divided by the total sum of the row (the number of out-links of the page). The resulting matrix is (almost) stochastic, so its entries may be viewed as proba- bilities. A nice interpretation of this for- mulat ion is a random surfer traveling along the web, following the links be- tween nodes, and uniformly choosing his next destination from among the links in the current node. It turns out that the ranking list we need is just the stationary vector of the Markov chain associated to matrix H ' . In fact, some adjustment must be done in order to have stochasticity, because H could have zero rows (pages with no out- links). In these cases we can replace these 0-rows with I/n entries. Call this adjusted and normalized matrix J.

But now we face a computa t ional problem: How can we determine this vector? Remember the matrix is ex- t remely large. The idea is now to make a second adjustment, to guarantee that the new matrix also be primitive. Then the stationary vector exists, is unique, and, as it is the e igenvector associated to the dominant e igenvalue of the ma- trix, can be calculated with a s imple and fast numerical p rocedure such as the p o w e r method. The reader should be aware that the Per ron-Frobenius Theorem on positive (or non-negat ive) matrices plays a key role in all these arguments. The precise adjustment is given by

1 G = o ~ . / + ( 1 - ez) n e . e "v

where e stands for the vector of all ones and oe is a number be tween 0 and 1. G is k n o w n as Google ' s matrix. Page- Rank output is just its dominant eigen- vector, which can be ob ta ined with an iterative method such as c~ t'+J = a k G. In terms of the random surfer, this ad- justment brings in a new possibility, namely that with probabi l i ty I - a the surfer gets bored of following the links and "teleports" to any page of the web. The reader could argue that this is an "artificial" matrix (for instance, Google ' s choice of the value of the teleportat ion constant, a, is a round 0.85). And in- deed it is, but it al lows effective com- putat ion of the ranking vector and, above all, it works! The resulting out- put is incredibly good at assigning rel- evance to web pages.

So this is the model. Not too com- plicated, is it? Of course, there are many details to complete: computat ional as- pects (such as convergence rates of the iterative scheme, sensitivities to the pa- rameters), possible improvements of the numerical p rocedures (or the model itself) . . .

Most of this can be found in the fif- teen chapters of the book under re- view. The first three chapters introduce the reader to the main features of w e b searching, including some review of the traditional methods of information re- trieval. Chapters 4 to 10 deal with the mathematics behind Google ' s algo- rithm: the Markov chain model, nu- merical procedures , sensitivities to pa- rameters, convergence issues, methods for updat ing the rankings, etc. All the mathematical concepts used in the book are t reated in detail in the "Math- ematical Guide" of Chapter 15: l inear algebra, Markov chains, Per ron-Frobe- nius Theory, etc. Chapter 11 includes a brief review of Kleinberg 's HITS algo- rithm; other ranking methods are men- t ioned in Chapter 12. Chapter 13 dis- cusses some quest ions related to the future of web information retrieval, including spam and personal ized searches. I would have liked to see a more comprehens ive discussion of eth- ical issues such as privacy and censor- ship. Considering that Google has be- come the standard source for information (you appear in Google or you are nothing!), these are really dis- mrbing topics. But p robably a whole new book could be written on this.

The book under review is excel- lently written, with a fresh and engag- ing style. The reader will particularly enjoy the "Asides" interspersed throughout the text. They contain all kind of entertaining stories, practical tips, and amusing quotes. "How do search engines make money?," "Google bombs," "The Google dance," and "The ghosts of search" are some of these st imulating asides. The book also con- tains some useful resources for com- putation: Pieces of Mattab code are scattered throughout the book, and Chapter 14 contains a guide to web re- sources related to search engines.

Despite the technical sophistication of the subject, a general science reader can enjoy much of the book--certainly

Chapters 1 to 3, and also Chapter 13. With some basic knowledge of linear al- gebra, the description of the model (Chapters 4 and 5) can be fol lowed with- out problem. Chapters 6 to 12 are more technical, and they are intended for ex- perts. The authors provide a webpage (h t tp : / /pagerankandbeyond.com/) that includes a list of errata for this edition.

Departamento de Matemfiticas Facultad de Ciencias Universidad Aut6noma de Madrid Ciudad Universitaria de Cant�9169 28049 Madrid, Spain e-mail: pablo.fernandez@uam.es

Mathematical Form: John Picl<ering and the Architecture of the Inversion Principle by Pamela Johnston (ed.)

