An interview with Professor Robert V. Moody, one of the greatest Canadian mathematicians, Professor Emeritus of the University of Alberta, Edmonton, Alberta, Canada.




ON DIFFERENT WAYS OF USING ONE'S MIND





An interview with

Professor Robert V. Moody, FRCS

by Andrew M. Kobos





Robert V. Moody
(Photo: Andrew Kobos, Edmonton, AB, 2005)



On Mathematics, Symmetries and Quasicrystals

–   Robert, you are a great mathematician. Please let us go back to your young years. When did you know that you would become mathematician? Was it the ease with which you tackled math at school or something deeper?

–   At high school when we began study Euclid's elements, which was in Grade 11  in those days – I was in Ottawa at that time, and I realized that mathematics was something that was easy for me to do. I enjoyed it so – that it was a beginning. But at that time I did not know whether I wanted to be in physics or mathematics. My Dad was a scientist. He was an electrical engineer. He was very much more on the practical side of things, an experimentalist, but knew I was more theoretician than experimentalist. I did not know between physics and mathematics what I would do. By the time I got out of high school, I wanted to do mathematics – pure mathematics, even though I did not know what pure mathematics was ! I just thought it was what I wanted to do.

–   Before we talk at greater length about mathematics, let me ask you a few questions about your formative years. You were born in England during the war. You probably do not remember much from the wartime, and from the years just after the war?

–   That's right. I don't have really many memories back in England. I was born just outside London. My Dad was working on radar during the war. We were moved to Great Malvern that was the place where scientists worked on radar development.

–   When did your parents and you immigrate to Canada?

–   We came to Canada in 1947, quite soon after the war. My Dad, as I said, was a scientist. We moved out to Deep River, Ontario, a small town for scientists attached to Chalk River Laboratories. He was working on instrumentation. I began to be surrounded by scientists early in my life. Deep River was a good place for a kid to grow up, among the forests and beaches along Ottawa River.

–   Later  your family and you settled in Saskatchewan where you obtained your first degree in Mathematics at the University of Saskatchewan in Regina…

–   Yes. You ask about going to Saskatchewan. To some extent the war made my father. He never had a chance to have a formal education, but he was deeply interested in science even as a kid. He was fascinated with chemistry, building things, and finally got into electronics, radios. Before the war he worked with private companies building radios and so forth. Then the war came and they needed people like that. He worked with scientists and learned, and he became a scientist too.

Back to Saskatchewan: my Dad had was offered a professorship at the University of Saskatchewan in Saskatoon as Head of the Department of Electrical Engineering. This is a story in itself since he did not have any degree. The President of the University was a little reluctant about hiring somebody who had no university education whatsoever, but eventually he was persuaded by the Dean of Engineering (who was one of Dad's colleagues during the war years) that he was the right man for the job.

I got my first degree in mathematics at the University of Saskatchewan in 1962.

–   Then you went to University of Toronto to do doctoral studies. What made you widely known as mathematician is – I think – the so-called Kac-Moody algebras, which you introduced in 1966 independently of Victor Kac. Would you please elaborate in short and possibly simple terms on this algebra?

–   I did this work in 1965-66 for my PhD. thesis. It was totally independent of Kac. Victor Kac is Russian. He was then in Moscow. Once it became possible to get out of Russia he went to Israel and then to MIT. We both discovered these things for totally different reasons. Another mathematician, D.-N. Verma, was also at the point of discovering them. If it were not for the fact that my thesis went to be examined by his supervisor, he probably would have go on to discover them also. I was at the right place at the right moment.

What are these algebras about? It is very hard to describe in a few simple words, but it has to do with symmetry. In the world of continuous symmetry the chief mathematical carriers of the symmetry are the so-called Lie groups (named after Sophus Lie). In turn they are largely controlled by their infinitesimal algebras, called the Lie algebras. A great achievement of the early part of the 20th century was the classification of all the simple Lie groups – those with no simpler factors. The internal structure of these is beautiful and highly combinatorial. My contribution was to realize that this structure had some beautiful generalizations, and that there was a whole new class of (now infinite dimensional) Lie algebras that could be constructed on the basis of them. So, these new algebras classified new forms of symmetry and these new forms have had many fascinating uses in mathematics and physics.

–   So, your love to symmetry had started already then?

–   Oh yes, I guess so. When you look back at my mathematical life you can say that it has all been around symmetry though I had no conscious awareness of trying to make it so.

–   Most of your scientific work in mathematics revolves around the questions of symmetry. The majority of the 80 plus scientific papers you have authored deal with mathematical aspects of symmetry. I have an impression that you consider symmetry one of the most important aspects of the Universe. Is it so?

