A detailed mathematical and historical analysis of four of the challenges encountered in the game L – A Mathemagical Adventure

Introduction

L – A Mathemagical Adventure (hereinafter referred to as L) is a text-based single-player role-play computer game produced in 1984 by the Association of Teachers of Mathematics (ATM, no date), aimed at Key Stages 2 to 4. Originally designed to run on the BBC platform then more or less common in schools, the game is similar in format to other text-based games, such as Zork (Barton, 2007): the player has a quest – in this instance, to rescue a fair maiden named Runia; to achieve this objective, the player must overcome obstacles, defeat fearsome enemies, solve puzzles, and so forth; and the play is controlled by means of a set of relatively simple commands. Despite its single-player setup, there are opportunities for shared gameplay, discussion and co-operation in solving the puzzles presented.

The scenario contains references throughout to the literary works of Lewis Carroll, who – as Charles Dodgson – was an eminent Victorian mathematician with a predilection for games and puzzles. The puzzles are, naturally, mathematical. Some are relatively easy, others less so; and a few seemingly innocent tasks conceal problems of considerable depth and antiquity.  This paper will explore the solution and pertinent history of four of the tasks: the telephone; the bat room; the code room; and baking a cake.

My route through the palace was fairly systematic. In each room or area, I initially travelled east, then explored other routes in an anticlockwise direction. Having selected a direction, I explored it as far as possible, retracing my path backwards in stages to the first room after attempting all the tasks.

The Telephone

The telephone room is at the north end of a corridor to the west of the workshop. The possible phone numbers run from 000 to 999 (see Figure 1). The telephone itself is a red herring in the quest: no useful information is revealed in solving this puzzle. It is the chest on which the telephone sits that is important. Due to the route taken, this was one of the last places I looked: I was laden with objects and it is only thanks to a friendly games guru that I realised I had to drop them all to move the chest. I initially tried a few likely numbers (000, 111, etc.) to no effect. I then tried prime numbers, and got messages for 002, 003 and 005. When 007 yielded nothing, I tried 008, the next Fibonacci number, and on its success I tried the remaining Fibonacci numbers below 999 – 16 in all (see Appendix 1).

The Fibonacci sequence is named after Leonardo of Pisa [1] (O’Connor & Robertson, 1998) -nbsp;undoubtedly the 13th century Italian mentioned in one of the telephone messages. It appears in his Liber Abaci, in which he also introduces place-value decimal numbers. Fibonacci came to the attention of the Holy Roman Emperor, Frederick II, and his court. He was set a series of problems, including a problem regarding rabbit populations (see Figure 2). Starting with a pair of rabbits, and assuming they mature at one month old and produce young at two months old, how many pairs of rabbits would there be at the end of any given month? Fibonacci determined that the number of pairs in any one month was equal to the sum of the pairs in the previous two months: 1, 1, 2, 3, 5, 8 being the number of pairs in each of the first six months.

Had the sequence remained in the sphere of population dynamics, it would have been of limited use: none of Fibonacci’s rabbits ever die, for example, and continue to breed unabated by considerations of food supply, or, indeed, the walls surrounding their habitat as described in the original problem. However, the sequence has found applications in such a wide arena that there is a scholarly journal, the Fibonacci Quarterly, is devoted to its academic study. An examination of the L telephone messages reveals some of the areas in which the Fibonacci sequence has been applied:

001, 377:  meteorology (Swinbank & Purser, 2006);
013:          botany and biology (Knott, date unknown);
089:          economics (Frost & Prechter, 1998);
114:          sports and betting (O’Connell, 2008).

An unusual property of the Fibonacci sequence is that, when each number in turn is divided by its predecessor, the results converge on a curious number known as ϕ (phi), or the Golden Ratio. This number is found in nature, in architecture, art, music, geometry, and visual perception; it appears to underlie our notions of beauty; and has even inspired authors such as Dan Brown. Its pervasiveness is such that some see it written by the hand of God (Meisner, date unknown).

Surprisingly then, under the circumstances, I have been unable to establish a link between Lewis Carroll’s Alice and the Fibonacci sequence, except for the rather trivial coincidence that both began with rabbits.

[1]: Leonardo was a member of the Bonacci family: Fibonacci may be a contraction of figlio Bonacci – son of Bonacci.

The Bat Room

The Bat Room occurs towards the end of the game. The solution to this puzzle, any triangular number between 20 and 90, is one of the most clearly signalled in the game: the room has a triangular floor and walls. If this were an insufficient clue, any mistake causes the large bat to write out a triangular list of numbers, declaring that he hates anything that is not triangular (see Figure 3). This hint is repeated every time an error is made; any attempt to leave without completing the task causes the bats to swarm around the door, making escape impossible.

