and that o n e must have 2 balls to jug- gle "312". This "average theorem" has a nice corollary: A finite sequence of inte- gers will give rise to a juggling sequence only if the average is an integer. How- ever, this condi t ion is not sufficient. As an e x a m p l e , cons ider 321. The average is 2, but the assoc ia ted funct ion n ~-+ n + g(n) is not o n e - t o - o n e .

Surprisingly, it can be s h o w n that the c o n d i t i o n is nearly sufficient: If the av- erage of a finite sequence a of n o n -

nega t ive integers is an integer, then there exists a permutat ion of a which is a juggling sequence. The proof is rather involved, but never the less - -as it is to be expec ted- -e lementa ry .

Now suppose that someone is jug- gling a certain juggling sequence a 0 . . . ap_~ of length p. After some time it might be desirable to pass without in- ter~tpt ion to another pattern. Fix an i and an in teger d such that 0 < d-< a~. If a i is rep laced by a~+d + d and a~+a by a ~ - d, then this will be again an

admissible sequence. For example, the ball thrown at time unit i will be caught at that moment when in the original pat- tern the ball thrown at time unit i + d is going to land. (For example, in the c a s e p = 3 , i = 0 , and d = l one will pass from 642 to 552.)

The operat ion of generating a new sequence in this way is called a site swap. As another, more elementary operation, one can cons ider the wclic sbifi where o n e replaces c~ ap_l by a l a 2 " " a p - l a o . It is rather surprising that site s w a p s and cyclic shifts suffice to pass from any juggling sequence to any other provided that length p and the number of balls (ao + �9 " �9 + ap_ 1)/P co- incide. For example, one could start with bb" �9 �9 b (which is identical with the cas- cade b) and arrive at ( p b ) O " " 0 (this corresponds to the pattern where at every p t h time unit a ball is thrown so strongly that it is in the air for pb time units).

The book also contains many results cen- tering on counting: What is the number of juggling sequences with a prescribed prop- erty? As a sample theorem, consider the case of minimal juggling sequences. (A juggling sequence a0 �9 �9 �9 a/,-a is called minimal if it cannot be written as a repetition of smaller juggling sequences; so 441 is minimal, but 44444 is not.) It can be s h o w n that the number of minimal jug- gling sequences of per iod p with aver- age b is precisely

here/ . , is the usual MObius funct ion, a n d

juggling sequences which are identical

up to cyclic permutations are identified.

M o r e S o p h i s t i c a t e d De f in i t ions Juggling sequences are the bas ic objects of interest; their study covers the first chapters of Polster 's book. More com- plex notat ion is needed when weaken- ing the assumptions of the first ap- proach. How can o n e deal with the possibility of throwing and catch ing

more than o n e ball in a given time unit (multiplex juggling)? What modi f i ca t ions

are necessary if not o n e but several jug- glers are involved (mult ihand juggling)?

Formally, a multiplex juggling se- quence is a finite sequence of finite nonempty ordered sets of non-negat ive integers. These sets encode what kinds of throws are made on every beat. For example, {1,4}{1} has to be realized as follows: On the first beat, two balls are caught, they are immediately thrown again, the first one to height one, the s e c o n d to height four; on the s e c o n d

beat, one ball is caught and thrown to

height one; the act ions o n beat o n e a n d two are repeated aga in and again.

Mult ihand juggling n e e d s an even more elaborate notation, one has to pass to matrices, the columns of which prescr ibe what has to be done at a cer- tain time unit. For example, o n e can

learn from the s e c o n d entry of such a column what action "hand" number two is assumed to perform at the corre- sponding time: catch 5 balls, throw two of them such that they arrive at "hand" number one two units later and throw the remaining balls such that they are at "hand" four at the next time.

The emphasis is similar to that in the case of the above juggling sequences of a single player. One can prove average theorems, it is possible to c o u n t the number of essentially different patterns,

the mutual de pe nde nc e of these pat- terns can be visualized by graphs, etc.

With the notations of mult ihand jug- gling at hand it is also possible to

change the point of view. If b balls are juggled by b hands in a certain way, one may interchange the roles of balls and hands: Fix the posi t ion of the balls and let the hands move! Now the balls juggle the hands using a pattern which

is in a sense dual to the original one . Claude Shannon, the famous informa- tion theorist, w a s o n e of the first to in- vestigate this "duality theory".

The book contains much more . A survey of the history of juggling, hints for jugglers, the connect ions with bell ringing, to ment ion a few. For me it was very stimulating. I tried to jugg le s o m e of the simpler juggling s e q u e n c e s , a n d I a l so w r o t e a computer p rogram to visualize even very complicated a 0 , . . . , ap 1 (which no human being will ever manage). The mathematics involved are

very interesting: I h a d n o t e x p e c t e d to see so many connect ions with algebra, graph theory, and combinatorics.

In a review of this b o o k in the No- tices" o f the AMS (January 2004) Allen Knutson argued that the historical part is not free of errors. I d o not have the b a c k g r o u n d informat ion to decide whether this criticism is justified. For me, it is a fascinating book from which I learned a lot.