–   I think it is a philosophical question. First of all, the notion what the word symmetry means has changed in the course of time. I think what the Greeks meant by symmetry, or what people meant two hundred years ago by symmetry, or in the last century are not the same as we mean now. It is a changing, or maybe better, evolving, concept embracing more and more. In the 20th century it was very much involved with the notion of groups – groups are the algebraic objects, which are the carriers of the symmetry, and then the representations of these groups are the ways by which the symmetry is manifested in a particular situation. So, we have a group, the carrier, and a representation, which makes it explicit.

But my feeling is, that perhaps it is pattern that's the essence. Pattern in some sense means something is repeating. You don't have an extended pattern if you do not come across some aspect of the same thing again. Pattern has something to do with repetition, so do the symmetries.

Symmetry means that something is unchanged as you transverse space or you transverse time – go to a different place and something has not changed. Something remains invariant. In that sense symmetry and pattern are much the same notion. Some quantity is conserved. All these things fall in the same ballpark. Where would we be if the sun did not rise again tomorrow in much the same way as it did today, if the laws of physics changed every hour, or if the fabric of Nature were utterly formless? So if you ask is this one of the most important things in the Universe – yes, sure, invariance, pattern, and the repetition of a structure, yes, for sure it is.

–   In elementary particles and in the snake skin…

–   Yes, for example. In everything. And I think we'll find symmetry in this wider sense in brain too. The brain seems to be built around the spawning and recognition of patterns. It's then natural we feel the Universe that way.

–   Symmetries at the most fundamental level are present in elementary particle physics. Yet in the Standard Model symmetries must be broken to arrive at non-zero particle masses. The much expected Higgs boson is supposed to be carrying the mechanism of breaking symmetry at that level. High energy physicists are talking about supersymmetries in the particle “world”, string theories, etc. Do you see a link between the mathematics of symmetry and particle physics?

–   Yes, for sure. This type of mathematics has been fundamental in particle physics. Most of the particle physicists know an awful lot about representations of groups. The whole theory of quarks is built around representation theory. The Standard Model itself is built around representation theory of certain groups. String theories also borrow a lot from the theory of infinite-dimensional Lie algebras and from conformal symmetry. So, I think it is inevitable that the symmetry theories will appear in these physical theories and it is also inevitable that physical theories will lead to new mathematics.

But there is a flip side to symmetry. Perfect invariance would lead to world of no change or evolution – as dead as one that is utterly random. So that symmetry breaking should be required as part of physical theories is not so surprising.


* * *

–   For the last 15 years or so, you have been creating mathematical foundations for aperiodic order in quasicrystals. You said in your web page that these are "almost periodic structures that permit normally forbidden symmetries to appear in Nature." Would you please elaborate on "forbidden symmetries appearing in Nature"?

–   This is an old thing. Well before the time that people knew what crystals really are, scientists interested in crystals had begun to study lattices… Let's backup a little bit. What does it mean to be a crystal? Crystal has a structure that has the underlining symmetry of periodic repetition in three independent directions. That is the usual definition of crystal. So, what we have underneath basically is a building block and that building block is simply repeated along the crystal lattice. It is like piling up bricks in three directions to fill up space. This is the way the Nature does it and it is an extremely common mechanism of forming solids – crystallization. Even complicated molecules, including molecules like DNA, crystallize.

So, as I was starting to say, it was discovered long ago – well before the atomic theory of matter could be confirmed – that with this type of symmetry one cannot have a fivefold rotational symmetry. It is "forbidden" or rather it's just not possible for the lattice to permit the fivefold rotational symmetry. Although the fivefold symmetry is pretty prevalent in Nature in things like starfishes and flowers, there can't be an extended repetitive structure, which is built on a fivefold symmetry, at least not in a strictly periodic way. It became sort of one of the fundamental ideas in crystallography that crystals were the same as lattices and some types of symmetry were not possible.

So, it was a big surprise when in fact such forbidden structures were discovered by Dan Shectman in Israel in 1982. He was studying certain metallic alloys that he was creating by rapid cooling techniques. One of these alloys produced a diffraction pattern that showed that what he was looking at was built on the icosahedral symmetry. The fivefold symmetry was there staring him in the face!

Here was the situation. The signature of crystals is a diffraction pattern with sharp Bragg peaks. Shectman's diffraction pattern had the same tell-tale signature of sharp Bragg peaks, except that actually it had a fivefold symmetry. This was not supposed to be able to happen. He doubted it, everyone doubted – at least at first. It took several years before the paper was finally accepted for publication. There was a lot of controversy around it. Linus Pauling was one of the chief objectors to it. At present, a couple of hundred of these quasicrystals are known, some of which are beautifully perfect. There is a lot mystery around them, because it is not easy to know how Nature makes them. They are definitely not crystals in the usual sense of the word since they do not have the periodic structure. But they definitely have incredible long-range internal order – and also underlying fivefold axes of symmetry.

