Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters tu the editor should be sent tu either of the editors-in-chief, Chandler Davis or Marjorie Senechal.
Mathematics and Narrative Marjorie Senechal's article [1] is a de lightful account of a meeting organised by the group Thales and Friends, at tempting to explore the relationship he tween mathematics and narrative. What could he more beguiling than dis cussing this on the Aegean? But it seemed strange to read a paper on mathematics and narrative that doesn't mention the most commercially suc cessful hooks ever written by a mathe matician: Alice 's Aduentures in Won derland and Through the Looking Glass. Neither is there any reference to the most successful paramathematical hook of recent years (and here I'm guessing even more wildly than in the previous sentence), Mark Haddon's The Curious Incident of the Dof!, in the Night-Time. Neither Carroll/Dodgson nor Haddon, who clearly know how to write some thing that people actually want to read, was trying to put mathematics across. They both produced beautifully written stories with a mathematical theme. Car roll just couldn't help being playful with mathematical and logical ideas: the mathematics in the background shines through. Haddon, who isn't a mathe matician, has written a lovely story about a strange boy with some mathe matical talent that (as it seemed to me) has captured a little of the feeling of what doing mathematics is like. Thales and Friends' website [2] is also a pleasure to browse through. I was par ticularly interested in the papers by Mazur [3] and Chaitin [4] . Mazur has made a heroic attempt to classify the different ways in which stories can he used in "mathematical exposition". Rut I think there is a problem here in his use of the word exposition. Stories and exposition don't seem to go together naturally: stories are surely more about exploration than exposition. Carroll is exploring logic in his Alice hooks: set up a crazy situation, apply the rules of logic, and see where we get to. And Haddon is explorinJ< the relationship he tween mathematics and autism, which, by the way. is exposed by James in [5]. Chaitin, in his paper, contrasts two views of mathematics: Hilbert's attempt to describe it as a closed, formal sys tem of axioms, rules of deduction, and so on; and the Lakatos-Chaitin approach
to mathematics as quasi-experimental. The Hilbert viewpoint demands expo sition: here is mathematics all wrapped up in this formal system, now we must expose it. The Lakatos-Chaitin view point suggests exploration: let's look around us, move off in an interesting direction, and see where it takes us. My suggestion is that, following Carroll and Haddon, you are much more likely to write a readable narrative if you can adopt the Lakatos-Chaitin-exploration point of view. The contrast between exposition and exploration, between a formal system and a quasi-experimental approach, seems very similar in spirit to the de sign/evolution dichotomy discussed in [6] . If you take a design point of view, then narrative, if it has any role at all , is merely a pedagogical device to sugar the pill or to set the mathematics in a wider context: a taxonomy for this is very well set out by Mazur. But with an evolutionary/exploration viewpoint, narrative is at the centre of the action. How could one provide any under standing of an evolutionary world bet ter than by telling stories' No theory of economic development is going to give one a better idea of what happens in a market-place than the story of the evo lution of the internet. And Haddon's story offers something that no formal theory of autism will ever give. Some businesspeople have adopted the idea of writing stories about the fu ture to help them and their colleagues to understand possible developments in the business environment that may be of vital importance. These stories are called scenarios. (I would prefer to use the simple word story under all cir cumstances, but I have often encoun tered resistance unless I adopted more bureaucratic words like narrative, fic tion, and scenario. ) This approach was pioneered in the energy company Shell International (see [7]) in the early 1970s as a way of opening the minds of se nior management to the possibility that the price of crude oil might one day rise above $2 per barrel , and has been used in Shell ever since. It has a num ber of advantages including: Having more than one scenario re minds people that the future is un predictable. Before reading Chaitin's •
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paper, I had thought this might not be relevant to mathematics, but now I'm not so sure. His idea of adding new axioms such as the Riemann Hypothesis (RH) in a quasi-experi mental way seems rather analogous to the writing of different scenarios to explore the future. In fact I would like to challenge mathematicians to write two narratives about mathe matics in one of which RH is true and in the other of which RH is false. If this turns out to be not possible, then I suppose the obvious next question is: does RH really matter? (I would also very much like to read a story set in a world in which the Continuum Hypothesis is false.) • The scenarios help to liberate people from "common sense" and from their prejudices. The different stories allow unusual ideas to be put forward and discussed as pieces of fiction rather than matters of life and