The triangular numbers are one of a class of numbers known as ‘figurate’, meaning that each term can be represented as a figure: graphically as a pattern of dots, or physically using counters, bottle tops, etc., where they can be used to teach younger students about the relationships between numbers, patterns and graphical representation. Such figures are regular geometric shapes – triangles, squares, pentagons, and so forth (Weisstein, date unknown–a). The figurate numbers are widely studied in number theory, but the triangular numbers attract considerable interest as they pop up in a variety of different equations, such as the sum of consecutive integers, square numbers, Pascal’s Triangle, and even integrals (Weisstein, date unknown – b). The most famous triangle number is the infamous 666, the so-called Number of the Beast. Sadly, the true Number of the Beast according to modern Biblical scholarship is the rather less inspiring 616.

The link to Lewis Carroll’s work in this task is again somewhat tenuous. The Bat was the nickname for Bartholomew Price, a professor of mathematics at Oxford known to both Lewis Carroll and Alice Liddell, the model for Alice, and Carroll parodies a well-known nursery rhyme through the voice of the Mad Hatter:

Twinkle, twinkle, little bat
How I wonder what you’re at!
Up above the world you fly
Like a tea-tray in the sky.
Twinkle, twinkle, little bat
How I wonder what you’re at!

The Code Room

The code room is located to the south of the billiards room, through an anteroom, and was thus located at an early point in my quest. In the code room, the screen fills with apparently random letters, digits and punctuation marks (see Figure 4). I solved this by making a codebook: typing each line of keys on the keyboard, and writing down the resulting text (see Appendix 2). Fortunately, it turned out to be a simple substitution code. Despite my lack of gaming knowledge, I knew that there should be a device in the room which would return its appearance to normal, and this turned out to be the case. I kept the glasses with me for the rest of the quest: this was not necessary for all the tasks, but a few did turn into code if I had to set the glasses down.

Codes have a history beyond the scope of this paper, ranging from military ciphers to Victorian flower language. Much ingenuity has gone into producing unbreakable codes for military and political uses, where a variety of techniques and devices have been employed – once-only codes, and the Enigma machine are examples. While many early codes were simple substitution codes such as found in the code room, there has been a progressive move toward mathematically-based codes and ciphers – and the use of mathematical techniques in cracking them. Naturally, as codes became more complex, computers are needed: it is arguable that computers would not be the household item they are today, if they had not been necessary for code breaking in the Second World War. Today, encryption is a major area in computer and internet security.

Dodgson was greatly interested in cryptography both recreationally and academically, and is known to have produced several ciphers, including a matrix cipher (Abeles, 2005). Some of these ciphers cunningly included nulls – non-code characters, or code characters used randomly – to disguise the meaning further. He used these codes to write letters to friends, and to remember dates.

Baking a Cake

The new kitchen is situated on the east of the palace, almost opposite the old kitchen near the game entrance. A cook needs to bake a cake at least 25cm high, using three ingredients, TOLT, FIMA and MUOT, in grams. The scenario glosses the soup-making episode in Alice in Wonderland, substituting a salty cake for the over-peppered soup.

Initially, I attempted to use the codebook on the ingredient names. I tried to find another code or language that might convert the letters into digits, sums, four-letter number names, ingredients with four letters (e.g., eggs or soda), or even four-letter acronyms. Finally, I tried simply putting in random numbers. More than 100g was declared ‘wasteful’ by the cook. By systematically changing one number at a time, I found a rough relationship similar to a Bell curve between the ingredients and the cake: that is, up to a point, increasing the ingredients increased the size of the cake, but beyond that, the cake decreasing in size. However, I was unable to find any distinct mathematical relationship, such as Pythagorean Triples.

Then I had a happy accident. Having found that 6g TOLT, 10g FIMA and 8g MUOT produced a 25cm cake, I decided to leave the game but forgot to save my position. When I went back in, I accidentally typed 10g MUOT instead of 8g – and it worked. After another 20 or 30 tries, I determined that 5, 6 or 7g of TOLT, 10-18g inclusive of FIMA, and 1-100g inclusive of MUOT, in any combination, produced a cake of the desired height.

It would be disappointing if this task, which must on average take players more time to complete than any other, were merely some kind of trial and error problem. I searched for more possibilities, but with little success. However, amongst the extra pieces of information I gleaned was that Dodgson had done some research on matrices, or ‘blocks’ as he called them, and had produced a method for finding determinants, known as Dodgson’s Condensation, which remains one of the most efficient to date (Dodgson, 1867). I also discovered, quite incidentally, that a contemporary of his, Sylvester, worked on the determinants of rectangular matrices (Weisstein, date unknown-c). It then occurred to me that the ingredients could be arranged into a rectangular 3 x 4 matrix of letters, with 25 perhaps representing a determinant. Unfortunately, I could go no further forward with this idea – partly because the mathematics is currently beyond me, and partly because there is still a crucial element missing: the key to the matrix. I was unsuccessful in finding a copy of Dodgson’s matrix cipher, which may be the key – assuming, of course, that I am not seeing patterns where none exist.