I was attracted to the subject when I saw a picture of a quasicrystal diffraction pattern. Beautiful spots, a beautiful pattern. But the special interest for me came out in Lie theory. In Lie theory, in the classification of simple groups, the ones that we mentioned before, a restriction occurs. As I said, the underlying structure of a simple Lie group is a discrete combinatorial structure, and to go further, the basic object there is a finite group generated by reflections – a so-called Coxeter group. And it turns out that the orders of the products of the pairs of the generating reflections are restricted to 2, 3, 4, and 6. Note, no 5! The same crystallographic restriction occurs as we saw above.

So, when I saw the quasicrystal diffraction pattern, I thought, oh, this is very interesting! Here the fivefold symmetry does occur in Nature in an extended structure. Maybe there is a way in which Lie theory would extend somehow to include these objects. This was my original idea. But I was never successful in doing that. On the other hand I have grown more and more into the area of aperiodic order and I am spending a lot of time studying diffraction. It is very interesting mathematically.




Selected Area Electron Diffraction pattern of a decagonal phase
of the Al70Co11Ni19 quasicrystal alloy.


–   It is, I understand, a new area of mathematics. Diffraction is very common in atomic physics. Since quasicrystals demonstrate themselves by electron diffractive structures, there must be a link of your mathematical papers to atomic physics?

–   Definitely there is a strong physical side to it. Actually people who are in quasicrystalline physics and material science are interested in these things. A fair number of experimentalists are working on the theory and producing some marvellous results. Diffraction is of course the primary way in which they tend to get information. However, with diffraction there is always an inverse problem – the diffraction pattern does not tell us what the structure is that it came from. It is not possible to take a diffraction pattern and unambiguously say where it came from. Of course, as an experimentalist you always have additional physical information and it's usually when you tie that together with the diffraction, you can hope to deduce the position of atoms and so forth.

So here we are: it is over twenty years since the first quasicrystals were discovered. I was at the ninth international conference on quasicrystal just a few weeks ago and they say that of the quasicrystals they know much about, the best they know is about 80% of the atomic positions. After twenty years of work! So, it is still obviously a very difficult problem.

But it is one of the few areas, I think, where experimentalists and mathematicians are talking together, because certainly some of the mathematical approaches to the subject are critical for experimentalists. The key method has always been to use projection of a lattice from higher dimensions. Talking about projections from six dimensions does not sound very physical, but the interesting thing is that the experimentalists do actually talk in these terms. They are thinking of projections of lattices from four to six dimensions in order to make models of what they are doing. There is definitely a real need for the abstraction in order to even do the experimental work. You have to have some sort of model in order to make the experiments, and they use these higher dimensional models that have been really worked through both by physicists and mathematicians.

–   More generally, I guess, your papers constitute a link between fundamental mathematics and physics?

–   I believe they could – maybe I hope they would! However, it is probably incorrect that you said I was one of the people who founded some mathematical basis for the mathematical theory of quasicrystals. I am certainly interested in it and I've been working on it, but I can't take much credit for being the initiator or having really great ideas. There are some other really good people involved.

–   Some theoretical physicists claim that the last mathematically fully correct theory in physics was quantum mechanics some eighty years ago. All newer major theories, they say, involve dubious mathematics, introduce renormalizations, new constants of Nature, like in string theory, etc. Would you agree with such an opinion on the current fundamental physical theories?

–   A lot of people think that theoretical physics have moved too far away from experiment at the moment, and so far string theories have not produced many results that could be tested experimentally. In that sense, I think, there are some difficulties. It's very hard for me to speculate – I'm not fluent in these areas. But let me say something – even if it is nonsense. At the level of the Planck scale we know that usual ideas about geometry break down. Most of mathematics is based on set theory, and most of geometry on the idea of sets of points. But points are not observable things, even in theory. So it seems suspicious! And likewise the fundamental role of real numbers has always seemed questionable to me. The real numbers are such an artificial and complicated construct. Why should the real numbers be the right object to parameterize space and time? So perhaps what's missing is a totally new view of the geometry. It would take some tremendous imagination to create it. I think that string theorists would say that that is what they are doing. But I could imagine something far more iconoclastic than that.


* * *

–   You have been in fundamental mathematical research and in academic teaching for many years. You have had many very talented graduate students. On the perspective, what does it take to be an outstanding mathematician?

