# Big Think Interview With Benoit Mandelbrot

**Benoît B. Mandelbrot**is a French and American mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was born in Poland and educated in France, and is now a dual French and American citizen. His books include the classic "The Fractal Geometry of Nature" (1982).

**Benoit Mandelbrot:**Benoit Mandelbrot, Sterling Professor, Emeritus at Yale University, IBM Fellow Emeritus at IBM Research Center.

**Question: **What is fractal geometry?

**Benoit Mandelbrot:** Well, regular geometry, the geometry of\r\nEuclid, is concerned with shapes which are smooth, except perhaps for corners\r\nand lines, special lines which are singularities, but some shapes in nature are\r\nso complicated that they are equally complicated at the big scale and come\r\ncloser and closer and they don’t become any less complicated. Closer and closer, or you go farther or\r\nfarther, they remain equally complicated. \r\nThere is never a plane, never a straight line, never anything smooth and\r\nordinary. The idea is very, very\r\nvague, is expressed – it’s an expression of reality.

Fractal geometry is a new subject and each definition I try\r\nto give for it has turned out to be inappropriate. So I’m now being cagey and saying there are very complex\r\nshapes which would be the same from close by and far away.

\r\n\r\n**Question: **What does it mean to say that fractal shapes are\r\nself-similar?

**Benoit Mandelbrot: **Well, if you look at a shape like a\r\nstraight line, what’s remarkable is that if you look at a straight line from\r\nclose by, from far away, it is the same; it is a straight line. That is, the straight line has a\r\nproperty of self-similarity. Each\r\npiece of the straight line is the same as the whole line when used to a big or\r\nsmall extent. The plane again has\r\nthe same property. For a long\r\ntime, it was widely believed that the only shapes having these extraordinary\r\nproperties are the straight line, the whole plane, the whole space. Now in a certain sense, self-similarity\r\nis a dull subject because you are used to very familiar shapes. But that is not the case. Now many shapes which are self-similar\r\nagain, the same seen from close by and far away, and which are far from being\r\nstraight or plane or solid. And\r\nthose shapes, which I studied and collected and put together and applied in\r\nmany, many domains, I called fractals.

**Question:** How can complex natural shapes be represented\r\nmathematically?

**Benoit Mandelbrot:** Well, historically, a mountain could not\r\nbe represented, except for a few mountains which are almost like cones. Mountains are very complicated. If\r\nyou look closer and closer, you find greater and greater details. If you look away until you find that\r\nbigger details become visible, and in a certain sense this same structure\r\nappears at those scales. If you\r\nlook at coastlines, if you look at that them from far away, from an airplane,\r\nwell, you don’t see details, you see a certain complication. When you come closer, the complication\r\nbecomes more local, but again continues. \r\nAnd come closer and closer and closer, the coastline becomes longer and\r\nlonger and longer because it has more detail entering in. However, these details amazingly enough\r\nenters this certain this certain regular fashion. Therefore, one can study a coastline **** object because the\r\ngeometry for that existed for a long time, and then I put it together and applied\r\nit to many domains.

**Question: **What was the discovery process behind the\r\nMandelbrot set?

**Benoit Mandelbrot:** The Mandelbrot set in a certain sense is\r\na **** of a dream I had and an uncle of mine had since I was about 20. I was a student of mathematics, but not\r\nhappy with mathematics that I was taught in France. Therefore, looking for other topics, an uncle of mine, who\r\nwas a very well-known pure mathematician, wanted me to study a certain theory\r\nwhich was then many years old, 30 years old or something, but had in a way\r\nstopped developing. When he was\r\nyoung he had tried to get this theory out of a rut and he didn’t succeed,\r\nnobody succeeded. So, there was a\r\ncase of two men, Julia, a teacher of mine, and Fatu, who had died, had a very\r\ngood year in 1910 and then nothing was happening. My uncle was telling me, if you look at that, if you find\r\nsomething new, it would be a wonderful thing because I couldn’t – nobody\r\ncould.

I looked at it and found it too difficult. I just could see nothing I could\r\ndo. Then over the years, I put\r\nthat a bit in the back of my mind until one day I read an obituary. It is an interesting story that I was\r\nmotivated by an obituary, an obituary of a great man named Poincaré, and in that obituary this\r\nquestion was raised again. At that\r\ntime, I had a computer and I had become quite an expert in the use of the\r\ncomputer for mathematics, for physics, and for many sciences. So, I decided, perhaps the time has\r\ncome to please my uncle; 35 years later, or something. To please my uncle and do what my uncle\r\nhad been pushing me to do so strongly.

\r\n\r\nBut I approached this topic in a very different fashion than\r\nmy uncle. My uncle was trying to\r\nthink of something, a new idea, a new problem, a new way of developing the\r\ntheory of Fatu and Julia. I did\r\nsomething else. I went to the\r\ncomputer and tried to experiment. \r\nI introduced a very high level of experiment in very pure\r\nmathematics. I was at IBM, I had\r\nthe run of computers which then were called very big and powerful, but in fact\r\nwere less powerful than a handheld machine today. But I had them and I could make the experiments. The conditions were very, very\r\ndifficult, but I knew how to look at pictures. In fact, the reason I did not go into pure mathematics\r\nearlier was that I was dominated by visual. I tried to combine the visual beauty and the\r\nmathematics.

\r\n\r\nSo, I looked at the picture for a long time in a very\r\nunsystematic fashion just to become acquainted in a kind of physical fashion\r\nwith those extraordinary difficult and complicated shapes. Two were extraordinarily\r\ndifficult. Computer graphics did\r\nnot exist back then, but to have a machine which was – made it seem\r\ndoable. And I started finding\r\nextraordinary complications, extraordinary structure, extraordinary beauty of\r\nboth a theoretical kind, mathematical, and a visual kind. And collected observations of my trip\r\nin this new territory. When I\r\npresented that work to my colleagues, it was an explosion of interest. Everybody in mathematics had given up\r\nfor 100 years or 200 years the idea that you could from pictures, from looking\r\nat pictures, find new ideas. That\r\nwas the case long ago in the Middle Ages, in the Renaissance, in later periods,\r\nbut then mathematicians had become very abstract. Pictures were completely eliminated from mathematics; in\r\nparticular when I was young this happened in a very strong fashion.

\r\n\r\nSome mathematicians didn’t even perceive of the possibility\r\nof a picture being helpful. To the\r\ncontrary, I went into an orgy of looking at pictures by the hundreds; the\r\nmachines became a little bit better. \r\nWe had friends who improved them, who wrote better software to help me,\r\nwhich was wonderful. That was the good\r\nthing about being at IBM. And I\r\nhad this collection of observations, which I gave to my friends in mathematics\r\nfor their pleasure and for simulation. \r\nThe extraordinary fact is that the first idea I had which motivated me,\r\nthat worked, is conjecture, a mathematical idea which may or may not be\r\ntrue. And that idea is still\r\nunproven. It is the foundation,\r\nwhat started me and what everybody failed to **** prove has so far defeated the\r\ngreatest efforts by experts to be proven. \r\nIn a certain sense it’s a very, very strange because the object itself\r\nis understandable, even for a child. \r\nIf the object can be drawn by a child with new computers, with new\r\ngraphic devices, and still the basic idea has not been proven.