Conclusion

As a disclaimer, I should say that I have an intense dislike of computer games. This dates back to my early experience of programming in the mid to late 1980s, when running silly games on a computer was irresponsible and wasteful. However, I do acknowledge that some games can be educational, and therefore worthwhile. L – A Mathemagical Adventure has thus been an interesting look at the possibilities of such games. I found there is plenty of food for more advanced thought – up to A Level and beyond in some tasks. The literary allusions are amusing, and could be useful in broadening students’ views on mathematics, too.

When drawing the map, I noticed that the palace is missing much of its ground floor. Perhaps it is a fully-working evaluation copy, missing some additional ‘levels’: some objects are of no use; some tasks are unconnected to the quest; and there are unresolved issues at the end of the game. If so, then I would love to play a full version – which is high praise indeed.

Reflective Processes

Once upon a time there was an academic discipline. Its practitioners were learned men, its advocates the wealthy and wise. It sought to explain the nature of man and of his actions. In this, it was successful: all agreed the profiles were highly accurate, the prognostications pleasing. No one noticed the over-detailed bland generalisations – or if they did, only so far as to feel validated to have their own analysis confirmed by such learned men.

That discipline was astrology. Today, it is called social psychology, American-style.

The methodology is quite simple:

1.      Find a normal social process;

2.      Give it a fancy name;

3.      Describe it to death, in lieu of real evidence (Coffield et al., 2004);

4.      Market it to hell and back;

Should an ‘undiscovered’ process prove elusive, you have two options: a) find some piffling absence in a ‘discovered’ process; or b) tweak the existing model (bigger words are good) - and book your spot on Oprah[1].

The result is a proliferation of near-identical theories and instruments, with muddled and over-reaching claims (that learning is the same as knowledge, for example), and little empirical evidence – and that contradictory at best (Coffield et al., 2004, pp 61-69). Kolb’s Experiential Learning Theory (Kolb & Fry, 1975, in Smith, 2001a) is merely Lewin’s Action Research model (Smith, 2001b) turned into a circle, with bigger words for the stages. Gibbs’ Reflective Cycle (1988, in University of Brighton, no date) covers similar ground, but requires examination of one’s emotions: explicitly in one stage and implicitly throughout. However, this rehashes much of Boud et al.’s (1983) Reflection Model, albeit in a simplified form. Little information is available on Gibbs’ model: the book itself is out of print, and I found no mention of any empirical work. Nonetheless, many online professional development guides are based on Gibbs’ model, often aimed at student nurses, teachers, and social workers. Coupled with the lack of any reference to Gibbs’ work in Coffield et al.’s (2004) review, it is as if Gibbs[2] merely wrote a student guide - as many lecturers do - neither intending to pursue the ideas therein academically nor imagining that it would take on such impetus.

A final point is that, while both Kolb’s and Gibbs’ models are used for reflective practice, reflection per se is only part of each cycle, and is little elaborated. Does reflection arise from the full cycle, or only from these one or two stages? Is it necessary to complete every stage, or can they be combined, or skipped entirely?  Is there a set of procedures embedded in these stages, and if so, what are they – and is the rest of the cycle necessary at all? For answers, or at least clues, it might pay to review afresh Dewey’s original discussion of reflection (1910, ch. 6, pp 68-78); unfortunately, I must now move on.

******  Junior School: Reflection using Kolb’s Reflection Cycle.

Concrete Experience

***** ******** and I led a mathematics session with six mixed-ability Year 5 students. There were two main tasks: number skills, using Factor Bugs; and shape naming. The tasks were introduced by themed Bingo games respectively. At the end, the students were awarded certificates and given nets to take away.

Reflective observation

Positive:

1.      Students said they understood factors, squares and primes better.

2.      Students demonstrated a good grasp of shape names and their meanings.

3.      The amount of work planned was almost right for the time slot.

4.      The work lent itself well to the wide range of abilities.

5.      The certificates and nets.

Negative:

1.      We did not plan detailed timings. As a result, we did not fully cover the second task.

2.      We did not stick to the rules for the Bingo games, which overran and impacted on the task time.

3.      The tasks and activities were rather sedentary.

Abstract Conceptualisation

We had the advantage of previous groups’ experience and knew roughly how much work to prepare. However, the slight timing issue suggests that on another visit, we should plan one major theme rather than two, with shorter tasks and perhaps more games and breakout activities.

Rules for games should be planned in advance and adhered to as far as possible. This is important if the number of games and activities were increased.

We did not have competitions this time, as we could not gauge its impact on the dynamic on an unknown group.