With contributions bY Mobsen

Mostafavi, George L. Legendre, .John

Picketing, Chris Wise, John Silver,

and John Sharp

LONDON: ARCHITECTURAL ASSOCIATION, 2006,

96 PP., s ISBN 978-1-902902-37-1

REVIEWED BY KIM WILLIAMS

"] ohn Pickering's models----<~r build- / ings----of geometrical forms, der ived

- / t h r o u g h the process of inversion, are the subject of this little book. The book is essentially a catalogue of the 2002 exhibit of his works in the gallery of London's Architectural Association. Picketing is an artist (the exhibit in- c luded his earlier studies of the human form, but this book does not), who at a certain point, in his own words, "be- came anti-nature" and dedica ted him- self to the derivation and visualization of complex three-dimensional geomet- rical forms. To create his forms, Pick- ering employs the "inversion principle," transformations of either plane or solid

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Ellipsoid consisting of one half of the hyperboloid of two sheets and one half of the hyperboloid of one sheet. 1971-1973. Card. 14 cm • 27 cm • 19 cm. Photograph by Sue Barr. Reproduced by permission of AA Publications, http://www.aaschool.ac.uk/ publications.

figures first studied intensively in the 1820s through the 1840s by mathemati- cians such as Jakob Steiner and Jean- Victor Poncelet.

Mathematician John Sharp's concise and clear essay that concludes the book should have opened it. His explanation (p. 87) of the inversion principle makes clear the basis of Pickering's work as well as its process:

Inversion is simple. In the plane it is a kind of reflection in a circle, which maps all points inside a cir- cle to points outside and vice versa; with the centre of the circle being a special point where all points at in- finity map. In space, the mapping is of the inside and outside of a sphere. �9 . . The distances of the points from the centre are related by the formula:

MP. MQ = r 2, where r is the ra- dius of the circle.

The term inversion arises be- cause, to find the length MQ in or- der to find the inverse point to [a given] point P, the formula is re-

r 2

arranged so that MQ = 3 " Understanding the inversion principle on which Pickering's forms are based,

and placing them in an historic con- text, our appreciation of both the metic- ulous process of derivation and the fi- nal product are greatly enhanced. I hope in any case that architects are bet- ter at designing buildings than they are at designing books (don't get me wrong; those of you who know me will know that this is the pot calling the ket- tle black: I am an architect who designs books). The page numbers and some captions in this one are lost in the bind- ing rather than being on the outside margins for easy reading. It would be helpful if the book came with a mag- nifying glass, as the OED does, for reading the captions and viewing the figures that are such a necessary part of John Sharp's essay. (I have heard it said, as far as required courses in math- ematics in the architecture curriculum are concerned, that the architects would like to throw out the mathe- maticians but keep the math, and per- haps this attitude is reflected here.) The photographs of the objects, mostly by Sue Barr with some by Arthur Picker- ing, are very good.

That Pickering's w o r k - - a very de-

liberate and rigorous exercise in de- velopment and realization of geomet- ric fo rm--shou ld be the subject of an exhibit aimed primarily at architects says more about the present state of ar- chitecture than about the artist. To his credit, Pickering himself does not say that his models are intended to be seen as architecture. None of the essays in the book says exactly how the archi- tect might use the inversion principle as a design tool. But much space in the essays, by architects Mohsen Mostafavi, George Legendre, John Silver, and en- gineer Chris Wise, is dedicated to the possible relationship between Picker- ing's models and architecture. The models are "prototypes of giant struc- tures waiting to be realised" (p. 7) and "look like buildings" (p. 22). Silver is the most unequivocally enthusiastic: "If ever conceived on a civic scale, John's sculptures would be incredible!" (p. 83). Only the dour engineer--always the architect's party-pooper--expresses skepticism: "As a basis for fledgling buildings, I feel Pickering's inversion principle is not really so good" (p. 80).

Why do Pickering's models look like buildings? For one thing, they are mass- ings of geometric volumes that appear as though they could be containers of space. For another, they look- -and are--engineered, in the sense that be- hind the built form is an equally im- pressive mass of manually executed calculations. For still another, the sur- faces are only intimated; what we ac- tually see is the framework that sup- ports the surface, and that framework looks like floors, columns, and beams. But perhaps the biggest reason of all is that contemporary architects--first of all Frank Gehry- -have taught us that buildings can have weird shapes; any- thing you can imagine can become a building. (Is this why Pickering envi- sions one of his models "as a large built structure 'perhaps in Spain' " (p. 7), the country that brought us Gehry's Guggenheim, where obviously any- thing can be built?)

As Sydney Pollack's 2005 documen- tary "Sketches of Frank Gehry" showed, Gehry's process of form generation is the exact opposite of Pickering's. Gehry starts with the model, subjects it to his very personal tastes and instincts ("I don' t like this, cut it, bend it, fold it,

"TO THE MATHEMATICAL INTELLIGENCER

corrugate it . . . there!"), then lets his technicians scan it and digitalize it in order to a l low the compute r to gener- ate the necessary instructions for its full-scale construction. Picketing, in contrast, begins with an equation, man- ually calculates a series of vector lengths, maps the coordinates in space, then painstakingly cuts out and assem- bles the required pieces to construct his final model.