\r\n\r\nBut the development of it has been extraordinary, then it\r\nwas slowed down a bit, and now again it is going up. New people are coming in and they prove extraordinary\r\nresults which nobody was hoping to prove, and I am astonished and of course,\r\nvery pleased by this development.

\r\n\r\n**Question: **What is the unproven conjecture that drove you?

Benoit Mandelbrot: The conjecture itself consists in two\r\ndifferent issues in Mandelbrot set – two alternative definitions which are too\r\ntechnical to describe without a blackboard, but which are both very simple and\r\nwhich I assumed naively to be equivalent. \r\nWhy did I assume so? \r\nBecause on the pictures I could not see any difference. Obtaining pictures in one way or\r\nanother way, I couldn’t tell them apart. \r\nTherefore, I assumed they were identical and I went on studying this\r\npiece. I found that, again, many\r\ninteresting observations of which most were very preferred by many other very,\r\nvery skilled mathematicians. But\r\nthe idea that these two conditions, definitions, are identical is still\r\nopen. So there are two definitions\r\nin Mandelbrot set, the usual one and another one, and they may theoretically be\r\ndifferent. People are getting\r\ncloser, but have not proven it completely.

\r\n\r\n**Question:** Why do people find fractals beautiful?

**Benoit Mandelbrot: **Well, first of all, one explanation of\r\nthat is that the feeling for fractality is not new. It is one very surprising and extraordinary discovery I made\r\ngradually, very slowly by looking again at the paintings of the past. Many painters had a clear idea of what\r\nfractals are. Take a French\r\nclassic painter named Poussin. \r\nNow, he painted beautiful landscapes, completely artificial ones,\r\nimaginary landscapes. And how did\r\nhe choose them? Well, he had the\r\nbalance of trees, of lawns, of houses in the distance. He had a balance of small objects, big\r\nobjects, big trees in front and his balance of objects at every scale is what\r\ngives to Poussin a special feeling.

Take Hokusai, a famous Chinese painter of 1800. He did not have any mathematical\r\ntraining; he left no followers because his way of painting or drawing was too\r\nspecial to him. But it was quite\r\nclear by looking at how Hokusai, the eye, which had been trained from the\r\nfractals, that Hokusai understood fractal structure. And again, had this balance of big, small, and intermediate\r\ndetails, and you come close to these marvelous drawings, you find that he\r\nunderstood perfectly fractality. \r\nBut he never expressed it. \r\nNobody ever expressed it, and then the next stage of Japanese image\r\nexperts did some other things.

\r\n\r\nSo humanity has known for a long time what fractals\r\nare. It is a very strange\r\nsituation in which an idea which each time I look at all documents have deeper\r\nand deeper roots, never (how to say it), jelled. Never got together until I started playing with the computer\r\nand playing with topics which nobody was touching because they were just\r\ndesperate and hopeless.

\r\n\r\n**Question:** How has computer technology impacted your work?

Benoit Mandelbrot: Well, the computer had been sort of spoken about since the\r\nearly 19th century, even before. \r\nBut until the electronic computers came, which was in reality during\r\nWorld War II, or shortly afterwards. \r\nThey could not be used for any purpose in science. They were just too slow, too limited in\r\ntheir capacity. My chance was that\r\nI was myself a very visual person. \r\nAgain, a mathematician who had started a very unconventional career\r\nbecause my interest was both mathematics and in the eye. And with IBM very primitive\r\npicture-making machines became available and we had to program everything. It was heroic. And my friends at IBM who helped me\r\ndeserve a great thank you. With\r\nthese two, I could begin to do things which before had been impossible. I could begin to implement an idea of\r\nhow a mountain looked like. To\r\nreduce a mountain, which is something most complicated to a very simple idea –\r\nhow do you do it? Well you make a\r\nconjecture, have positives about shapes of mountains, and you don’t think about\r\nthe mathematics of it, you must make a picture of it.

\r\n\r\nIf the picture is – everybody to be a mountain, then there’s\r\nsomething true about it. Or a\r\ncloud. It was astonishing when at\r\none point, I got the idea of how to make artifical clouds with a collaborator,\r\nwe had pictures made which were theoretically completely artificial pictures\r\nbased upon that one very simple idea. \r\nAnd this picture everybody views as being clouds. People don’t believe that they aren’t\r\nphotographs. So, we have certainly\r\nfound something true about nature. \r\n

\r\n\r\n**Question: **What does the word “chaos” mean to mathematicians?

**Benoit Mandelbrot:** The theory of chaos and theory of\r\nfractals are separate, but have very strong intersections. That is one part of chaos theory is\r\ngeometrically expressed by fractal shapes. Another part of chaos theory is not expressed by fractal\r\nshapes. And other part of fractals\r\ndoes not belong to chaos theory so that two theories which overlap very\r\nstrongly and do not coincide. One\r\nof them, chaos theory, is based on behavior of systems defined by\r\nequations. Equations of motion,\r\nfor example, and classical mathematics, and around 1900, Poincaré and ****, two great\r\nmathematicians at the time, have realized that sometimes the solution of very\r\nsimple looking equations can be extremely complicated. But in 1900, it was too early to\r\ndevelop that idea. It was very\r\nwell expressed and very much discussed, but did not – could not grow.

Much later, of course, with computers this idea came to life\r\nagain and became the very important part of science. So both chaos theory and fractal have had contacts in the\r\npast when they are both impossible to develop and in a certain sense not ready\r\nto be developed. And again, they\r\nintersect very strongly but they are very distinct.

\r\n\r\n**Question:** Do mathematical descriptions of chaos define some\r\norder within chaos?

**Benoit Mandelbrot:** Well a very strong distinction was made\r\nbetween chaos and fractals. For\r\nexample, the rules which generate most of natural fractals, models of\r\nmountains, of clouds, and many other phenomena involve change. And therefore they are not at all\r\nchaotic in the ordinary sense of the word, in an ordinary, current, modern\r\nsense of the word. Not chaotic in\r\nthe old sense of the word, which doesn’t have any specific meaning. But I don’t\r\nlike to discuss the question of terms. \r\nThe term chaos came, but you know something which was very confused, it\r\nhelped it jell, but the use of a biblical name in a certain sense forces us to\r\nfind the implications which were not important in mathematics. That’s why when the time came to give a\r\nname to my work, I chose the word fractal which was new. Before that, there was no need of a\r\nword at all because again there were only a few undeveloped ideas in the very\r\nmany great minds. But when a word\r\nbecame necessary, I preferred not to use an old word, but to create a new one.