–   Well, there are some beings who just seem to live and breathe mathematically naturally. But for the rest of us – I don't know – perhaps a good deal of passion for the subject and a good deal of luck! Many people of only modest talent achieve a lot in their lifetimes through just plain hard work driven by love of the subject, together with a some good luck.

It's interesting when you see students. Every student is different and even the brightest of them have deficiencies to overcome. If they don't work at those they don't go anywhere. It's not only a question of the sheer brilliance, it's sometimes sticking with problems, it's not giving up, it's working years and years on things. And no doubt luck! Serendipity is a huge element of development of science. Somebody says the right thing or you meet the right person, you make a stupid mistake that turns out to be totally revealing of something else, something clicks and off you go.

Recently John Gribbin, in his book The Scientists, wondered how many geniuses there really have been in the world of science. And he decided not many! Science is a wonderful area of human endeavour because a person with an ordinary talent can do it, and if lucky, can make a really important progress. The same does NOT hold for art!

Much of science is a community effort. Many people have to work on things and it is really an accumulation of ideas of many people over a long period of time. Once in a while somebody comes with totally brand new ideas about things. On the whole most of us, you know, play the ideas they're given, add a little bit and, if we are lucky, something falls into place and we make some progress.

–   Even Albert Einstein took borrowed from Hénri Poincaré and later admitted it. Furthermore, Bose-Einstein Statistics was probably based on the ideas of Wladyslaw Natanson that Einstein, it seems, knew...

–   Yes, that's right; the same with Newton. Now it seems that a lot of Newton's ideas were seen by Robert Hook first. Yet one can't deny the brilliance of those people, one cannot belittle people of statures of Newton or Einstein. These people were fantastic. But they did not come out of a vacuum.

–   Andrzej Bialas, a particle theoretical physicist from Cracow, told me that his supervisor, Jan Weyssenhoff, had taught him 45 years ago a very important thing: you think about physics all time, day and night, at least until some age. Could such an advice be given to young mathematicians too?

–   Yes. A lot of people have spoken about learning to live with mathematics, not only with mathematical problems. I think that is one of the things that as a young mathematician you have to learn. Mathematics is taught in a particularly bad way in certain sense, because you tend to see a theory presented to you, one that has been worked over very carefully by many people and it is now very smooth, very slick, everything comes at exactly the right moment, everything fits together. So when you do measure theory, or analysis, or group theory, everything seems to be perfect. But when you do things yourself you realize that you have no idea which way to go next and nothing is simple and often you get things backwards. So as a young mathematician you have to learn how hard it is, how long the periods are when you seem to be getting nowhere. And you wonder, oh how did always these great mathematicians do it so easily? But then you look at the history of the subject and realize how hard it was for them too.

–   Research in the contemporary cut-edge physics, not only the experimental one, involves, or rather is done in large teams of scientists. Has mathematics changed too in this respect? Or is it still done by the few chosen as before?

–   Mathematics used to be really a solo type of subject, at least as far as writing papers was concerned. But there are few soloists these days. Now people do mathematics much more in groups. Most of papers are co-authored by two, three authors. It has become very common, it's an enjoyable way to do science, and often very productive, too. So, yes, I would say, the culture is changing in this respect. I still write the occasional paper for myself, but this is rather an exception.


* * *

–   In the last a few years you have organized the Banff International Research Station for Mathematical Innovation and Discovery, now known by its acronym BIRS. You were its founding scientific director. Would you please describe briefly the objective and the mode of operation of BIRS?

–   That was really the idea of Nassif Ghoussoub, the founding director of PIMS (The Pacific Institute for Mathematical Sciences) in Vancouver. He was looking for a new venture that Canadian mathematicians could do. The reason for this was partly because we are forced every five years to go through NSERC's reallocation of funds exercise. NSERC is the Canadian federal funding agency for science. Every five years it goes through a process in which every sector of science that they are funding has to put ten percent of its budget into a pot and this money is reallocated according to what are believed to be the evolving directions of science. In this process some areas loose up badly every time, some gain.

Nassif wanted mathematics to go in with a fresh new vision. He played upon a long felt need in North America. There is a famous mathematics research institute in Germany in Oberwolfach in the Black Forest and it has been a centre for mathematical workshops for some fifty years now. It has evolved in time, but now what it does principally is to run workshops, one for a week, probably fifty of them a year. People come by invitation of the people organizing the event. This has been enormously successful. So, we thought, maybe we should create a similar thing in the Canadian Rockies.