Active Experimentation

Our plan for the next session is as follows:

1.      A single theme with related tasks and games, and practical activities get the students moving and doing.

2.      Planned differentiation for tasks.

3.      Game and activity rules agreed in advance and adhered to.

4.      Detailed timings, with some built-in flexibility.

5.      An optional competitive element.

6.      More prizes!

---

[1] Lest this critique be thought unnecessarily harsh, some authors have used the terms ‘disease’ (Coffield, 2008) and ‘snake oil’ (Atherton, 2009) to describe some of the work reviewed briefly here, and associated research.

[2] English, not American, as it happens. A clear case of American cultural imperialism. 

Bibliography

Atherton, J.S. (2008); Reflection; an idea whose time is past. [on-line] UK: Doceo. [Cited: 24/03/2010]. Available at: <http://www.doceo.co.uk/lincoln/index.htm >. 

Atherton, J.S. (2009) Learning and Teaching; Experiential Learning [On-line] UK: Doceo.  [Cited: 25/03/2010]. Available at:  <http://www.learningandteaching.info/learning/experience.htm>.

Boud, D., Keogh, R., & Walker, D. (eds.) (1985). Reflection. Turning experience into learning. [online]. London: Kogan Page. [Cited 24/03/2010]. Partial copy available at Google Books: <http://books.google.co.uk/books?id=xBshIryFdr0C&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false >.

Coffield, F. (2008). Just Suppose Teaching and Learning Became the First Priority... [online]. London: Learning and Skills Research Centre. [Cited 25/03/2010]. Available at: <https://crm.lsnlearning.org.uk/user/order.aspx?code=080052 >.

Coffield, F., Moseley, D., Hall, E., & Ecclestone, K. (2004). Learning styles and pedagogy in

post-16 learning: A systematic and critical review. [online].  London: Learning and Skills Research Centre. [Cited 24/03/2010]. Available at: <https://crm.lsnlearning.org.uk/user/order.aspx?code=041543 >.

Dewey, J. (1910). How We Think. [online]. New York: Dover Publications. [cited 25/03/2010]. Partial copy available at Google Books: <http://books.google.co.uk/books?id=zcvgXWIpaiMC&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false >

Smith, M. K. (2001a). David A. Kolb on experiential learning. [online]. London: The Encyclopedia of Informal Education. [Cited 24/03/2010]. Available at: < http://www.infed.org/b-explrn.htm >.

Smith, M. K. (2001b) Kurt Lewin, groups, experiential learning and action research. [online]. London: The Encyclopedia of Informal Education. [Last updated November 04, 2009] [Cited 24/03/2010]. Available at: <http://www.infed.org/thinkers/et-lewin.htm>.

University of Brighton. (no date). Reflection. [online]. University of Brighton Staff Central. [Cited 24/03/2010]. Download at: < http://staffcentral.brighton.ac.uk/CLT/events/documents/Ramage%20Example%202.doc >.

Resources in Brief

Factor Bugs: http://www.teachers.tv/video/37869, accessed 07/03/2010.

Shapes: http://www.firstschoolyears.com/numeracy/shape/worksheets/name%20regular%20polygons.pdf, accessed 07/03/2010.

Nets: http://www.senteacher.org/Worksheet/12/Nets.xhtml, accessed 07/03/2010.

Polya, G. (1973) How To Solve It: A New Aspect of Mathematical Method. Princeton, New Jersey: Princeton University Press.

George Polya, or Pólya György, was a Hungarian mathematician who spent much of his career at the Federal Institute of Technology, Zurich (ETH Zurich) and at Stanford University. His is a family which suffered, or courted, difficulty. His father, perhaps best known for his Hungarian translation of Adam Smith’s The Wealth of Nations (Polya, 2006), was a lawyer whose academic ambitions were thwarted by institutional anti-Semitism, until he converted to Roman Catholicism. His brother Eugen (Jenö) was a surgeon, famous for a type of gastrointestinal bypass surgery for the treatment of stomach ulcers, who died at the hands of the Nazis; another gifted brother, Laszlo, was killed in the First World War. His great-nephew is the controversial artist and biochemist, Gideon Polya. George seems to have limited his misfortunes to an early punch-up with a student with royal and high level political connections in Göttingen: fortunately for mathematics, this incident partly led to his appointment at ETH Zurich.

Polya performed poorly in mathematics at school, which he later attributed to bad teaching. It was not until he was at university when, following qualifications in literature, he began to be interested in philosophy, that he began his mathematical studies. Despite this late start, he went on to make contributions in a range of topics so diverse as to call to mind the polymaths of ancient and mediaeval times, in a career that extended well beyond his retirement in 1953: remarkably, he was still teaching at Stanford University in 1978, at the age of 91. From the first, his real interest in mathematics seems to have lain in proof and mathematical discovery (Albers & Alexanderson (1985), in O’Connor & Robertson (2002)). Indeed, it is arguably that this underlies his contributions to mathematical education, for which he is most remembered. This legacy comprises a series of books on problem-solving in mathematics, the first of which, How To Solve It, is the subject of this review.