**Question: **How did you come up with the word “fractal”?

**Benoit Mandelbrot: **\r\nWell, it was a very, very interesting story. At one point, a friend of mine, an older person, told me\r\nthat he saw a paper of mine on a new topic. And he said, “Look Benoit, I tell you, you must stop writing\r\nall of these papers in that field, that field, that field, that field. Nobody knows where you are, what you\r\nare doing. You just sit down and\r\nwrite a book. A short book, a\r\nclear book, a book of things which you have done.” So, I sat down and wrote the book. Now, the book had to title, why? Because the topics I had been studying had not been the\r\nobject of any theory whatsoever. \r\nAnd there are many words which mean nothing, but many fields which have\r\nno name because they don’t exist. \r\nSo, the publisher didn’t like this very ponderous title, said, “Look.” And a friend of mine, another friend,\r\ntold me, “Look, you create a new field, you are entitled to give it a\r\nname.” So, I had Latin in high\r\nschool and it turned out that one of my son’s was taking Latin in the United\r\nStates, and so there was a Latin dictionary in our house, which was an\r\nexception.

I went in there and tried to look for a word which fitted\r\nwhat I had been working on. And\r\nwhen I was playing with the word fraction, and looked in the dictionary for a\r\nword where fraction came from. It\r\ncame from a Latin word which meant, how to say disconnect – rough and\r\ndisconnected, it was a very general – the idea of roughness originally in Latin. So, I started playing with *fractus*, which I named it that and coined\r\nthe word fractal. First of all, I\r\nput it in this book, *Objets Fractals*,\r\nin French as it turned out, and then the English translation of the book, and\r\nthen the word took off. First of\r\nall, people applied it in ways in which I didn’t find sensible, but there was\r\nnothing I could say about it. So,\r\nthen the dictionary started defining it, each a little bit differently. And in a certain sense the word became\r\nalive and independent of me. I\r\ncould scream and say, I don’t like it, but it made no difference.

I had once a curiosity of looking on the Web in different\r\ncountries having different languages, what is a fractal, and found that in one\r\ncountry, I will not mention, it’s a word that has become applied to some\r\nnightclubs. A fractal nightclub is\r\na kind of nightclub. I don’t know\r\nwhich, because I haven’t been there, but, and I don’t know the language, but I\r\nguess, from what I could guess, what it was. It’s a word which has its own life. I gave it a definition, but that\r\ndefinition became too narrow because some objects I want to go fractal did not\r\nfit the old definition.

\r\n\r\nSo some people asked me would I still believe the definition\r\nof whatever – 40 years ago. I\r\ndon’t. But I have no control. It’s something which works by\r\nitself. The fact that very many\r\nadults I know never heard of it, but the children have, is what gives me\r\nparticular pleasure because high school students, even the bright ones, are\r\nvery resistant to, how to say, imposed terms. And the combination of pictures and of deep theory, you can\r\nlook at the picture and find something, some idea about this picture is\r\nsensible, and then be told that very great scientists either can’t prove it, or\r\nhas taken 40 years to prove it, or had to be several of them together to prove\r\nit because it was so difficult. \r\nAnd it can be seen by a child, understood by a child. That aspect is one which very many\r\npeople find particularly attractive in the field.

\r\n\r\nIn mathematics and science definition are simple, but\r\nbare-bones. Until you get to a problem which you understand it takes hundreds\r\nand hundreds of pages and years and years of learning. In this case, you have this formula,\r\nyou track in a computer and from a simple formula, in a very short time\r\namazingly beautiful things come out, which sometimes people can prove instantly\r\nand sometimes great scientists take forever to prove. Or don’t even succeed in proving it.

\r\n\r\n**Question: **How can we understand financial market\r\nfluctuations in fractal terms?

**Benoit Mandelbrot:** Well, what I discovered quite a while ago\r\nin fact, that was my first major piece of work is that a model of price\r\nvariation which everybody was adopting was very far from being applicable. It’s a very curious story.

In 1900, a Frenchman named Bachelier, who was a student of\r\nmathematics, wrote a thesis on the theory of speculation. It was not at all an acceptable topic\r\nin pure mathematics and he had a very miserable life. But his thesis was extraordinary. Extraordinary in a very strange way. It applied very well to Brownian\r\nmotion, which is in physics. So,\r\nBachelier was a pioneer of a very marvelous essential theory in physics. But to economics, it didn’t apply at\r\nall, it was very ingenious, but Bachelier had no data, in fact no data was\r\navailable at that time in 1900, so he imagined an artificial market in which\r\ncertain rules may apply. \r\nUnfortunately, the theory which was developed by economists when\r\ncomputers came up was Bachelier’s theory. \r\nIt does not account for any of the major effects in economics. For example, it assumes prices are\r\ncontinuous when everybody knows the prices are not continuous. Some people say, well, all right, there\r\nare discontinuities but they are a different kind of economics that we are\r\ndoing, not because certain discontinuities become too complicated and only will\r\nthe **** look more or less continuous. \r\nBut it turns out that discontinuities are as important, or more\r\nimportant than the rest.

\r\n\r\nBachelier assumed that each price changes in compared of the\r\npreceding price change. It’s a\r\nvery beautiful assumption, but it’s completely incorrect because we know very\r\nwell, especially today that for a long time prices may vary moderately and then\r\nsuddenly they begin to vary a great deal. \r\nSo, even we’re saying that the theory changes or you say the theory\r\nwhich exists is not appropriate. \r\nWhat I found that Bachelier’s theory was defective on both grounds. That was in 1961, 1962, I forgot the\r\nexact dates and when the development of Bachelier became very, very rapid. Since nobody wanted to listen to me, I\r\ndid other things. Many other\r\nthings, but I was waiting because it was quite clear that my time would have to\r\ncome. And unfortunately, it has\r\ncome, that is, the fluctuation of the economy, the stock market, and commodity\r\nmarkets today are about as they were in historical times. There was no change which made the\r\nstock market different today than it was long ago. And the lessons which are drawn from **** peers do represent\r\ntoday’s events very accurately. \r\nBut the situation is much more complicated than Bachelier had\r\nassumed. Bachelier, again, was a\r\ngenius, Bachelier had an excellent idea which happened to be very useful in\r\nphysics, but economics, he just lacked data. He did not have awareness of discontinuity which is\r\nessential in this context. Not\r\nhaving an awareness of dependence, which is also essential in this context. So, his theory is very, very different\r\nfrom what you observe in reality.

\r\n\r\n**Question: **As you write your memoirs, which memories are the\r\nmost fun and the most difficult to look back on?