Why does it really work? The idea is that you have an institute that offers complete local support for people once they come here. They don't actually pay for anything. You ask potential organizers to submit proposals to a scientific committee, and it selects the proposals it likes. It says, "OK guys, you're funded. All you have to do is to give us the names of your participants." BIRS contacts all the suggested participants, arranges their arrival and departure dates, special needs, and so forth, and once the participants are in Banff, it is free for them. It is very, very attractive. If you are a top-notch scientist, you don't want to spend a lot of time organizing conferences, looking around for money, or negotiating for banquets, hotels, etc. Now all you need do is submit a proposal for the research area of mathematics that you want a meeting for, and a list of names. And that's it. It leaves everyone the maximum amount of time for the exchange of ideas. It has been very successful.

–   Are these more some sort of brainstorming sessions on a particular problem in mathematics or some sort of presentation sessions?

–   Sometimes they are brainstorming sessions, sometimes organizers tell us that what they really need is to get to different groups of mathematicians or scientists together who normally would never meet. For example, some people working on hydrogen fuel cells organized a workshop. They invited some biologists, some people from Ballard Systems (a leading Canadian group developing fuel cells), some chemists and engineers, as well as some numerical analysts, all of them involved in one way or another with membranes. They had a fantastic time, because most of these people had not met before, yet had lots to offer each other. It was very, very productive.

So this is one sort of thing. The other is when you have a pretty well recognized area, such as some part of number theory and people come to exchange their latest results. It's very up-to-date and it's pretty intense. Five days, they work from dawn to dusk. As we used to say, they work, eat, drink, and sleep together! This type of atmosphere makes quite a difference.


* * *

–   To finish this part of our conversation: Robert, you have been awarded a number of different distinctions, professorships. You've been made Officer of the Order of Canada. I know you are an extremely modest and unassuming man, but which of your distinctions do you value most?

–   Oh, I am sure of all those The Order of Canada is one I value most. I have lived my life in Canada, and it is a great honour to have your country distinguish you in that way. Of course there are a lot of people who are equally or more deserving it, but it is the way things turned out. I'm proud of that particular thing. Canada has been good to me and my family.


On Far Eastern Spirituality

–   Bob, you are very much interested in the culture and the tradition of the Far East. Buddhist spirituality is close to you. For you, it must have been a long way to go, almost from one civilization to another, quite a different one. Would you mind telling me how have you arrived at all that?

–   It's hard to know how I have arrived at that. I felt a definite attraction to oriental art early on. But probably it wasn't until the age of thirty that I came to some sort of crisis in my life when I began looking for a deeper understanding of the nature of things. I was aware of my spiritual side earlier, but had made no effort to understand it more deeply. I read a book on Zen Buddhism, and was immediately attracted to its poetic vision of life and its determination not to get hung up on ideas and concepts.

But it was to take many years to come to terms with the scientific, cultural, and spiritual sides of my mind. Probably it is not complete yet!

The Orient is appealing in many ways because it has a very organic viewpoint on things, much less dualistic than Western ideas. The seamless interconnection of all things is a foundational point of oriental philosophy, and of course it is very appealing to the scientific mind. Oriental spirituality is not founded in the great struggles of good and evil, but rather on the questions of realization of the ignorance and illusion that so limit our concept of self.

Most of us in the West draw on our roots in the music, art, literature, and architecture of European culture, as well as from the scientific revolution that emerged out of the Renaissance. In mathematics all of us look back to the world of the Greeks. Being part of that, the Western tradition has a strong attraction for me. But its concept of the nature of reality (i.e. Judeo-Christian philosophy) is impossible for me. I really did try to embrace it!

In the Orient one finds a totally different approach. Look at the very first words of the Tao Te Ching:

"The Tao that can be spoken of is not the eternal Tao,
Words that can be spoken are not eternal words".

Already the realization that words can never describe ultimate reality. Or from the Upanishads, "the self in man and in the sun are one". It seems strangely prophetic, in a way, given what we now know about cosmology! But that is not the point. It is the fact that there is nothing that is not intimately connected with us.

–   The following can be asked about any spiritual model of a human being in the physical Universe, about any religion. But in this particular case – how do you reconcile science, mathematics with Buddhism?

–   Buddhist spirituality is, I would say, close to me, even though I've never become a Buddhist as such. But let me back up a little bit. Buddhism is many-faceted. It has spread into many countries across the world and it's hard to know exactly what it means. But if you go back as close to the original teachings of Buddha as you can find, they all point at the same direction: that one should carefully observe the mind, how it makes its values, shapes concepts, and in general creates the world that we experience. The Dhammapada opens with:

"We are what we think. All that we are arises with our
thoughts. With our thoughts we make the world."

As I understand it, Buddha never asked anyone to accept his teachings on faith. He pointed out what he perceived to be the truth and encouraged people examine themselves and see for themselves that it is true.