How To Solve It is an unusual book. There are in fact only 36 pages in all which outline Polya’s method: these form Parts I and II of the book, In The Classroom and How To Solve It – A Dialogue, respectively. These chapters are preceded by a two-page outline of the method, presumably intended as an aide memoire, and an Introduction, which outlines and explains the book’s rather unique structure. Part III, the largest section of the book, is entitled a Short Dictionary of Heuristic, essentially an appendix expounding in detail on the problem-solving techniques and ideas mentioned in Parts I and II, together with potted biographies of a few mathematicians and definitions of a few pertinent terms. The final section, Part IV, is the aptly-titled Problems, Hints, Solutions. Throughout the book, subsections within the chapters are numbered, rather more like a textbook or business document. Initially a little disturbing to the reader expecting something a little more literary, this does however reinforce the fact that this book is very much a guide: a textbook in mathematical guidance.

Of the book, Part I – In The Classroom – is the most important. It is this chapter that sets out the purpose of the book: to help improve and develop problem-solving in students. Five main issues are raised and explored briefly:

• Helping the student
• Questions, recommendation, mental operations
• Generality
• Common sense
• Teacher and student. Imitation and practice.

The emphasis is very much on finding problems suited to, yet challenging at, the student’s level of knowledge and ability, and then on unobtrusively guiding the student to a solution. Polya then explains that his model, for reasons of convenience, is split into four phases: understanding the problem; making a plan; carrying out the plan; and looking back. Each phase is then examined in detail, with a running example. Polya recommends a method of a quasi-Socratic, positive questioning – posing a variety of leading questions, in a variety of ways, to steer students gently towards finding ‘their own’ solution. On completion, the solved problem can then be examined – mined, as it were – in the same manner to explore connections to other problems and concepts under the guise of checking, incidentally reinforcing the learning that has just occurred. 

A significant difficulty with the book is the language in which it is written. Originally published in 1945, it has something of the flavour of F. Scott Fitzgerald and the more serious works of P.G. Woodhouse and Dorothy Parker – a convoluted phrasing, an emotionally-detached remoteness, that makes for difficult reading. Polya’s literary and philosophical background shows strongly, although the fact that the text is readable at all by the layman indicates that he must have tried to play both down considerably. Some of the mathematical terminology is outdated: in the first running example, he uses a problem concerning a parallelepiped. Fortunately, I vaguely remembered seeing the –epiped suffix somewhere as meaning a 3D figure, so I did not have to interrupt my reading to look up a dictionary. There is a diagram, but this appears later in the exposition of the problem-solving phases – perhaps too late for another reader. However, I had a difficulty with the printed text: the suggested guidance questions were identified by italicised font, but unfortunately, so are all Polya’s emphases. For the teacher skimming through for ideas, this is unhelpful. Since the edition I read was published in 1973, it is odd that some alternative was not found.

In all, this is an immensely useful book which fully repays the endeavour of reading, despite the difficulties of language. Polya makes a heroic attempt to explain, in albeit condensed and simplified terms, a difficult task. As anyone who has ever attempted to solve a difficult mathematical problem can attest, the thought processes that lead to an answer are proof (sic) against analysis: occasionally lightening fast, at other times tortuously slow, and then again, suddenly productive after perhaps weeks of drought. Polya takes this amorphous thing, and supplies a structure, a logic, and a process from which something might reliably emerge. It is not a book to be read through, short as it is. It is a book that should be kept on a convenient bookshelf, to be referred to, dipped into, and mulled over. Frequently.

References
J J O'Connor, J.J., & Robertson, E.F. (2002). MacTutor: George Pólya. [online]. University of St Andrews: MacTutor. [Cited 26/03/2010].

Polya, G. (1973) How To Solve It: A New Aspect of Mathematical Method. Princeton, New Jersey: Princeton University Press.

Polya, G. (no date). Personal Profile. [online]. Media With Consciousness News . [Cited 26/03/2010].

Polya, G. (2006). Global Avoidable Mortality. [online]. Blogger. [Cited 26/03/2010].

Who Named It? contributors (no date). Eugen (Jenö) Alexander Pólya. [online]. Oslo: Who Named It? [Cited 26/03/2010].

What is this thing called Mathematics?