**Benoit Mandelbrot: **Well, my life has been extremely\r\ncomplicated. Not by choice at the\r\nbeginning at all, but later on, I had become used to complication and went on\r\naccepting things that other people would have found too difficult to\r\naccept. I was born in Poland and\r\nmoved to France as a child shortly before World War II. During World War II, I was lucky to\r\nlive in the French equivalent of Appalachia, a region which is sort of not very\r\nhigh mountains, but very, very poor, and Appalachia we are poorer even, so\r\npoorer than Appalachia of the United States. And for me, I was in high school where things were very easy. It was a small high school way up in the\r\nhills and had mostly a private intellectual life. I read many books; there were many books, a very good\r\nlibrary. I had many books and I\r\nhad dreams of all kinds. Dreams in\r\nwhich were in a certain sense, how to say, easy to make because the near future\r\nwas always extremely threatening. \r\nIt was a very dangerous period. \r\nBut since I had nothing to lose, I was dreaming of what I could do.

Then the war ended. \r\nI had very, very little training in taking an exam to determine a\r\nscientist’s life in France. There\r\nwere two schools, both very small. \r\nOne tiny, and one small, which in a certain sense was the place that I\r\nwas sure I wanted to go. I had\r\nonly a few months of finding out how the exam proceeded, but I took the exam\r\nand perhaps because of inherited gifts, I did very well. In fact, I barely\r\nmissed being number one in France in both schools. In particular I did very well in mathematical problems. The physics I could not guess, other\r\nthings I could not guess. But then\r\nI had a big choice, should I go into mathematics in a small and ****\r\nschool. Or should I go to a bigger\r\nschool in which, in a certain sense would give me time to decide what I wanted\r\nto do?

\r\n\r\nFirst I entered the small school where I was, as a matter of\r\nfact, number one of the students who entered then. But immediately, I left because that school, again, was\r\ngoing to teach me something which I did not fully believe, namely mathematics\r\nseparate from everything else. It\r\nwas excellent mathematics, French mathematics was very high level, but in\r\neverything else it was not even present. \r\nAnd I didn’t want to become a pure mathematician, as a matter of fact,\r\nmy uncle was one, so I knew what the pure mathematician was and I did not want\r\nto be a pure – I wanted to do something different. Not less, not more but different. Namely, combine pure mathematics at which I was very good,\r\nwith the real world of which I was very, very curious.

\r\n\r\nAnd so, I did not go to École Polytechnique. It was a very rough decision, and the\r\nyear when I took this decision remembers my memory very, very strongly. Then for several years, I just was lost\r\na bit. I was looking for a good\r\nplace. I spent my time very nicely\r\nin many ways, but not fully satisfactory. \r\nThen I became Professor in France, but realized that I was not – for the\r\njob that I should spend my life in. \r\nFortunately, IBM was building a research center, I went there for a\r\nsummer thing, for a summer only. I\r\nknew this summer, decided to stay. \r\nIt was a very big gamble. I\r\nlost my job in France, I received a job in which was extremely uncertain, how\r\nlong would IBM be interested in research, but the gamble was taken and very\r\nshortly afterwards, I had this extraordinary fortune of stopping at Harvard to\r\ndo a lecture and learning about the price variation in just the right way. At a time when nobody was looking, was realizing\r\nthat either one needed, or one could make a theory of price variation other\r\nthan the theory of 1900 at which Bachelier had proposed, which was very, very\r\nfar from being representative of the actual thing.

\r\n\r\nSo, I went to IBM and I was fortunate in being allowed – to\r\nbe successful as to go from field to field, which in a way was what I had been\r\nhoping for. I didn’t feel\r\ncomfortable at first with pure mathematics, or as a professor of pure mathematics. I wanted to do a little bit of\r\neverything and explore the world. \r\nAnd IBM let me do so. I\r\ntouched on far more topics than anybody would have found reasonable. I was often told, “Settle down, stay in\r\none field, don’t go all the time to another field.” But I was just compelled to move from one thing to\r\nanother.

\r\n\r\nAnd fractal geometry was not an idea which I had early on,\r\nfor something was developed progressively. I didn’t choose to go into the topic because of any\r\ncompelling reason, but because the problems there seemed to be somehow similar\r\nto the ones I knew how to handle. \r\nI had experienced this kind of problem and gradually realized that I was\r\ntruly putting together a new theory. \r\nA theory of roughness. What\r\nis roughness? Everybody knows what\r\nis roughness. When was roughness\r\ndiscovered? Well, prehistory. Everything is roughness, except for the\r\ncircles. How many circles are\r\nthere in nature? Very, very\r\nfew. The straight lines. Very shapes are very, very smooth. But geometry had laid them aside\r\nbecause they were too complicated. \r\nAnd physics had laid them aside because they were too complicated. One couldn’t even measure roughness. So, by luck, and by reward for\r\npersistence, I did found the theory of roughness, which certainly I didn’t expect\r\nand expecting to found one would have been pure madness.

\r\n\r\nSo, one of the high points of my life was when I suddenly\r\nrealized that this dream I had in my late adolescence of combining pure\r\nmathematics, very pure mathematics with very hard things which had been long a\r\nnuisance to scientists and to engineers, that this combination was possible and\r\nI put together this new geometry of nature, the fractal geometry of nature.

\r\n\r\n**Question:** Which honor means more to you: your Légion\r\nd'Honneur medal or the “Mandelbrot Set” rock song?

**Benoit Mandelbrot**: Well, I happen to know this song, it was\r\nsent to me and I was very impressed by it and by its popularity. In a certain sense, it is not which\r\none, but the combination. When\r\npeople ask me what’s my field? I say,\r\non one hand, a fractalist. Perhaps\r\nthe only one, the only full-time one. \r\nOn the other hand, I’ve been a professor of mathematics at Harvard and\r\nat Yale. At Yale for a long\r\ntime. But I’m not a mathematician\r\nonly. I’m a professor of physics,\r\nof economics, a long list. Each\r\nelement of this list is normal. \r\nThe combination of these elements is very rare at best. And so in a certain sense, it is not\r\nthe fact that I was a professor of mathematics at these great universities, or\r\nprofessor of physics at other great universities, or that I received, among\r\nother doctorates, one in medicine, believe it or not. And one in civil engineering. It is the coexistence of these various aspects that in one\r\nlifetime it is possible, if one takes the kinds of risks which I took, which\r\nare colossal, but taking risks, I was rewarded by being able to contribute in a\r\nvery substantial fashion to a variety of fields. I was able to reawaken and solve some very old problems. The problems are just so old that in a\r\ncertain sense, they were no longer being pursued. And nobody – I didn’t know anybody who was trying to define roughness\r\nof ****. It was a hopeless subject. \r\nBut I did it and there’s a whole field by which has been created by\r\nthat.

In a certain sense the beauty of what I happened by\r\nextraordinary chance to put together is that nobody would have believed that\r\nthis is possible, and certainly I didn’t expect that it was possible. I just moved from step to step to step. Lately I realized that all these things\r\nheld together, and very lately I see that in each field very old problems could\r\nbe if not solved, at least advanced or reawakened, and therefore gradually very\r\nmuch improved in your understanding.