Golden Reclining Buddha.
Wat Po, Bangkok.
(Photo: Robert V. Moody)


For example, consider his teaching that all existence is suffering. The idea of suffering, the cause of suffering, the way to live with suffering, you can find it all out for yourself. You have lived long enough to realize that these things are true. Of course that is nothing new! But Buddhism moves away from usual philosophy in its insistence on moving beyond intellectual thought and into practice. In this case his teaching was a practice (the 8-fold way) aimed at finding happiness in the midst of this suffering. Namely by nonattachment, the realization that the small self that most of us live with and attach to most of the time is only fragment of who we really are. Actually this teaching was probably not particularly new at that time (500 BC). In Patanjali's Yoga sutras, which come from the same tradition, say the same thing – including the emphasis on practice.

I don't think this is at odds with science. It doesn't come forth on stone tablets and doesn't have any creed. It is directly based on your own personal experience. I think, it allows you to find what you find for yourself by living.

Often people think that religion has to do with beliefs, specific beliefs, which come from somewhere else. But this idea of belief is too narrow. I think I am a believer in the sense that I really feel something about the world I live in, I really have faith in it. But that faith is one that sees me as a product of evolution, related to all the creatures on this planet, with common ancestors way back. Life is a unity and I don't exist save in the interaction with all other things. In that sense I have a great faith that I belong here.

A lot of people think: "I don't belong." A friend of mine says: "It's unfair that I was born, I was never asked if I wanted to be born." Such things are now totally foreign to me. There have been moments earlier in my life I thought I did not belong. But I realized there is no other life than the life I'm living. I do belong and it could not be any other way.

My favourite statement about belief is from the Zen literature:

"If you do not believe, look at September, look at October!
How the leaves fall to fill valley and stream."

So, I don't see any conflict whatsoever.




Robert V. Moody
(Photo: Andrew Kobos, Edmonton, AB, 2001)


–   Yet mathematics is a way of describing reality…

–   Yes. I must say mathematics is peculiar in the sense that it is applicable and it has that notion of being “right”. I don't know, however, what existence it has outside the human mind. They say that every mathematician is doing mathematics as a Platonist: you have to believe that somehow the result is there and you just try to find it. On the other hand, I really think that outside the human mind we don't know anything. It's is a problem to which I have no answer!

–   You just said you had been very interested in Buddhist and Hindu art even before you started to look at it in a more spiritual way. On the occasions of your scientific travels to India and East Asia, you visit Hindu and Buddhist temples, take pictures of the sculptures. How do you perceive Oriental art? What else, apart from the linkage to your spirituality, does attract you in this art? After all we come from quite different, Western roots, say those of our aesthetics.

–   It's a good question. I think, one of the notions might be simplicity. Of course not all Oriental art is simple, but definitely in Chinese and Japanese art where there is an effort to keep very simple lines, very clean. And often the subtlety lies in the power of suggestion from a few brush strokes. Then of course, it comes out of the philosophical spirit that I have already said appeals to me so much.




Krishna playing his flute.
Wooden figure, Cochin, India.
(Photo: Robert V. Moody)


If you look at pictures I take, I don't take pictures of symmetry. It's somewhat amazing to me. I don't find things that are totally symmetric appealing. Why is that? My mathematics is all around symmetry, but I don't find symmetry artistically so pleasing! That's something I can't really reconcile in myself.

–   Recently, you wrote a scientific paper on the geometry of the temple cave art in India1). Would you please tell me how you came to this subject, rather unusual for a mathematician?

–   In 2003, I was in Varanasi, India, for three weeks. I was at the city's university, visiting material scientists. Varanasi, also called Benares, is an old city – several thousands of years old in fact – a famous site where the people go down the stone steps and into the Ganges every morning at sunrise to worship. It is a very fascinating place to be. It is like being immersed in another era.

I went to the museum on the campus there. In there is a little gallery for the person called Alice Boner, who was a Swiss artist, painter and sculptress who lived for 48 years in Varanasi, into her nineties. She was a brilliant woman and did a lot of drawings of Indian art in caves. These are not caves in usual sense. Whole temples were carved out of the sides of mountains. This goes down to the smallest details. For example, it is remarkable that the many stone sculptures were not put place there after the fact – they were carved from the solid rock, in situ, out of the mountain as the temple was carved . Huge marvellous things these temples are.

Alice Boner studied a lot of these sculptures – she was in fact drawn to them over and over again – and she realized that there was a geometric foundation to this art form. It doesn't look like it and you can't readily see it even when you know. In fact, European scholars – really good ones – had expressly declared there was no geometric basis to this form of Indian art.

But Boner discovered there was – and eventually she wrote a book on it. It is considered as an important contribution to the understanding of Indian art, and in fact it is still in print today – 40 years or so after it was first published.

