking countries, and indeed, the main international mathematics competitions had their genesis in those regions.

The best established series of such books in the English language emanate from the Mathematical Association of America, their New Mathematical Library commenced in 1961 (at that time through Random House) and has since put out over 30 titles. Their Dolciani series of expositions, dominated by Ross Honsberger books (Mathematical Gems, Morsels, etc.) is equally outstanding. In Australia, the Australian Mathematics Trust has its own series, which includes the highly regarded Mathematical Toolchest, the unofficial olympiad syllabus, edited by Neil Williams and Ash Plank.

A new series is now emerging from America, published by Stanley Rabinowitz's Mathpro Press. Rabinowitz is a mathematician and computer scientist with much experience and association with the major American competitions. He had published a compendium of problems from many of the world's major contests and a further one is anticipated. But in the meantime he has published two outstanding volumes in a new series Contests in Mathematics''.

The first, Leningrad Mathematical Olympiads 1987-1991 , is by Dmitry Fomin, a prodigious problem composer who has recently moved to America, for a while at least, and Alexey Kirichenko. Together, they ran a Leningrad circle'' which produced many members of the USSR Olympiad team. This has many problems of classical Russian style (like those of the Tournament of the Towns). The problems are original, often stated clearly and concisely, and often profound, stated in a context to which students can relate with interesting stories and situations.

The Leningrad Olympiad is special as it was the first in Russia (founded in 1934) and became an integral part of the All-Union Olympiad when it developed. It is also unusual in that in the last stages (those used in the book) the students must present the solutions orally. This may suggest that the problems should be used with some care, but my analysis of the problems suggests that they would have also been quite appropriate in a traditional (written) paper.

The authors are listed in the index and include a who's who'' of Russian problem composing. In addition to Fomin and Kirichenko are found names such as Konstantinov, Nazarov and the geometer Sharygin. The translations are always clear although sometimes they could have been made a little smoother.

Here is a typical problem from the book:

{\sl The shah's guard is searching for the thief of Baghdad, who has broken into the palace. The palace consists of 1000 rooms, connected in such a way that there is only one route from each room to another (in terms of graph theory the plan of the palace is a tree). Prove that

• (a) {ten guards can catch the thief if the plan of the palace is known;}
• (b) {it could happen that five guards can not catch the thief;}
• (c) {in fact six guards are enough to ensure that the thief can be caught.} }

The other book, ARML-NYSML Contests 1989-1994 , by the experienced New York teachers Lawrence Zimmerman and Gilbert Kessler, is also based on a contest with non-standard format. The American Regions Mathematics League (in earlier days I understand A stood for Atlantic, but it grew) is a team event with many teams of 15 competing at the one venue. The event has four components, a Power question with a number of related components, an individual event, a relay and a team event in which the team divides the questions among them as suits their talents. The questions are therefore of varied format and are not usually typical of an olympiad'' format but contain much useful material for circles''. The syllabus is at least that of the olympiad style. The NYSML (a version for New York state with identical format) is also covered in the book. The questions are all interesting and challenging and carry the hallmarks of a problems committee of high quality and excellent moderation. Here is a sample from the Team section:

{\sl Lewis and Carol travel together on a road from A to B, then return on the same road, with the entire trip taking 3 hours. Sometimes that road goes uphill, sometimes downhill, and sometimes it is level. When the road goes uphill, their rate is 40 mph, downhill their rate is 60 mph; on level road their rate is $x$ mph.''

Even if you were given a numerical value for $x$, the distance from A to B would (in most cases) not be uniquely determined. But there is one value for $x$ that would determine that distance uniquely. Compute this value of $x$. [Note: Uphill going is downhill returning.] }

One final note. Rabinowitz is an excellent \TeX practitioner and this has led to excellent presentation of both books. The diagrams have been accurately drawn electronically and match well with the text.