**Recorded on February 17, 2010**

Interviewed by Austin Allen

A conversation with the mathematician and Professor Emeritus at Yale University.

## Big ideas.

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## To the very beginning: going back in time with Steven Weinberg (Part 2)

What was the universe like one-trillionth of a second after the Big Bang? Science has an answer.

- Following Steven Weinberg's lead, we plunge further back into cosmic history, beyond the formation of atomic nuclei.
- Today, we discuss the origin of the quark-gluon plasma and the properties of the famous Higgs boson, the "God Particle."
- Is there a limit? How far can we go back in time?

Last week, we celebrated the great physicist Steven Weinberg, bringing back his masterful book *The First Three Minutes: A Modern View of the Origin of the Universe*, where he tells the story of how, in the first moments after the Big Bang, matter started to organize into the first atomic nuclei and atoms. This week we continue to follow Weinberg's lead, plunging further back in time, as close to the beginning as we reliably can.

But first, a quick refresher. The first light atomic nuclei — aggregates of protons and neutrons — emerged during the very short time window between one-hundredth of a second and 3 minutes after the bang. This explains Weinberg's book title. Recall that atoms are identified by the number of protons in their nuclei (the atomic number) — from hydrogen (with a single proton) to carbon (with six) and all the way to uranium (with 92). The early cosmic furnace forged only chemical elements 1, 2, and 3 — hydrogen, helium, and lithium (as well as their isotopes, which contain the same number of protons but different numbers of neutrons). All heavier elements are forged in dying stars.

The hypothesis that the universe was the alchemist responsible for the lightest elements has been beautifully confirmed by numerous observations during the past decades, including improving a lingering discrepancy with lithium-7. (The "7" represents three protons and four neutrons for this lithium isotope, its most abundant in nature.) This primordial nucleosynthesis is one of the three key observational pillars of the Big Bang model of cosmology. The other two are the expansion of the universe — measured as galaxies recede form one another — and the microwave background radiation — the radiation leftover after the birth of hydrogen atoms, some 400,000 years after the bang.

## The primordial soup of particle physics

At about one minute after the bang, the matter in the universe included light atomic nuclei, electrons, protons, neutrons, photons, and neutrinos: the primordial soup. What about earlier? Going back in cosmic time means a smaller universe, that is, matter squeezed into smaller volumes. Smaller volumes mean higher pressures and temperatures. The recipe for the soup changes. In physics, temperature is akin to motion and agitation. Hot things move fast and, when they cannot because they are stuck together, they vibrate more. Eventually, as the temperature increases, the bonds that keep things together break. As we go back in time, matter is dissociated into its simplest components. First, molecules become atoms. Then, atoms become nuclei and free electrons. Then, nuclei become free protons and neutrons. Then what?

Since the 1960s, we have known that protons and neutrons are not elementary particles. They are made of other particles — called quarks — bound together by the strong nuclear force, which is about 100 times stronger than electric attraction (that is, electromagnetism). But for high enough temperatures, not even the strong force can hold protons and neutrons together. When the universe was a mere one-hundred-thousandth of a second (10-5 second) old, it was hot enough to dissociate protons and neutrons into a hot plasma of quarks and gluons. Gluons, as the name implies, are the particles that stitch quarks into protons and neutrons (as well as hundreds of other particles held together by the strong force commonly seen in particle accelerators). Amazingly, such strange quark-gluon plasma has been created in high-energy particle collisions that generate energies one million degrees hotter than the heart of the sun. (Here is a video about it.) For a fleeting moment, the early universe re-emerges in a human-made machine, an awesome scientific and technological feat.

## Remember the Higgs boson?

Credit: NASA

Is that it? Or can we go further back? Now we are contemplating a universe that is younger than one-millionth of a second old. For us, that's a ridiculously small amount of time. But not for elementary particles, zooming about close to the speed of light. As we keep going back toward t = 0, something remarkable happens. At about one-trillionth of a second (10-12 second or 0.000000000001 second) after the bang, a new particle commands the show, the famous Higgs boson. If you remember, this particle became both famous and infamous when it was discovered in 2012 at the European Center for Particle Physics, and the media decided to call it the "God Particle."

For this, we can blame Nobel Prize Laureate Leon Lederman, who was my boss when I was a postdoc at Fermilab, the biggest particle accelerator in the U.S. Leon told me that he was writing a book about the elusive Higgs, which he tried to but could not find at Fermilab. He wanted to call the book *The God-Damn Particle*, but his editor suggested taking out the "damn" from the title to increase sales. It worked.

The Higgs goes through a strange transition as the universe heats up. It loses its mass, becoming what we call a massless particle, like the photon. Why is this important? Because the Higgs plays a key role in the drama of particle physics. It is the mass-giver to all particles: if you hug the Higgs or (more scientifically) if a particle interacts with the Higgs boson, it gets a mass. The stronger the interaction, the larger the mass. So, the electron, being light, interacts less strongly with the Higgs than, say, the tau lepton or the charm quark. But if the Higgs loses its mass as it gets hotter, what happens to all the particles it interacts with? They also lose their mass!

## Approaching t = 0

Think about the implication. Before one-trillionth of a second after the bang, all known particles were massless. As the universe expands and cools, the Higgs gets a mass and gives mass to all other particles it interacts with. This explains why the "God Particle" nickname stuck. The Higgs explains the origin of masses.

Kind of. We do not know what determines the strengths of all these different hugs (interactions), for instance, why the electron mass is different from the quarks' masses. These are parameters of the model, known as the Standard Model, a compilation of all that we know about the world of the very, very small. These all-important parameters determine the world as we know it. But we do not know what, if anything, determines them.

Okay, so we are at one-trillionth of a second after the bang. Can we keep going back? We can, but we must dive into the realm of speculation. We can talk of other particles, other dimensions of space and superstrings, the unification of all forces of nature, and the multiverse. Or we can invoke a pearl the great physicist Freeman Dyson once told me: most speculations are wrong. Readers are best served if we stick to what we know first. Then, with care, we dive into the unknown.

So, we stop here for now, knowing that there is much new territory of the "Here Be Dragons" type to cover in this fleeting one-trillionth of a second. We will go there soon enough.

## Surprisingly modern lessons from classic Russian literature

Though gloomy and dense, Russian literature is hauntingly beautiful, offering a relentlessly persistent inquiry into the human experience.

- Russian literature has a knack for precisely capturing and describing the human condition.
- Fyodor Dostoevsky, Leo Tolstoy, and Aleksandr Solzhenitsyn are among the greatest writers who ever lived.
- If you want to be a wiser person, spend time with the great Russian novelists.

In Fyodor Dostoevsky's 1864 novella *Notes from Underground*, an unnamed narrator asks the following question: "What can be expected of man since he is a being endowed with strange qualities?" The answer: "Even if man were nothing but a piano-key and this were proved to him by science, even then he would not become reasonable, but would purposefully do something perverse out of simple ingratitude. He would contrive destruction and chaos only to gain his point!"