death (which they can literally be, for example when this technique was used in a meeting between warring parties in South Africa towards the end of the apartheid era). This idea is reminis cent of the development of non Euclidean geometries, a story that often inspires me when I write sce narios. • Occasionally, the same outcome crops up in the course of two quite different scenarios. This strengthens belief that this outcome might actu ally happen. Compare and contrast this with the famous result of Skewes and Littlewood that can be proved in two quite different ways depend ing on whether RH is true: far from having their belief in the result strengthened, some mathematicians, of course, have refused to accept that this constitutes a proof at all. The most significant benefit of sce narios, in my view, is in the under standing they bring to thinking about developments in the business environ ment, and hence an enormous im provement in the quality of discussions about unfolding events. (This is why [8] is called The Art of Strategic Conversa tion.) As events and trends happen, one can look at them and say, "Ah, yes, this is what one would expect in scenario A, but on the other hand that looks like scenario B". If you can say something like that, you have understood what is
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going on. Chaitin says that understand ing means compression, "the fact that you're putting just a few ideas in, and getting a lot more out". I think he's right in this, and that in practice the com pression is often (always?) in the form of a narrative. When I say that I under stand why somebody became angry, I mean that there is a narrative starring certain characters and featuring events and motivations, and that the anger fits into this narrative. When we say that we understand why the planets move round the sun in ellipses, we mean that this fits into the narrative of Newton, gravity, cal culus, and so on. (This narrative will be more or less sophisticated for different people.) Here is an example from math ematics, which I think is archetypal. In [9] , Singer says, "We should not be too surprised that mathematics has co herent systems applicable to physics. It remains to be seen whether there is an already developed system in mathemat ics that will describe the structure of string theory. [At present we do not even know what the symmetry group of string field theory is.]" Singer is trying to fit string field theory into a narrative. The narrative is called "Symmetry Groups", and Singer might think of this narrative as starring Galois and Einstein and a cast of thousands, or he might think of it, as with Mazur's story about rational points of elliptic curves, as a narrative of ideas. But it's a narrative. And if he finds out what the symmetry group of string field theory is, he will be justified in saying, "Now I understand!" in just the same way as a businessperson faced with the prospect of new environmental legisla tion can say, "Yes, I understand what's happening, it fits into one of my sce narios". If the symmetry group of string field theory is never found, then either string field theory will be abandoned or an entirely new narrative will need to be written. I can't stop without taking issue with a statement in [1]. (I think it is a remark made by a participant at the Thales meet ing rather than necessarily the opinion of the author. ) The statement is, "Popu lar math books must not mislead. They must tell the whole truth and nothing but the truth". If this were taken literally (and I imagine that advocates of the whole truth and nothing but the truth would like to be taken literally) it would simply mean the death of popular math-
ematics. For the whole truth includes all the gory details, technical background, and arcane exceptions. This isn't popu lar mathematics, it's mathematics. Popu lar mathematics should certainly not mis lead, but it can't afford to be cluttered up with the "whole truth". So what should popular mathematics do? It should be .faitf?ful to the narrative. The astute reader will have noticed that I have managed to write 1 ,500 words on mathematics and narrative without ever saying what I think a nar rative is. But I'm in good company be cause as far as I can tell none of the Thales people has defined a narrative either. (Mazur does partly.) So on the principle of rushing in where angels fear to tread, here is a stab at a definition. "A narrative is a sequence over time of related specific events, emotions, or ideas designed to hold the attention of the reader, listener, or viewer. It is not the laying out of a general situation or theory in its entirety: but a good narra tive will help people to gain a better un derstanding of the general situation. " Eric Grunwald Mathematical Capital 1 87 Sheen Lane London SW1 4 8LE UK e-mail:
[email protected] REFERENCES
1 . Marjorie Senechal, "Mathematics and Nar rative at Mykonos", Mathematical lntelli gencer, Vol 28 (2), 2006 2. http://www.thalesandfriends.org 3. Barry Mazur, "Eureka' and Other Stories", June 29, 2005, on [2] 4. Gregory Chaitin, "Irreducible Complexity in Pure Mathematics", on [2] 5. loan James, "Autism in Mathematicians", Mathematical lntelligencer, vol 25 (4), 2003 6. Eric Grunwald, "Evolution and Design Inside and Outside Mathematics", Mathematical lntel/igencer, vol 27 (2), 2005 7. Pierre Wack, "Scenarios, Uncharted Waters Ahead", Harvard Business Review, Sep-Oct 1 985, and "Scenarios, Shooting the Rapids", Harvard Business Review, Nov-Dec 1 985 8. Kees van der Heijden, "Scenarios, The Art of Strategic Conversation", John Wiley & Sons, 1 996 9. Martin Raussen and Christian Skau, "Inter view with Michael Atiyah and Isadore Singer", EMS, September 2004