Mathpro have set a high standard with these two books which leads to confidence that the future books will be well worth buying. They will prove value to students developing their problem solving skills and provide excellent source material for the circle'' teacher. In addition they are just interesting to read!
University of Canberra

## ONE DIMENSIONAL SPLINE INTERPOLATION ALGORITHMS

H. Späth
Publisher: Wellesley, MA: A.K. Peters, 1995
Reviewed by: Rick Beatson

This book contains a large number of algorithms for fitting polynomial, rational and exponential splines to one dimensional data. Algorithms are presented for constrained problems with positivity, monotonicity or convexity constraints and for least squares and histospline fitting. The algorithms are accompanied by FORTRAN code. Constrained rational spline interpolants of the type pioneered by Delbourgo and Gregory, and splines under tension as investigated by Cline and separately Pruess, are a particular feature. The book is clearly written and very accessible. As such it will be very useful to anyone wanting to gain some understanding of constrained spline techniques, or wanting to use these techniques without writing their own code.

The book does not attempt to be an introduction or a text on univariate splines in general. Central topics in spline theory, such as error estimates and B-splines, are given either very cursory attention or no mention at all. In this reviewer's opinion the fundamental purpose of spline interpolation is to approximate at points where there is no data. From this point of view error estimates deserve a place in any book on splines. Furthermore the reference list is far from complete. For example only one paper by the father of splines, I.J. Schoenberg, is listed and that not one of his best. Those seeking a broader view of univariate spline theory are advised to consult any of the texts [1], [2], [3] listed below.

## References

• [1] {C. deBoor, A Practical Guide to Splines , Springer Verlag, New York, 1978.}
• [2] {M.J.D. Powell, Approximation Theory and Methods , Cambridge University Press, Cambridge, 1981.}
• [3] {L.L. Schumaker, Spline Functions: Basic Theory , Wiley, New York, 1981.}
University of Canterbury
New Zealand

## TWO DIMENSIONAL SPLINE INTERPOLATION ALGORITHMS

H. Späth
Publisher: Wellesley, MA: A.K.Peters, 1995
Reviewed by: Rick Beatson

This book is divided into two parts. Part~1 concerns spline interpolation on rectangular grids. Firstly, interpolation by coordinate degree and total degree polynomials on rectangular and triangular grids is discussed. Then the book moves on to various tensor product schemes. Schemes for interpolating subject to constraints are also discussed. Unfortunately, as in the previous book important topics such as B-splines and error analysis are omitted.

Part~2 concerns spline interpolation for arbitrarily distributed, that is scattered, data. Topics treated include Shepard's interpolation method, interpolation with Hardy multiquadrics, Delauney triangulations, Powell-Sabin quadratic elements and Clough-Tocher elements. This part of the book is well worth reading by anyone wishing to gain some familiarity with a variety of scattered data interpolation techniques. Furthermore, a lot of the material contained in it has not previously been collected together in one place.

Unfortunately, once again, many important topics are omitted. For example, there is no mention of Micchelli's (1986) or Schoenberg's (1940s) proof of the invertability of certain radial basis function interpolation matrices (including that for multiquadrics). Also thin-plate splines, which have the striking property of minimizing the energy functional
E(s) = \int \int_{R^2} \left( \frac{\partial^2 s}{\partial x^2}\right)^2 +2 \left( \frac{\partial^2 s}{\partial x \partial y}\right)^2 +\left( \frac{\partial^2 s}{\partial y^2}\right)^2 dxdy ,
over all interpolants for which E(s) exists, are mentioned only in a postscript. This is despite the fact that they have been the subject of intensive pure and applied research worldwide for the last ten years. They have been at least as successful as multiquadrics in applications ranging from image warping to cranioplasty. Furthermore, error estimates, preconditioning, fast fitting and fast evaluation techniques have all been developed recently.

In conclusion, those interested in scattered data interpolation will find the book well worth reading. However, it should be supplemented by further reading on radial basis functions and in particular on thin-plate splines.
University of Canterbury
New Zealand

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