After reading another handful of equally puzzling paragraphs, chances are you will find yourself seriously considering whether or not to put down this 100-page riddle. Chances are, plenty of readers will have beaten you to it already. Keep on reading, however, and you might just find that the second half of the story is not only much, much easier to understand, but can also make you look back at the first half from a radically different perspective.

## A small person with big power

This narrator, it turns out, is a proud but spiteful bureaucrat. Dissatisfied with his career, he uses the trivial bit of power his position bestows upon him to make life hell for those he interacts with. Eclipsed by former classmates who successfully climbed the ladders of the military and high society, he spends his days alone — lost inside his own head — thinking of reasons for why the world has yet to notice the extraordinary talents he believes he possesses.

After the narrator finishes his incoherent diatribe about society's discontents, we get a glimpse at his everyday existence and the events that have made him so embittered. In one scene, he invites himself to a party for a recently promoted colleague he despises, only to spend the rest of the night complaining about the fact that everyone but him is having a fun time. "I should fling this bottle at their heads," he thinks, reaching for some champagne and defeatedly pouring himself another round.

Angsty college students will recognize this kind of crippling social anxiety in an instance, leaving them amazed at the accuracy with which this long-dead writer managed to put their most private thoughts to paper. Dostoevsky's unparalleled ability to capture our murky stream of consciousness has not gone unnoticed; a century ago, Sigmund Freud developed the study of psychoanalysis with *Notes* in the back of his mind. Friedrich Nietzsche listed Dostoevsky as one of his foremost teachers.

To an outsider, Russian literature can seem hopelessly dense, unnecessarily academic, and uncomfortably gloomy. But underneath this cold, rough, and at times ugly exterior, there hides something no thinking, feeling human could resist: a well-intentioned, deeply insightful, and relentlessly persistent inquiry into the human experience. Nearly two hundred years later, this hauntingly beautiful literary canon continues to offer useful tips for how to be a better person.

## Dancing with death

Credit: Jez Timms via Unsplash

Some critics argue that the best way to analyze a piece of writing is through its composition, ignoring external factors like the author's life and place of origin. While books from the Russian Golden Age are meticulously structured, they simply cannot be studied in a vacuum. For these writers, art did not exist for art's sake alone; stories were manuals to help us understand ourselves and solve social issues. They were, to borrow a phrase popularized by Vladimir Lenin, mirrors to the outside world.

Just look at Dostoevsky, who at one point in his life was sentenced to death for reading and discussing socialist literature. As a firing squad prepared to shoot, the czar changed his mind and exiled him to the icy outskirts of Siberia. Starting life anew inside a labor camp, Dostoevsky developed a newfound appreciation for religious teachings he grew up with, such as the value of turning the other cheek no matter how unfair things may seem.

Dostoevsky's brush with death, which he often incorporated into his fiction, was as traumatizing as it was eye-opening. In *The Idiot*, about a Christ-like figure trying to live a decent life among St. Petersburg's corrupt and frivolous nobles, the protagonist recalls an execution he witnessed in Paris. The actual experience of standing on the scaffold — how it puts your brain into overdrive and makes you wish to live, no matter its terms and conditions — is described from the viewpoint of the criminal, something Dostoevsky could do given his personal experience.

Faith always played an important role in Dostoevsky's writing, but it took center stage when the author returned to St. Petersburg. His final (and most famous) novel, *The Brothers Karamazov*, asks a question which philosophers and theologians have pondered for centuries: if the omniscient, omnipotent, and benevolent God described in the Bible truly exists, why did He create a universe in which suffering is the norm and happiness the exception?

To an outsider, Russian literature can seem hopelessly dense, unnecessarily academic, and uncomfortably gloomy. But underneath this cold, rough, and at times ugly exterior, there hides something no thinking, feeling human could resist: a well-intentioned, deeply insightful, and relentlessly persistent inquiry into the human experience. Nearly two hundred years later, this hauntingly beautiful literary canon continues to offer useful tips for how to be a better person.

It is a difficult question to answer, especially when the counterargument (that is, there is no God) is so compelling. "I don't want the mother to embrace the man who fed her son to dogs," Ivan, a scholar and the novel's main skeptic, cries. "The sufferings of her tortured child she has no right to forgive; she dare not, even if the child himself were to forgive! I don't want harmony. From love for humanity, I don't want it. I would rather be left with unavenged suffering."

Yet it was precisely in such a fiery sentiment that Dostoevsky saw his way out. For the author, faith was a never-ending battle between good and evil fought inside the human heart. Hell, he believed, was not some bottomless pit that swallows up sinners in the afterlife; it describes the life of someone who is unwilling to forgive. Likewise, happiness did not lie in the pursuit of fame or fortune but in the ability to empathize with every person you cross paths with.

## On resurrection

No discussion of Russian literature is complete without talking about Leo Tolstoy, who thought stories were never meant to be thrilling or entertaining. They were, as he wrote in his 1897 essay *What is Art?*, "a means of union among men, joining them together in the same feelings." Consequently, the only purpose of a novel was to communicate a specific feeling or idea between writer and reader, to put into words something that the reader always felt but never quite knew how to express.

Tolstoy grew up in a world where everything was either black or white and did not start perceiving shades of grey until he took up a rifle in his late teens. Serving as an artillery officer during the Crimean War, he found the good in soldiers regardless of which side of the conflict they were on. His *Sevastopol Sketches*, short stories based on his time in the army, are neither a celebration of Russia nor a condemnation of the Ottomans. The only hero in this tale, Tolstoy wrote, was truth itself.

It was an idea he would develop to its fullest potential in his magnum opus, *War and Peace*. Set during Napoleon's invasion of Russia, the novel frames the dictator, who Georg Hegel labeled "the World Spirit on horseback," as an overconfident fool whose eventual downfall was all but imminent. It is a lengthy but remarkably effective attack aimed at contemporary thinkers who thought history could be reduced to the actions of powerful men.

Semantics aside, Tolstoy could also be deeply personal. In his later years, the writer — already celebrated across the world for his achievements — fell into a depression that robbed him of his ability to write. When he finally picked up a pen again, he did not turn out a novel but a self-help book. The book, titled *A Confession*, is an attempt to understand his increasingly unbearable melancholy, itself born from the grim realization that he — like everyone else — will one day die.

In one memorable paragraph, Tolstoy explains his situation through an Eastern fable about a traveler climbing into a well to escape from a vicious beast, only to find another waiting for him at the bottom. "The man, not daring to climb out and not daring to leap to the bottom, seizes a twig growing in a crack in the wall and clings to it. His hands are growing weaker and he feels he will soon have to resign himself to the destruction that awaits him above or below, but still he clings on."