What Gehry and Pickering share is a fascination and predi lect ion for form. Indeed, most of the architectural pro- fession focuses on the issue of form these days (or so it seems from the pop- ular architectural press, and from the buildings that most capture the public 's attention). Such a preference is p roper to the artist, but improper to the ar- chitect, because if formal quest ions predomina te , they do so to the detri- ment of function; that is, the building's program, or functional requirements, are made to fit inside the form con- ceived by the architect rather than con- tributing to its determination. It is in- teresting to hear Geh~ T descr ibe the rectangular rooms where art is hung in the Bilbao G u g g e n h e i m - - p r e s u m a b l y the function of an art m u s e u m - - a s "ba- nal space," whereas his much more heroic spaces tend to overwhelm the art d i sp layed in them. This is not a new di lemma in museum design: Wright 's Guggenhe im Museum in New York City, a top-heavy, descending spiral, w a s - - a n d i s - - severe ly criticized be- cause the d o w n w a r d force of gravity a long the ramp tends to pull the spec- tator past the art wi thout giving him time to l inger over it. If the architect adopts a subjective method of design, he may willfully subjugate function t o

form, but he at least has the oppor tu- nity to let the bui lding 's form morph if function requires it. Adopt ing a rigor- ous me thod of form generat ion only ex- acerbates the problem, for if the form is to remain true to the generat ing prin- ciple, the functional requirements are necessari ly overpowered . This is why engineer Chris Wise writes (p. 80)

Mutation, good or bad, has no place in Pickering's world. Deliberate in- tervent ion does not feature either. There 's no chance to "add a bit here" if the vo lume ends up a bit mean or looks a bit wobbly: if one little p iece

is tweaked, everything else has to follow the inversion principle and change too, whether it wants to or not.

The inversion principle is not the only form genera tor that imposes the im- possibility of invariance; a rigorous sys- tem of propor t ions does the same thing. Although rigor is a healthy prin- ciple for mathematics, it is not always healthy for architecture.

The obsess ion with form in archi- tecture may arise out of the fact that today 's architects have such powerful tools to work wi th - - r ende r ing tools such as AutoCAD that allow architects almost instantly to see their designs in three dimensions, and other more mathematical tools such as CATIA (originally deve loped for the French aerospace industry), which models forms.

Pickering, however, uses no such software, so his fascination with form is not der ived from the possibili t ies pre- sented by the tools, but rather out of a love of form for form's sake. As an artist, he has no building program to respect; his forms don' t have to house specific functions.

But architects ' will ingness to con- centrate on form for form's sake may in itself explain why "the status of ar- chitectural composi t ion is presently at an all-time low" (p. 25); that is, by con- centrating on form alone, architects are shirking their other responsibilit ies, those of providing buildings that are functional and meaningful for the users. This is not to imply that mathematical principles shouldn ' t be used in the gen- eration of architecture, but simply that in themselves they are not sufficient.

I think this presentat ion of Picker- ing's work is misdirected. Is Pickering's work architecture, as the subtitle of this book suggests? It is not. (I wonde r if it is even art, but I 'm not open ing that can of worms here.) It is geometry: beautifi.d, complex, spine-tingling, never- before-seen geometry. Trying to make it what it is not diminishes his accom- plishment, which is to al low us to see forms that we perhaps cannot even imagine. In this he follows the footsteps of Leonardo da Vinci when illustrating the solids for Luca Pacioli 's De divina proportione. Unfortunately, we are never told how Picketing discovered

the inversion principle nor why he chose to explore it. In his own essay on "Music and the Inversion Principle," he says that he is now moving toward fractals, so we await the results of this new mathematical explorat ion by the artist.

The main value of the book is this: Pickering shows that advanced mathe- matics does not d e p e n d on advanced tools, only on advanced thinking.

Mathematicians, enjoy. Architects, beware.

Kim Williams Books--P.l.05056220485 Via Cavour 8-10123 Turin, Torino Italy e-maih kwilliams@kimwilliamsbooks.com

Fearless Symmetry: Exposing the Hidden Patterns of Numbers by Avner Ash and Robert Gross

PRINCETON, NEW JERSEY, PRINCETON UNIVERSITY

PRESS, 2006, 302 PP., US $24.95, ISBN-IO:

0-691-12492-2; ISBN-13:978-0-691-12492-6

REVIEWED BY PAMELA GORKIN

I n an article in the Bulletin of the Lon- don Mathematical Society entitled "Generalized non-Abelian Reciproc-

ity Laws: A Context for Wiles ' Proof," Avner Ash and Robert Gross expla in general ized reciprocity to mathemati- cians who know only "basic algebra" and the definition of homology groups. The b o o k Fearless Symmetry: Exposing the Hidden Patterns of Numbers is Ash and Gross 's a t tempt to reach an even broader audience. Fearless Symmetry, as we discern from the cover of the book, is a imed at "math buffs." The au- thors' goal is not to explain the history of Fermat 's Last T h e o r e m - - t h a t has been done, and done well, before. Though the authors often refer to Si- mon Singh's book, Fermat's Enigma, they do not repeat parts of Singh's b o o k in Fearless Symmetry. Instead, Ash and Gross focus on descr ibing the proof of Fermat 's Last Theorem and, in doing

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