There is an apocryphal tale of the French mathematicians who published collectively under the name Nicolas Bourbaki, who apparently wrote 200 pages on the number one alone (Bergamini, 1972). I cannot say I believe this story completely, having heard similar anecdotes of topics broad (psychology) and narrow (the modern Italian historical novel). The experts shake their heads and chuckle wryly at the naivety of asking for a definition. Thus forewarned, I shall not fall into this trap: I shall say only what mathematics means to me. Mathematics is, to me, the science of number; its uses, both practical and theoretical (Courant, Robbins & Stewart, 1996); and associated notations. I will explain my take on these in reverse order. Notations I do not regard as especially important: Shakespeare is still Shakespeare, whether in the original Elizabethan English, translated into Norwegian, or performed in British Sign Language. It is unfortunate that notation is often the first thing people think of on hearing the word ‘mathematics’. Perhaps more effort is needed in schools to describe this as a code, shorthand, or language.

The uses of mathematics I regard as practical, meaning applied or descriptive, or theoretical. The practical element is relatively easy to explain: a shepherd counts his sheep to keep track of them, plan grazing, and so on; that number is also a handy descriptor for others, rather than taking them to the hills to see for themselves! This practical element also elucidate processes and phenomena which cannot be observed or comprehended directly, such as the detection of orientation in vision (Anderson, 2001) or the origin of the universe. The phrase ‘theoretical uses’ is something of an oxymoron - perhaps another synonym of ‘use’ would be better here – service, practice, exercise… However, I am inclined to think that theory – the study of a subject purely for the subject’s sake, as it were – is a ‘use’ whose day is not yet come. Reading a popular science book on mathematics some years ago, I was amazed (briefly, and I really shouldn’t have been. It won’t happen again.) that so many ‘pointless’ theories were discovered to have practical applications, some discovered long after the theory was first promulgated (Stewart, 1998). A true application may appear later, or suggest itself immediately: the effort is not wasted.

I define number very broadly and inclusively: the integer 6, a triangle, or the algebraic notation x², are to me number. Similarly, the word ‘happy’, Hals’ Laughing Cavalier, and a smiley icon convey the same emotional information - in informational, pictorial/geometric, and symbolic forms. Where does number come from? While I don’t want to come over all Pythagorean, I feel that number is present in the structure and ordering of the Universe. Not as a separate and discoverable entity in itself, but as our code for the otherwise incomprehensible and ineffable. It should not have surprised me that the Fibonacci Sequence can be seen in the different petal arrangements on flowers – I should have been asking how could petal arrangements be modelled, and what are the biological processes underlying the Fibonacci Sequence’s fit to the data. God may or may not play dice, but is almost certainly a mathematician.

References
Anderson, R. (2001). Detection and Representation of Oriented Contours in Human Vision. Ph.D. Thesis, University of Birmingham.
Bergamini, D. (ed.) (1972). Mathematics. 3^rd ed., Netherlands: Time-Life International.
Courant, R., Robbins, H., & Stewart, I. (1996). What is mathematics?: an elementary approach to ideas and methods. 2^nd ed., Oxford University Press Inc.
Devlin, K., (2000) The Maths Gene, Why Everyone Has It, But Most People Don't Use It. London: Weidenfeld & Nicolson
Richer, É. (no date). PlanetMath [online]. [cited 12^th January 2010]. http://planetmath.org/encyclopedia/NicolasBourbaki.html
Stewart, I. (1998). Nature's Numbers: Discovering Order and Pattern in the Universe.London: Phoenix.

Stewart, I. (1997) Nature’s Numbers: Discovering Order and Pattern in the Universe. 2nd ed., London: Phoenix.

Ian Stewart has recently retired as a Professor of Mathematics at the University of Warwick: he has been made an Emeritus Professor and Digital Media Fellow, with special responsibility for raising public awareness of mathematics (Dudhnath, 2009). He has been awarded the Royal Society’s Michael Faraday Medal (Buescu, 2004), and the Christopher Zeeman medal (The Guardian, 2009), for raising public awareness of and engagement in mathematics. But of his many accolades, arguably the most important to his many readers is his 1999 appointment as an Honorary Wizard of the Unseen University for his collaborations with that great sage of our times, Sir Terence Pratchett (University of Warwick, Public Affairs Office, 1999). He has written around 140 scientific papers and 70 books, of which about one-third are popular science: he claims his secret is that he ‘writes fast’ (Buescu, 2004). This review concerns one of those popular science books, Nature’s Numbers.

The prologue opens with a dream in which a Yahweh-like narrator issues Genesis-style commands that create a universe. This segues into a sequence reminiscent of the virtual reality computers in Minority Report - though not without a hint of Homer Simpson's Halloween trip to 3D-land - as the narrator manipulates his universe. The dream is then revealed to be but an average morning's work for a mathematician. Stewart explains that, with or without the help of advanced computers, this dream is how mathematicians 'see' their subject. Moreover, he promises to try to show his readers the universe through mathematicians' eyes.