*Confession *is by no means an easy read, yet it is highly recommended for anyone feeling down on their luck. Tolstoy not only helps you understand your own emotions better but also offers inspiring advice on how to deal with them. What makes us humans unique from all other animals, he believes, is the ability to grasp our own impending and inevitable death. While this knowledge can be a terrible burden, it can also inspire us to focus on what is truly important: treating others with kindness.

## Urge for action

Credit: Julia Kadel via Unsplash

Because 19th century Russia was an autocracy without a parliament, books were the only place people could discuss how they think their country should be run. While Tolstoy and Dostoevsky made conservative arguments that focused on personal growth, other writers went in a different direction. Nikolay Chernyshevsky, a progressive, treated his stories like thought experiments. His novel, *What is to be Done?*, explores what a society organized along socialist lines could look like.

*What is to be Done?*, which Chernyshevsky wrote while he was in prison, quickly became required reading for any aspiring Russian revolutionary. Imbued with the same kind of humanistic passion you may find in *The Brothers Karamazov*, these kinds of proto-Soviet blueprints painted such a convincing (and attractive) vision for the future that it seemed as though history could unfold itself no other way than how Karl Marx had predicted it would.

"I don't know about the others," Aleksandr Arosev, a Bolshevik who saw himself as the prophet of a new religion, once wrote about his childhood reading list, "but I was in awe of the tenacity of human thought, especially that thought within which there loomed something that made it impossible for men not to act in a certain way, not to experience the urge for action so powerful that even death, were it to stand in its way, would appear powerless."

Decades later, another Aleksandr — Aleksandr Solzhenitsyn — wrote an equally compelling book about the years he spent locked inside a Siberian prison camp. Like Arosev, Solzhenitsyn grew up a staunch Marxist-Leninist. He readily defended his country from Nazi invaders in East Prussia, only to be sentenced to eight years of hard labor once the government intercepted a private letter in which he questioned some of the military decisions made by Joseph Stalin.

In the camp, Solzhenitsyn took note of everything he saw and went through. Without access to pen and paper, he would lie awake at night memorizing the pages of prose he was composing in his mind. He tried his best remember each and every prisoner he met, just so he could tell their stories in case they did not make it out of there alive. In his masterpiece, *The Gulag Archipelago*, he mourns the names and faces he forgot along the way.

Despite doing time for a crime he did not commit, Solzhenitsyn never lost faith in humanity. Nor did he give in to the same kind of absolutist thinking that led the Soviet Union to this dark place. "If only it were all so simple!" he wrote. "If only there were evil people somewhere insidiously committing evil deeds. But the line dividing good and evil cuts through the heart of every human being. And who is willing to destroy a piece of his own heart?"

## The mystery of man

"All mediocre novelists are alike," Andrew Kaufman, a professor of Slavic Languages and Literature at the University of Virginia, once told *The Millions*. "Every great novelist is great in its own way." This is, in case you didn't know, an insightful spin on the already quite insightful opening line from another of Tolstoy's novels, *Anna Karenina*: "All happy families are alike, but every unhappy family is unhappy in its own way."

While Russian writers may be united by a prosaic style and interest in universal experience, their canon is certainly diverse. Writing for *The New York Times*, Francine Prose and Benjamin Moser neatly sum up what makes each giant of literature distinct from the last: Gogol, for his ability to "make the most unlikely event seem not only plausible but convincing"; Turgenev, for his "meticulously rendered but ultimately mysterious characters"; Chekhov, for his "uncanny skill at revealing the deepest emotions" in his plays.

As distant as these individuals may seem to us today, the impact they made on society is nothing short of profound. In the cinemas, hundreds of thousands gather to watch Keira Knightly put on a brilliant ballgown and embody Tolstoy's tragic heroine. At home, new generations read through Dostoevsky's *Notes of Underground *in silence, recognizing parts of themselves in his despicable but painfully relatable Underground Man.

Just as Tolstoy needed at least 1,225 pages to tell the story of *War and Peace*, so too does one need more than one article to explain what makes Russian literature so valuable. It can be appreciated for its historical significance, starting a discussion that ended up transforming the political landscape of the Russian Empire and — ultimately — the world as a whole. It also can be appreciated for its educational value, inspiring readers to evaluate their lives and improve their relationships.

Most importantly, perhaps, Russian literature teaches you to take a critical look at yourself and your surroundings. "Man is a mystery," Dostoevsky once exclaimed outside his fiction, reiterating a teaching first formulated by the Greek philosopher Socrates. "It must be unraveled. And if you spend your whole life unraveling it, do not say you have wasted your time. I occupy myself with this mystery, because I want to be a man."

## 3,000-pound Triceratops skull unearthed in South Dakota

"You dream about these kinds of moments when you're a kid," said lead paleontologist David Schmidt.

- The triceratops skull was first discovered in 2019, but was excavated over the summer of 2020.
- It was discovered in the South Dakota Badlands, an area where the Triceratops roamed some 66 million years ago.
- Studying dinosaurs helps scientists better understand the evolution of all life on Earth.

David Schmidt, a geology professor at Westminster College, had just arrived in the South Dakota Badlands in summer 2019 with a group of students for a fossil dig when he received a call from the National Forest Service. A nearby rancher had discovered a strange object poking out of the ground. They wanted Schmidt to take a look.

"One of the very first bones that we saw in the rock was this long cylindrical bone," Schmidt told St. Louis Public Radio. "The first thing that came out of our mouths was, 'That kind of looks like the horn of a triceratops.'"

After authorities gave the go-ahead, Schmidt and a small group of students returned this summer and spent nearly every day of June and July excavating the skull.

Credit: David Schmidt / Westminster College

"We had to be really careful," Schmidt told St. Louis Public Radio. "We couldn't disturb anything at all, because at that point, it was under law enforcement investigation. They were telling us, 'Don't even make footprints,' and I was thinking, 'How are we supposed to do that?'"

Another difficulty was the mammoth size of the skull: about 7 feet long and more than 3,000 pounds. (For context, the largest triceratops skull ever unearthed was about 8.2 feet long.) The skull of Schmidt's dinosaur was likely a *Triceratops prorsus, *one of two species of triceratops that roamed what's now North America about 66 million years ago.

Credit: David Schmidt / Westminster College

The triceratops was an herbivore, but it was also a favorite meal of the T*yrannosaurus rex*. That probably explains why the Dakotas contain many scattered triceratops bone fragments, and, less commonly, complete bones and skulls. In summer 2019, for example, a separate team on a dig in North Dakota made headlines after unearthing a complete triceratops skull that measured five feet in length.

Michael Kjelland, a biology professor who participated in that excavation, said digging up the dinosaur was like completing a "multi-piece, 3-D jigsaw puzzle" that required "engineering that rivaled SpaceX," he jokingly told the New York Times.