The Hidden Mathematics of the Mars Exploration Rover Mission UFFE THOMAS JANKVIST AND 8J0RN TOLDBOD
0
n January 4, 2004, Mars Exploration Rover (MER) A, named Spirit, entered the Martian atmosphere. The spacecraft, weighing 827 kg, was travelling with a speed of 19,300 km/h. During the next four minutes the ve locity of the craft was reduced to 1 ,600 km/h at the meet ing between the Martian atmosphere and the aeroshell of the craft. At this point a parachute was deployed and the velocity decreased to about 300 km/h. At a point 1 00 m above the Martian surface, the retrorockets were fired to slow the descent, and finally the giant airbags were inflated. The airhag-covered spacecraft hit the surface of Mars with a velocity of ahout 50 krn/h. The airbag ball bounced and rolled for about 1 km on the Martian surface just as Mars Pathfinder had done seven years earlier. When the landing module finally came to a stop its airbags were deflated and retracted and its petals were open. After six months in space the encapsulated rover, Spirit, could at last unfold its solar arrays. Three hours later Spirit transmitted its first image of the Gusev Crater to Earth. On January 15 Spirit left its land ing module and drove out onto the surface of Mars. Ten days later, on January 25th, the entire scenario was repeated at Terra Meridiani with Mars Exploration Rover B, named Opportunity. 1
Introduction In March 2005 we spent a week at NASA's Jet Propulsion Laboratory QPL) as part of our joint master thesis 2 at the
Opportunitys heat shield and the shield's place of landing as seen from the rover. http: I /marsrovers. jpl.
Figure I.
nasa.gov/gallery/press/opportunity/20041227a/ 1NN325EFF40CYLA3P0685L000Ml-crop-B330Rl_br.jpg
mathematics department of Roskilde University, Denmark. The purpose of our stay was to conduct a small investiga tion of the mathematics in the Mars Exploration Rover (MER) mission being performed at JPL. Professor Emeritus Philip ]. Davis had more or less suggested such an investigation in an article published in 2004: Consider the recent flight to Mars that put a "laboratory vehicle" on that planet. [ . . . ] Now, from start to fin ish, the Mars shot would have been impossible without a tremendous underlay of mathematics built into chips
'This information in great part originates from http: I lnssdc. gsfc. nasa. gov ldatabaseiMasterCatalog?sc=2003-027 A and http: I lnssdcgs fc. nasa. govldatabaseiMasterCatalog?sc=2003-032A
2The thesis consists of the texts [5], [6], and [7] and can be found in its original Danish version as IMFUFA-text number 449 at imfufateksterlindex.htm
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http: 1 lrrunf. rue. dkl
and software. It would defy the most knowledgeable his torian of mathematics to discover and describe all the mathematics that was involved. The public is hardly aware of this; it is not written up in the newspapers. [2] Although we are not "the most knowledgeable historians of mathematics," we nevertheless decided to engage in Philip Davis's project. Unfortunately, newspapers are not the only place in which this wasn't written up. In fact, finding ex tensive literature on the mathematics of the mission was so difficult that we decided to base our investigation on inter views. Hence the long travel to Pasadena, California. While in the US we decided also to visit Davis at Brown Univer sity in Providence, Rhode Island, to discuss our pending in vestigation with him. Davis advised us to tly to gain an in sight into the employees' personal motivations for working in the aerospace industry, as well as an understanding of the nature of the mathematical work performed at JPL [ 1 ] . We also decided to look at what might be referred to as the ex ternal influences on the daily work, such as deadlines and economic limitations-the basic work context. One of the more interesting aspects of our investigation quickly turned out to be the invisibility of the mathematics in volved in the mission. The fact that the mathematics involved is hidden from the public may seem natural, hut parts of the mathematics of MER are also hidden from the scientists par ticipating in the mission. In fact the hiding, or invisibility, of the mathematics in MER occurs on several levels, some in tended and some not. The aim of this a1ticle is to present some of the mathematical aspects of the MER mission and to discuss the way they are hidden in the mission, as well as the effect the work context had on the process5 Much of the ac count is built by letting the JPL scientists speak for themselves, i.e. , by frequently quoting from our interviews. 1
JPL Scientists at Work We found that, in general, JPL's scientists arc people with the highest educational level who join the institution shortly after completing their university studies. They are driven by a desire to he part of the aerospace industry and a passion for planetary exploration. To some extent, they were also drawn to JPL by a fascination with the mathematical, phys ical, and engineering problems involved in space explo ration; hut as a motivating factor this seemed secondary.