She also painted. She spent a lot of years on a triptych: three paintings of Divinity. From Hindu perspective there are three aspects to the way the God manifests himself (or whatever 'self''!): the creative aspect, the preserving aspect that keeps things in bond, and the destructive aspect. In the West we don't tend to think of God as being destructive.

–   But he or she really is. Here is his or her kingdom…

–   Really! Look at the tsunamis… destruction is everywhere! Life doesn't exist without death and destruction.




Shiva – Dancing Torso.
Asian Museum, San Francisco.
(Photo: Robert V. Moody)


So, Alice Boner created these three paintings, one for each of the three aspects. I took photographs of them and I went back and read about her. I thought: "well, she spent all those years on them; did she apply the same principles in these paintings as she had discovered in the local cave art?". Nobody seemed to have asked this question! So, I started to draw the circles and the lines on these photos and it all was there. Surely, she did! You can't notice that when you see the paintings. So that was the origin of this little paper. It was published in a proceedings from a math–arts conference in Banff.


On Photography

–   Bob, the beginning of our friendship was through photography. You have been an accomplished photographer. Your, mainly black and white pictures reflect, I think, your spirituality. I know such a linkage only too well from my several decades long experience with photography. What does photography mean to you?

–   I love it. I like the notion of making images. I think the mathematics I do uses my mind in a certain way. Photography uses it in a different way – not only when I'm taking pictures, but also when printing them. I don't print very much, and when I print it's hard to be satisfied. But when I finally get an image I like, I spent a lot time with it. There is definitely some notion of feeling in there, and of trying to impart what that feeling is visually.

Perhaps the real linkage with mathematics is the level of the abstraction of black and white photographs. You're looking and you select something to be abstracted.

I believe most people appreciate this same abstraction in photographs. Actually, if you ask somebody to imagine some photograph that they have found striking and can bring to mind, then almost inevitably it will be some great black and white photograph – a Yousuf Karsh, an Ansel Adams, or a Cartier-Bresson.

Colour does not have quite the same impact. Black and white photography has a capacity to isolate, abstract, and bring into focus particular aspects of pattern, form, and beauty – and these are like mathematics itself. In a truly great black&white photograph the combination of light, form, and texture creates an indelible impression.




Haida Totem Poles.
Vancouver.
(Photo: Robert V. Moody)


–   Your black and white landscapes are quite reminiscent of Ansel Adams'…

–   I was very influenced by Ansel Adams, way back. I first learned about him in the 1960s. I was then living in New Mexico. He was very involved with Sierra Club, which is an environmental organization, and through them I saw a lot of his pictures. It was originally the landscape that I wanted to photograph, but once I began, I started to take pictures of places, which are linked with spirituality, e.g. churches, and so forth.




Near Mammoth Hot Springs, Yellowstone National Park.
(Photo: Robert V. Moody)




Santa Clara Canyon.
Big Bend National Park, Texas.
(Photo: Robert V. Moody)




Saint Francis of Assisi Church.
Ranchos de Taos.
(Photo: Robert V. Moody)


And I like taking pictures of sculptures and art. You start with something rather magnificent and you have a chance to put an angle on it to see it in a particular way.

–   Your photographs can clearly be divided into two different groups: landscapes and sculptures. In your landscapes you go to a larger perspective, not a detail. In your photographs of sculptures you do go to detail.

–   I suppose, I do. I live in the West here, in Western Canada. My wife and I had a choice at one point between going back to the East, to Queen's University in Kingston, Ontario, or coming to Alberta. The geography kept us here. We love the big spaces. I love Western North America, its vast areas. This is something I like and try take pictures of.




Looking East.
The Porcupine Hills, southern Alberta.
(Photo: Robert V. Moody)


And that's why I like the Hasselblad with its big format. The bigger the format the better. If I could carry around a bigger camera I would do it. And then you can make big prints that somehow portray the scope of the scene.

It's true. When it comes to sculptures, I like to get into details.

–   Let us please be a little bit more specific. Your photographs of sculptures comprise European, North American and above all Asian Buddhist sculptures. The latter seem most impressive. Is there anything else, apart from the beauty of the form, you want to convey with your pictures of sculptures?

–   My photographs represent my emotional response to what I see, whether that be the natural world or the handiwork of some craftsman or artist, a new intimacy of things, detail, structure, or the momentary shared understanding between artist and viewer.