Morrison Formation in Colorado

James St. John via Flickr

The Badlands aren't the only spot in North America where paleontologists have found dinosaurs. In the 1870s, Colorado and Wyoming became the first sites of dinosaur discoveries in the U.S., ushering in an era of public fascination with the prehistoric creatures — and a competitive rush to unearth them.

Since, dinosaur bones have been found in 35 states. One of the most fruitful locations for paleontologists has been the Morrison formation, a sequence of Upper Jurassic sedimentary rock that stretches under the Western part of the country. Discovered here were species like Camarasaurus, Diplodocus, Apatosaurus, Stegosaurus, and Allosaurus, to name a few.

Triceratops illustration

Credit: Nobu Tamura/Wikimedia Commons |

As for "Shady" (the nickname of the South Dakota triceratops), Schmidt and his team have safely transported it to the Westminster campus. They hope to raise funds for restoration, and to return to South Dakota in search of more bones that once belonged to the triceratops.

Studying dinosaurs helps scientists gain a more complete understanding of our evolution, illuminating a through-line that extends from "deep time" to present day. For scientists like Schmidt, there's also the simple joy of coming to face-to-face with a lost world.

"You dream about these kinds of moments when you're a kid," Schmidt told St. Louis Public Radio. "You don't ever think that these things will ever happen."

## Do we still need math?

We spend much of our early years learning arithmetic and algebra. What's the use?

- For the average person, math seems to play little to no role in their day-to-day life.
- But, the fanciest gadgets and technologies are all heavily reliant on mathematics.
- Without advanced (and often obscure) mathematics, modern society would not be possible.

*The following is an adapted excerpt from the book What's the Use? It is reprinted with permission of the author and Hachette Book Group.*

What is mathematics for?

What is it doing for us, in our daily lives?

Not so long ago, there were easy answers to these questions. The typical citizen used basic arithmetic all the time, if only to check the bill when shopping. Carpenters needed to know elementary geometry. Surveyors and navigators needed trigonometry as well. Engineering required expertise in calculus.

Today, things are different. The supermarket checkout totals the bill, sorts out the special meal deal, adds the sales tax. We listen to the beeps as the laser scans the barcodes, and as long as the beeps match the goods, we assume the electronic gizmos know what they are doing. Many professions still rely on extensive mathematical knowledge, but even there, we have outsourced most of the mathematics to electronic devices with built-in algorithms.

My subject is conspicuous by its absence. The elephant isn't even in the room.

It would be easy to conclude that mathematics has become outdated and obsolete, but that view is mistaken. Without mathematics, today's world would fall apart. As evidence, I am going to show you applications to politics, the law, kidney transplants, supermarket delivery schedules, Internet security, movie special effects, and making springs. We will see how mathematics plays an essential role in medical scanners, digital photography, ﬁber broadband, and satellite navigation. How it helps us predict the effects of climate change; how it can protect us against terrorists and Internet hackers.

Remarkably, many of these applications rely on mathematics that originated for totally different reasons, often just the sheer fascination of following your nose. While researching this book, I was repeatedly surprised when I came across uses of my subject that I had never dreamed existed. Often, they exploited topics that I would not have expected to have practical applications, like space-ﬁlling curves, quaternions, and topology.

Mathematics is a boundless, hugely creative system of ideas and methods. It lies just beneath the surface of the transformative technologies that are making the twenty-ﬁrst century totally different from any previous era — video games, international air travel, satellite communications, computers, the Internet, mobile phones. Scratch an iPhone, and you will see the bright glint of mathematics.

Please don't take that literally.

There is a tendency to assume that computers, with their almost miraculous abilities, are making mathematicians, indeed mathematics itself, obsolete. But computers no more displace mathematicians than the microscope displaced biologists. Computers change the way we go about doing mathematics, but mostly they relieve us of the tedious bits. They give us time to think, they help us search for patterns, and they add a powerful new weapon to help advance the subject more rapidly and more effectively.

In fact, a major reason why mathematics is becoming ever more essential is the ubiquity of cheap, powerful computers. Their rise has opened up new opportunities to apply mathematics to real-world issues. Methods that were hitherto impractical, because they needed too many calculations, have now become routine. The greatest mathematicians of the pencil-and-paper era would have ﬂung up their hands in despair at any method requiring a billion calculations. Today, we routinely use such methods, because we have technology that can do the sums in a split second. Mathematicians have long been at the forefront of the computer revolution — along with countless other professions, I hasten to add. Think of George Boole, who pioneered the symbolic logic that forms the basis of current computer architecture. Think of Alan Turing, and his universal Turing machine, a mathematical system that can compute anything that is computable. Think of Muhammad al-Khwarizmi, whose algebra text of 820 AD emphasized the role of systematic computational procedures, now named after him: algorithms.

Most of the algorithms that give computers their impressive abilities are ﬁrmly based on mathematics. Many of the techniques concerned have been taken "off the shelf" from the existing store of mathematical ideas, such as Google's PageRank algorithm, which quantiﬁes how important a website is and founded a multi-billion-dollar industry. Even the snazziest deep learning algorithm in artiﬁcial intelligence uses tried and tested mathematical concepts such as matrices and weighted graphs. A task as prosaic as searching a document for a particular string of letters involves, in one common method at least, a mathematical gadget called a ﬁnite-state automaton.

The involvement of mathematics in these exciting developments tends to get lost. So next time the media propel some miraculous new ability of computers to center stage, bear in mind that hiding in the wings there will be a lot of mathematics, and a lot of engineering, physics, chemistry, and psychology as well, and that without the support of this hidden cast of helpers, the digital superstar would be unable to strut its stuff in the spotlight.

The importance of mathematics in today's world is easily underestimated because nearly all of it goes on behind the scenes. Walk along a city street, and you are overwhelmed by signs proclaiming the daily importance of banks, greengrocers, supermarkets, fashion outlets, car repairs, lawyers, fast food, antiques, charities, and a thousand other activities and professions. You do not ﬁnd a brass plaque announcing the presence of a consulting mathematician. Supermarkets do not sell you mathematics in a can.

Dig a little deeper, however, and the importance of mathematics quickly becomes apparent. The mathematical equations of aerodynamics are vital to aircraft design. Navigation depends on trigonometry. The way we use it today is different from how Christopher Columbus used it, because we embody the mathematics in electronic devices instead of pen, ink, and navigation tables, but the underlying principles are much the same. The development of new medicines relies on statistics to make sure the drugs are safe and effective. Satellite communications depend on a deep understanding of orbital dynamics. Weather forecasting requires the solution of equations for how the atmosphere moves, how much moisture it contains, how warm or cold it is, and how all of those features interact. There are thousands of other examples. We do not notice they involve mathematics, because we do not need to know that to beneﬁt from the results.

Excerpted from WHAT'S THE USE?: How Mathematics Shapes Everyday Life by Ian Stewart. Copyright © 2021. Available from Basic Books, an imprint of Hachette Book Group, Inc.