Regardless of the dream's authenticity in the real world, the prologue sets the tone for the rest of the book: Stewart uses visual language throughout, deriving imagery from art and music, geography and - of course - nature, via household appliances, to make his points. The overall impression is of a book written for people who consider themselves to be 'arty', more interested in the humanities or the softer social sciences, or simply 'regular joes', for whom the word mathematics conjures up the worst memories of schooldays, with the word science not far behind. To allay further any fears, there follows a chapter on patterns in the natural world: with plenty of examples from stars to starfish, the idea of mathematics as a tool for “recognising, classifying and exploiting patterns” is slipped in almost incidentally in a discussion of snowflakes (Stewart, 1997, p1). The two subsequent chapters explain what mathematics is, and what it is for. Here mathematics is described as a tree, a landscape, a movie, even knitted fabric: clearly Stewart is aiming for the widest possible maths-phobic audience. Thus ends the first third of the book.

Chapter 4 begins the assault on real mathematics, out of our cosy trench into the no-man-in-the-street's-land of calculus. Having been coaxed over the top with propaganda about a hippy Newton dabbling alchemy, however, the promised enemy seems to have deserted the fray, taking most of their armaments: the word calculus itself appears only five times, two of those concealed in a caption to our first diagram. Nonetheless, it is a fairly painless introduction for the layman.

The same cannot be said for the chapter on symmetry (or rather breaking symmetry), a research interest of Stewart's – surprisingly, since symmetry in nature is so visual. I usually experience little difficulty in forming mental images, but I got lost in the prose: a few diagrams would have avoided this. Luckily, I recently watched a television programme about the Belousov-Zhabotinskii reaction, the memory of which helped. However, the chapter is hard to follow: too much is packed in, and the leap from symmetry to what appears closer to chaos (another of Stewart's research interests) is too long for this longest chapter in a short book.

Chapter 7, The Rhythm of Life, is essentially an annotated list of natural rhythms determined by a hypothetical neural oscillator circuit. Principally concerned with control systems in animal gait and firefly flash synchronisation, the question asked early in the chapter regarding why systems oscillate at all reminded me of the motor stereotypies displayed by people with a variety of behavioural and developmental disorders – rocking, tapping, drumming, and so forth. Presumably a similar oscillator circuit causes these behaviours. The remainder of the book continues the theme of occurrences of mathematical concepts such as chaos theory, quantum dynamics, and number sequences in nature. The epilogue is, oddly, a polemic calling for a new type of mathematics, another ‘dream’ called morphomatics. Stewart proposes that this new way of thinking will shed light on how nature’s patterns derive from simple rules, yet arise through networks of great complexity. It is not unusual for academics to introduce controversial ideas surreptitiously – via their PhD students’ theses, for example – but a popular science publication seems a strange choice.

Nature’s Numbers is certainly an enjoyable read, painting a comprehensible word picture of a wide range of mathematics. The language draws heavily on art, music, and animal life: a seemingly deliberate choice, to engage those who self-identify as artistic and see mathematics as entirely separate from their world and interests. Stewart has said in interview that his strategy is “don’t show the public a calculator or formulae” (The Guardian, 2004). True to this, there is only one barely-recognisable formula in the book, expressed in words rather than mathematical notation (Stewart, 1997, p62). This may suit the general public, but I found the overt avoidance of mathematical notation a little annoying: many lengthy descriptions could have been replaced with a few carefully chosen diagrams, or even a few worded equations such as that on page 62. Finally, much of the content reflects Stewart’s research interests over the years, and has been covered in greater detail and rigour elsewhere. For these reasons, I would recommend reading Science of the Discworld instead. The contrast between ‘Roundworld’ and Discworld makes the lack of equations less obvious, and provides a few good laughs along the way.

References
Stewart, I. (1997) Nature’s Numbers: Discovering Order and Pattern in the Universe. 2nd ed., London: Phoenix.
Dudhnath, K. (2009). The Interview: Bookbag Talks To Ian Stewart. [online]. [cited Sunday, 7 February 2010].
Buesco, J. (2009). An Interview with Ian Stewart. [online]. CIM Bulletin No. 16: Portugal, Centro Internacional de Matemática, June 2004 [cited Sunday, 7 February 2010].
Shepherd, J (2004). The magic numbers. The Guardian: London, Guardian News and Media Limited, Tuesday 8 June 2004 [cited Sunday, 7 February 2010].

University of Warwick, Public Affairs Office (1999). Terry Pratchett Receives Honorary Degree from University of Warwick. [online]. [cited Sunday, 7 February 2010].
Pratchett, T., Stewart, I., and Cohen, J.S. (2000). The Science of Discworld. London: Ebury Press.