Among the first to discuss the mathematical aspects of the work at JPL with us was Jacob Matijevic, a mathemati cian who had been with JPL for a long time. Particularly we discussed the modelling aspects of the work, which takes place before the actual mission is set in motion. A mission like MER is to a large extent about being able to predict how the technology onboard the craft is going to behave in space or in the Martian environment. Once the craft is flying it is impossible to make adjustments re quiring more than a radio signal. Everything must therefore function as expected. Take for instance the Mars environ ment's influence on the instruments onboard Spirit and Op portunity. You have to have very precise knowledge about the distribution of heat inside the rover and how this af fects the instruments. To acquire such knowledge, virtual models of the rovers are built in software so that the ther mic conditions can be simulated. Such thermic models are typically based on a number of differential equations which are solved within the programs. The work for the JPL em ployee consists of building the virtual model of the rover. The exact method of solution which the program imple ments is secondary, as long as it works and is not too slow. According to Matijevic [ 1 1 ] you also need models of how the environment depends on the seasons on Mars to he able to predict the concrete influence on the instruments. These models are partly based on data from the different Mars orbiters and partly on concrete measurements per formed on the Martian surface. The correctness of the sur face measurements depends on how good the description of the instrument's behavior in the Martian environment is, and it cannot be guaranteed. By comparing the data from the orbiters with the surface measurements, a more accu rate picture may arise; this may then be used to modify the models, so that they slowly become better and better. All of this is done in software. Regarding the models of how the seasons affect the Mars environment, it is probably fair to compare the work at JPL with that performed by an in stitute of meterology. Matijevic reported, When I first arrived here over twenty years ago there were still efforts to hand-implement certain mathemati cal models for certain applications. And there were spe cialist applications here for spec ial ists in the applied mathematical sciences who worked here to make those
3An expanded Danish version of this article with a slightly different angle has also appeared in the Nordic mathematical journal Normal (13]. 4Transcripts of these interviews in full, along with our conversation with Ph1hp Davis, can be found in (7].
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i n these old journals, some used pre cisely this feature. Perhaps the first 9 X 9 grid to be compl eted which had this breakdown into 3 X 3 subsquares was one published by H. Mary in the Revue des jeux in August 1 89 1 , reproduced here as Problem 3 (see also Figure 2). We may wonder if Colette (the famous author of the Claudine series, Cheri, and Gigi) attempted this problem, for she had a regular gossip column in the same weekly. Mary had earlier published a similar problem in the same journal, hut it was misstated, and no initial array was given. Afterward, many other 9 X 9 grids with 3 X 3 subsquares were pub lished by various authors in several pe riodicals. Such arrays were often called "squares with compartments."
Grids Decomposable into Two Sudokus A bimagic square is a magic square which remains magic when all its en tries are squared. After publishing the first-ever bimagic square, which was H X 8 [1][2][5] [ 1 1 ] , the Frenchman G.
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grid with 3 X 3 subsquares. (See Problem 3 . )
Pfeffermann carried on and published the first 9 X 9 bimagic square in 1891; see Problem 2 below. Like t h e other squares, these first himagic squares were published as games in which missing en tries had to be filled in. Once Pfeffer mann had opened the way, numerous other 8 X 8 and 9 X 9 bimagic squares appeared in various periodicals, for ex ample, Problem 7 by Bmtus Portier. A most interesting hidden feature of this example: unlike the predecessor by Pf effermann, is that it can be decomposed into two Sudokus, and this even helps in solving it! But a square need not be bimagic to be decomposable into two Sudokus, as witness Problem 4 from Le Siecle. This surprising decomposibility comes from the squares being eulerian squares, also called Greco-Latin Squares. Gaston Tarry, who met Brutus Portier in Algeria, became interested in magic squares and Latin squares. In 1 900, Tany gave the first proof [13] of the impossibility of the famous problem of 36 officers posed by Euler [9] in his 1782 paper: arrange a delegation from six regiments, each of which supplies
six officers of different ranks, in a 6 X 6 square in such a way that each row and each column contains one of
ficer from each regiment, and one of each rank. It is now known that for each n2 other than 4 and 36, there are solu tions to the problem of n2 officers, known as n X n eulerian squares; see for example Figure 3. Not every 9 X 9 eulerian square is obtained from two Sudokus-i.e., has the subsquare stmcture. But the solu tion of Problem 4 or Problem 7, when put in terms of two Sudokus, becomes a 9 X 9 eulerian square, a solution of the problem of 81 officers. Though this relationship was not mentioned in the
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Gaston Tarry (Villefranche de Rouergue 1834-Le Havre 1913).