La Toilette de Allante.
Sculpture by Jean-Jacques Pradier, 1850, Louvre, Paris.
(Photo: Robert V. Moody)




Mercy.
Sculpture by Oliver LaGrone, 1991. Albuquerque Museum, Albuquerque, New Mexico.
(Photo: Robert V. Moody)


I don't think, however, that at this point I've developed my photography sufficiently. I'm hoping in my retirement I'll have more time to do that, to concentrate more on it. Basically, I take a lot of photographs when my wife and I are travelling. This is the time when you can leave normal things behind and begin to look with “seeing eyes” at the things that are around you. It is a bit haphazard in this way, though, I would say.

–   You have also done some photographic compositions in colour, related to the City of Edmonton's Whyte Avenue, and old and a somewhat Bohemian street on the city's high, south bank of North Saskatchewan River. Were they merely your different attempts to search for forms, or did you have other thoughts leading to them?

–   I have experimented a bit with colour. I like colour. I fact, I've even recently been trying to take black and white pictures and colorize them afterwards. My work on south Edmonton… I like the area. I lived close by. It is very colourful, I took pictures there, but I can't say I had anything deep in mind about that.




Life is Beautiful!
Montage of scenes from Whyte Avenue, Edmonton-Strathcona.
(Photo: Robert V. Moody)


–   What about the portraits? You've done a number of them, a Gallery of Mathematicians. One of these portraits, i.e. that of Donald Coxeter, is superb. Light on his old face, he's looking down, slightly smiling as if absorbing that light to use it for a photosynthesis of ideas. Here is not a form, not a natural or a man-made structure or a texture, but a man in the Universe. If such a posture was not purposeful, you were extremely lucky when taking this picture.

–   I did not ask him to pose. He had just given a lecture in the year 2001 at the special conference on symmetry in Banff. He was in his 90's if you can believe it!

–   I remember this conference. It was on the occasion of your 60th birthday.

–   Yes. Coxeter came, it was the last time I saw him. He enjoyed himself a lot. He'd just finished giving a talk, someone was talking to him, and I started taking pictures. The picture technically is not perfect, but nevertheless it has caught something about him.




H. S. M. (Donald) Coxeter
(Photo: Robert V. Moody, Banff, AB, 2001)


It is interesting… when I gave a copy of this picture to his daughter she hated it because it showed how ravaged his face was. He was suffering a lot from various things, and it brought back all these memories to her. But most people see this picture the way you've described it. He had sense of wisdom. He was a distinguished man, for sure, one of the great Canadian mathematicians, one of the great geometers of the twentieth century. Yes, I was just lucky, just lucky.

–   From a superb Hasselblad camera you're going digital now. In B&W photography, while printing, you can basically play only with the darkness and the contrast of the image, which can produce great results anyway. In digital photography, apart from the ease and the cleanness of processing, you have much more freedom to extend your print composition by computer controlled imaging. The result may be a less than faithful copy of what you had seen. Do you perceive digital photography this way?

–   Some people would say they would not touch the photograph, they would not change any bit of it, that a digital print is a different print. To me this faithfulness to the actual negative or digital image is not compulsory at all. I think I follow Ansel Adams in that. His idea was that the whole process is that of making images, and it does not really matter what the means is if it is what you want to express through photography.

–   I think that is all about how you perceive the photograph. Is it to be more or less a faithful image of the reality, or a kind of your artistic expression, of your thoughts, your perception of the image's subject.

–   That's right. Absolutely. You know that Ansel Adams often said that a negative was for him like a musical score, that it comes with an artistic freedom to interpret it. Of course the negative must be pretty decent before you can do that. He thought the printer had a lot of control of the final artistic expression and that's why he left his negatives to future generations to try and print them again. I agree with him that the image you got with your camera is what you can work with and what comes later is important too.

So, I print a lot digitally, even if I continue to photograph with an ordinary film. I like the film camera. It is still a lot nicer than the digital camera. It's a beautiful, aesthetic thing to hold and use.

Digital printing definitely has the advantage of being much more convenient, though I would say that it is anything but easy! I'm moving now to Victoria, BC, and taking my darkroom with me, and I hope to do some ordinary, standard chemical printing as well as digital printing. I will be playing with both.

–   And employing your mind with symmetries, I'm sure.

–   With symmetries too.

Edmonton, AB, June 2005;
The transcript authorized on October 18, 2005.




Web pages of Robert V. Moody :

Home page :     http://www.math.ualberta.ca/~rvmoody/rvm/

Photography :    



Footnote:
  1. Robert V. Moody, "Alice Boner and the Geometry of Temple Cave Art in India". Proceedings of the 2005 Bridges: Mathematical Connections in Art, Music, and Science Conference (Renaissance Banff), Eds. Reza Sarhangi and Robert V. Moody, 2005.

    This paper can be downloaded as a PDF file from R.V. Moody's web page, section Papers to Download.   (return)




The Polish translation of this interview published in Zwoje – The Scrolls :



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