Charles Seife (2000). Zero: The Biography of a Dangerous Idea. London: Souvenir Press Ltd. ISBN-13 978 0 285 63594 4

As I settled to read this book, my son came running up to me, a look of mock consternation on his face: “Mommy, I have got only NINE fingers!” He proceeded to demonstrate, uncurling each in turn - and indeed, there were only nine fingers. I was called upon to locate the missing finger, which I did. But he counted again, and lo! Only nine fingers! It was all terribly confusing. He then ran off laughing at my befuddlement, shouting “Only joking, Mommy”.

This scenario has been repeated for weeks, ever since he learned about zero – or 'zewo' – at nursery. He happily counts forward and backwards, from and to zero – the latter often followed by “Blast-off!” He sings songs about zero, plays tricks on his poor old parents with zero, returns his plate to the kitchen when it has 'zero food' on it. If a 4-year-old is so comfortable with the idea of zero, why should anyone go to the trouble of writing a book about it?

Charles Seife clearly thought it worthwhile. Currently a professor of Journalism at Columbia, having written for some of the best-known science publications, he holds a Mathematics degree, and studied with Andrew Wiles, the mathematician who solved Fermat's Last Theorem (twice!). He has also done research relating to mathematics, so one can assume that he knows what he is talking about – and has something to say.

However, to the modern reader, there seems little point in writing a book on the subject. Zero, nought, nothing, nada, are all part of the common vocabulary. Yet, as Seife points out early in the book, zero is not a common number. Most of us can go through life without thinking about it. Palaces and pyramids have been built without it, inheritances divided, taxes reckoned, routes planned. Apart from the inconveniently empty state of bank accounts at the end of the month, we might never need to think about zero at all. So, why zero, and why is it dangerous, aside from the exorbitant bank charges for letters to tell us we are out of money?

I came to this book with what I considered a fair knowledge of zero. In addition to its mathematical uses, I was aware, for example, that the Romans and Greeks did not use zero, that it was introduced via Islam from India but probably pre-dated these civilisations, that its absence in the early Christian church is the reason for the arguments over the date of the millennium and more recently over the end of the decade - in short, I knew more than most. Seife covers this and more in a mathematics-free introduction. Writing clearly and with occasional flashes of Brysonesque humour, he uncovers the beginnings of human counting in a 30,000-year-old wolf-bone tally, and talks us through the development of written mathematical notation, taking in religion, mysticism, philosophy, the calendar, art, a smidge of comparative linguistics (zero and cipher the same word!), computer programming, astronomy and both classical and quantum physics along the way – including one of the nicest lay explanations of the uncertainty principle I have ever seen. No potential kittens in thought-boxes here - just the difficulty of measuring the length of a pencil (p 170). Amazingly, this is achieved with very few equations – the first appearing on p 95, almost halfway through the book. What little the author cannot explain in words is conveyed through well-chosen diagrams and images, many derived from contemporaneous sources, but even these support rather than elucidate the text.

The part of the book that worried me most was the section on Newton’s development of calculus (pp 114-126). This was clearly signalled in early chapters, adding to the apprehension. I had a niggling memory about strange notations; and while I have had little difficulty in using and understanding calculus, I was concerned about understanding proofs of the topic from first principles. However, I was pleasantly surprised. Not only is Seife’s explanation clarity itself, I actually recognised the notation and the vocabulary of fluxions and fluents, and even Newton’s ‘dirty trick’ of expunging the infinitesimals that allowed him to develop his proofs. I also seem to have learned L’Hôpital’s rule, though not by name: clearly, my maths teacher must have passed on more information than I remember!

The following chapter, Infinity’s Twin, is, I feel, the least successful part of the book, particularly the section on projective geometry and the complex plane. I came unstuck on the discussion of Riemann spheres, unable to visualise the complex rubber universe of his imagination. However, this may yield to a closer examination of the text.

The remainder of the book deals with zero – and infinity – in what is for me the more comfortable domain of physics: thermodynamics, relativity, quantum mechanics – nothing too strenuous. However, here I found one of the great surprises of the book. I have been myopic most of my life, but was not diagnosed until I was twelve because I had developed excellent coping mechanisms, one of which was looking through a ‘pinhole’ created by pinching my fingers together. I had always been aware of faint stripes when looking through this pinhole, for which I vaguely blamed my eyelashes. Thanks to Seife, I now know these stripes are interference!

In sum, I feel I could recommend this book to almost anyone who is interested in the history of mathematics, regardless of his or her subject knowledge. The mathematical treatments are light: there are few equations to scare off the layperson, and even these are offset by some truly outstanding explanations and examples. For those with an appetite for more, there is a wide-ranging selected bibliography to chew over. The appendices, usually the repository of the esoteric and arcane, are amusing little pieces – build your own time machine; why Winston Churchill is a carrot, and so on. Seife’s real gift is telling the stories behind the mathematics: his book is worth reading just for the image of Pythagoras as some crazed antediluvian guru, railing against the infamy of beans.