statement of these two problems in the 1 890s, it was well known and discussed, and in fact was the subject of Problem 6 by A. Huber. Its solution, published in 1 894 in Les Tahlettes du Chercheur, brings this out clearly: the 9 X 9 square is obtained from what we would call today two Sudokus, printed so as to dis play their 3 X 3 subsquares, each con taining numbers 1 through 9. It is probably by analysing the euler ian structure of these bimagic squares published by his friend Brutus Portier and others that Tarry thought of his marvellous general method of con structing multimagic squares [5]; Tany's first note on his method [ 1 4] was pre-
sented by Henri Poincare in 1906. The book of General Cazalas [6] from 1934 gives details of the method and many constructions of 8 X 8 and 9 X 9 bimagic squares using it, with slight im provements by Cazalas. An important remark which neither Tarry nor Cazalas made: all 9 X 9 bimagic squares constmcted with the Tarry-Cazalas method are combinations of two Sudokus, fully organized into sub squares [5]. Of course, all the pairs of Su dokus don't produce a bimagic square, and all the bimagic squares (those not constructed by the Tarry-Cazalas method, like Pfeffermann's Problem 2 here) can't be generated by a pair of Sudokus. The Viricel-Boyer method, which we use for tetra- and pentamagic squares (squares remaining magic when entries are raised to powers up to 4, resp., 5), as described in [1], is much inspired by the Tarry-Cazalas method.
All the Ingredients of Sudoku Were There All the problems discussed thus far are magic squares, and although they use the underlying idea of Sudokus, they don't directly ask the reader to deal with a simple Latin square involving only the numbers 1 through 9-which is what makes the charm and attraction of Sudoku. Nevertheless this purely Latin square feature was in use at that time. Thus Problem 5 by the same Bru-
tus Portier is a 9 X 9 Latin square to be completed, but it fails to be a Sudoku because it lacks the notion of 3 X 3 subsquare. Problem 9, from the daily La France, maybe comes closest: it is a 9 X 9 Latin square to be completed, and each of the 3 X 3 subsquares does have all the integers 1 through 9. Isn't it too bad that the author, B. Meyniel, didn't mention it! The array appeared in print without any indication of these subsquares-which, however, is done in other problems we have seen. This partitioning into 3 X 3 sub squares is used, by the way, in another problem not really related to Sudoku : Problem 8, a little goody from L 'Echo de Paris. I have given some examples, among the most pertinent I have been able to find, of the incredible proliferation around 1890 of diverse problems close to Sudoku: • 9 X 9 arrays with 3 X 3 subsquares • entries to be filled in with numbers • and sometimes only a single use of the numbers 1 through 9 in each row, each column, even each subsquare. Absolutely all the ingredients are there. In the wonderland of squares conceived by many different authors over several years, it is astonishing that none of them thought of combining all the ingredi ents and proposing the game of Sudoku exactly as we know it today. They came so close!
Figure 4. L'Echo de Paris, 1894. Is this a Sudoku? (for details, see Problem 8).
© 2007 Spnnger Sc,ence+Business Media, Inc . .
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Not that my research has been ex haustive. There are so many grids in so many dailies from the end of the 1 9th and beginning of the 20th cen turies in France. Maybe some lOOo/o Su dokus are hiding there somewhere? Readers who enjoy rummaging in li braries, take note. But consider how very close these problems are to Sudokus. Thus if you decompose Problem 4 or 7, or mark the 3 X 3 subsquares on Meyniel's Problem 6, you solve them just as you would a present-day Sudoku, yet they offer a taste of something created more than a century ago . . . and then forgotten!
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REFERENCES
Sudoku, Pour La Science, N°344, June
The various French newspapers and magazines published at the end of the 1 9th century and the beginning of the 20th century. And:
2006, 8-1 1 & 89 [5] Christian Boyer, Multimagic squares web site, with some Sudoku pages, www. multimagie.com/indexengl.htm [6] General Cazalas, Carres Magiques au de
[1 ] Christian Boyer, Les premiers carres tetra et pentamagiques, Pour La Science,
gre n, Hermann, Paris, 1 934 (7] Jean-Paul Delahaye, Le tsunami du Su
No286, August 2001 , 98-1 02
doku, Pour La Science, N°338, December
[2] Christian Boyer, Some notes on the magic squares of squares problem, The Mathe 27(Spring
2005, 1 44-1 49 [8] Jean-Paul Delahaye, The science behind
2005),
Sudoku, Scientific American, 294(June
[3] Christian Boyer, Letter- Magic squares,
[9] Leonhard Euler, Recherches sur une nou
matical lntelligencer, 52-64
2006), 70-77
Mathematics Today, 42(April 2006), 70
velle espece de quarres magiques, Ver
[4] Christian Boyer, Les ancetres fran