Fachbereich Mathematik und Informatik Freie Universit~t Berlin Arnimallee 2-6 D-14195 Berlin Germany behrends@math.fu-berlin.de

Flatterland by Ian Stewart

CAMBRIDGE, MASSACHUSETTS, PERSEUS

PUBLISHING, 2001, 301 PP. $25.00

ISBN 0-7382-0442-0

The Annotated Flatland A romance of many dimensions by Edwin A. Abbott

Introduction and notes by Ian Stewart

CAMBRIDGE, MASSACHUSETTS, PERSEUS

PUBLISHING, 2002, 239 PP. $31.00

ISBN 0-7382-0541-9

REVIEWED BY VAGN LUNDSGAARD HANSEN

i n 1884, Edwin Abbott Abbott (1838- 1926) publ ished his r enowned book "Flatland", which has amused read-

ers n o w for m o r e than a century. In 2002 Perseus Publishing reissued the

�9 2006 Springer Science+Business Media, Inc., Votume 28, Number 2, 2006 89

book under the title "The Annotated Flatland" with carefully written com- ments and notes by Ian Stewart, in- cluding a comprehensive biography of Abbott. Although he showed great tal- ent for intellectual work during his years of study at St. John's College, Univer- sity of Cambridge, England, 1857-1862, Abbott chose another career than uni- versity and became a priest in the An- glican Church in 1863. He spent most of his life as an enthusiastic and capa- ble educator and became headmaster of the City of London School in 1865, stay- ing in this position until his retirement in 1889. In mathematical circles, Abbott is known only for his book "Flatland".

In Flatterland, the distinguished math- ematician and popular science writer Ian Stewart takes on the almost impos- sible and daring task of writing a con- temporary sequel to Abbott's classic. In his charming joking style of exposition, Stewart draws on the contemporary fan- tasy world and phenomena that in many cases would not have made any sense in Abbott's times. In other cases, the jokes are slightly strained, such as the language joke on page 238: "the nat- ural logs in the hearth were burning brightly". In general, though, Stewart is entertaining and, admittedly, the re- viewer laughed even in the case men- tioned.

A review of The Annotated Flatland seems improper at this place since it will amount to a review of Abbott's clas- sic, for which time has already revealed its great merits. There is, however, good reason for an appraisal of Stewart's scholarly and elegantly written com- mentaries, many of which are pearls. This holds in particular for his comments on nine quotations in Flatland from Shakespeare (the only author quoted by Abbott), and for several short biogra- phies of mathematicians and physicists of importance for understanding the profundity in Abbott's book.

OOO

Stewart's book Flatterland is a de- manding one. The reader has to be ac- quainted with much of the language of modern physical theories to appreciate the elegance with which Stewart writes about the continuing attempts by sci- entists to understand the shape and ori- gins of the universe and the nature of space, time, and matter. Ever since Ein- stein formulated his revolutionary ideas of general relativity in 1915, these ef- forts have been intimately related to de- velopments in differential geometry and topology, and contemporary mathemat- ics altogether. It is difficult to tell how Stewart's exposition will appeal to the lay reader, but an educated reader with some background in science can gain insights into many recent concepts from cosmology that regularly pop up in the media, such as "big bang", "gravitational lenses", "antimatter", and "wormholes". A slightly annoying thing is Stewart's way of naming the persons in his nar- rative by concatenations like 'Haus- dorffbesicovich' (page 69); it is easier with 'Alberteinstein' (page 45); or is it?

Flatterland is a science-fiction story about Victoria Line, who comes upon her great-great-grandfather A. Square's diary, hidden in the attic. The diary helps her to contact the Space Hopper, who tempts her away from her home and family in Flatland and becomes her guide and mentor through 10 dimen- s ions- the manifold dimension of the extension of space-time describing the universe. During the journey with Space Hopper, we experience large parts of contemporary mathematics through the virtually enhanced eyes of the truly amazed Victoria Line and registered in her 'Diary Dear'.

It is impossible to mention all the things Victoria Line discovers during her journey, but here are a few" highlights. To interpret the diary of her great-great- grandfather, Victoria must first under- stand some basic elements of cryptog- raphy (Chapter 2). When she first meets the Space Hopper he gives her acom-

prehensive introduction to seeing 3-di- mensional things through 2-dimensional sections and to Kepler's sphere-stacking problem (Chapter 3). Stewart has earlier written about fractals, and lovely de- scriptions are given of fractal dimension and Mandelbrot sets in Chapter 5. In "The topologists' tea-party" (Chapter 6) we meet the MObius cow and milk in Klein bottles, and also the Alexander horned sphere. The projective plane is discussed as a one-sided surface, and there is a good introduction to finite projective planes (Chapter 8). The var- ious types of geometries as described by Felix Klein in terms of their trans- formation groups are beautifully ex- plained in Chapter 9. In "Cat Country" (Chapter 11), there is a well-thought- out exposition of the theory of ele- mentary particles and the duality be- tween waves and particles. On it goes with a description of general relativity in "The domain of the Hawk King" (Chapter 13) and a beautiful introduc- tion to Hawking's universe and the Penrose map in "Down the Wormhole" (Chapter 14). Through the remaining part of her journey, Victoria Line gets a stimulating introduction to theories about the shape of the universe, in- cluding an excellent explanation of the redshift and presentation of the argu- ments that the universe expands.

Ian Stewart has written many excel- lent books, mathematics textbooks as well as popular writings. Flatterland is not his best book, but it is still an en- joyable and stimulating book from which to capture the spirit of some ba- sic trends in contemporary mathemat- ics and science and to be entertained as well. Ian Stewart is indeed a master of exposition.

Department of Mathematics Technical University of Denmark Matematiktorvet, Building 303 DK-2800 Kgs. Lyngby Denmark e-mail: V.L.Hansen@mat.dtu.dk

90 THE MATHEMATICAL INTELLIGENCER