Kenneth Snelson Exhibition
The Nature of Structure
The New York Academy of Sciences, January - April 1989

Kenneth Snelson and Hans Christian von Baeyer: A Conversation

HANS CHRISTIAN VON BAEYER is Professor of Physics at The College of William and Mary in Williamsburg, Virginia. He is well known to Academy members for his column Physika which appears regularly in our magazine The Sciences. Attentive to the world Of art, he has long been acquainted with Snelson's sculpture. To prepare himself~for their conversation, however, he further familiarized himself~ with Snelson's work at the Galerie Zabriskie in Paris (where he was on sabbatical during the 1986-87 academic year) and read Snelson's unpublished manuscript, which is the most thorough nonvisual articulation of his ideas on the atom.
Von Baeyer's humanistic sensibilities are as well honed for a scientist as are Snelson's scientific sensibilities for an artist. He brings a special patience to the task at hand, as the following excerpt on physics from his dialogue with the artist suggests: "It turns out all the things we thought were solved fifty years ago were not. The world is always more complicated than one thinks. That's what's so nice about life, I suppose, not just physics. It keeps us going."
Artist and physicist met for the first time on Saturday, September 19, 1987, at the artist's Soho studio in lower Manhattan. Their spontaneous and informal conversation ranging throughout the day and evening, moving from studio to restaurant and back again, began at the artist's new graphics computer:

H.vB.: What does computer technology do to a sculptor when you've spent months and years building a monumental work? You have to do all the details, get them all right and then someone comes along and ...

K.S.: ... and [snaps fingers] does it like that.

H.vB.: But doesn't that disturb you as a sculptor? Sculpture is an ancient craft.

K.S.: No, I find ... no, I'm not even sure I'm a sculptor. I'm interested in three dimensional space and this kind of spatial experience is exciting.

H.vB.: That's a good answer ... if you don't classify yourself as a sculptor. Then that makes sense.

K.S.: A lot of people have questioned it. There was a Times article some years back where I said I'm not interested in sculpture, I'm interested in structure.

H.vB.: If you are interested in structure then all these things make eminent sense, but if you call yourself a sculptor in the tradition of Phidias then this computer art doesn't.

K.S.: But I find this too extraordinary!

H.vB.: I have always thought of sculpture as being something tangible, literally something you can touch. And now this computer generated image I can't touch, I can only look at. It's something else. K.S.: I've tried to create out of wood or steel or plastic or God knows-what the three-dimensional picture of an atom in my imagination and I've found it's quite impossible. The material objects don't convince you of anything because they're always so klutzy. They always look like hardware, whereas a proper atom enlarged to human scale should be jointless, glueless, stringless, and not encumbered much by gravity.

H.vB.: Ah, now I know even more why I was so bothered by this: because it shakes a preconception I had about your work. The preconception I had was that you are interested in atoms and, being a sculptor, you really can't deal with this thing we call an atom until you've made something tangible out of it - until you've made a model, a physical, tangible model, a three-dimensional thing. To a physicist the word model means something more general. It might be tangible like yours, or it might be much more abstract, like an equation. So I thought your fascination with atoms was that you didn't like the Schrödinger equation because that isn't tactile. You'd like something that you really could put your fingers on, right?

K.S.: Yes, I guess there are ...

H.vB.: ... but now you're moving away from that. You're moving into a middle ground because these graphics are sort of halfway between the sculpture and the physics of it. They are very abstract. K.S.: The question is: Where is that entry level of the quality we call solidity? Right there's one of the important questions about the atom: the critical question. And it's really been that the supporting or insistent evidence for me in making this atom model has come from my own experience of making sculpture out of solid stuff. How matter occupies space, especially in my sculptures of metal tubes and wire ... it's a very simple idea. There must be some logical system, some mechanism for solidity - as much a first principles mechanism as that of magnetism or of electrostatics. It must be a structural first principle.

H.vB.: Others call it the Pauli exclusion principle."

K.S.: I agree it's the same, except that the Pauli principle certainly isn't interpreted in this same way, spatially, in the atom. What I'm suggesting through my model is that Pauli's exclusion principle is really telling us that electrons in their atomic orbits carve out space, and totally use that space to exclude one another.

H.vB.: But, you know, the exclusion principle is a central mystery of physics. You substitute another central mystery, that of impenetrability. You're not telling me why two steel swords cannot interpenetrate, you're just saying they're solid. And I'm saying two electrons can't be in the same wave function. But those are very deep mysteries and we have no idea why the Pauli exclusion principle is there. And you don't have any idea why two swords can be ...

K.S.: Well, no, and I don't have any idea what a magnetic field really is either. But I don't have to know what a magnetic field is to experience a magnet's force.

H.vB.: OK, let's back up. It seems to me a place where we could start, where I think we would both absolutely agree, is that the first step of what you're doing, which is imagining an atom - imagining how it looks, imagining the forces that keep it together, and so on - that first step is absolutely identical between what you are doing and what all the great people who wrote these physics books were doing. That's how they all start, all of them, from the 19th-century folks who thought in very concrete models to 20th-century people like Heisenberg, who ultimately was an extraordinarily abstract thinker. Nevertheless, scientists - all physicists - start the same way you are starting, which is to wonder what's down in there and to imagine some kind of a model and adduce all kinds of evidence and bring it in there and build themselves a model which may be more mathematical or less mathematical, more tangible or less tangible. The starting point for us and you is identical. We would also like to know what atoms are made of. But then comes the next step.

K.S.: Yes, maybe there are two or three levels to this: One thing is that I'm not a scientist. In some other far-off civilization at a different time, perhaps I would be a scientist, but I don't think it would be possible for someone with my visual bent ever to wade through quantum mechanics, non-visual as it is. What troubled me when I was doing a lot of reading was not only the lack of pictures but the general insistence that pictures are out of the question. They were almost naughty! I'm not troubled by not being a scientist! People do ask what I'm doing worrying about the atom - that, after all, I'm an artist and art shouldn't be involved with scientific subjects. I notice that when art brushes into the realm of science, it's always the kiss of death as far as the art world is concerned.

H.vB.: Why do you suppose that's so?

K.S.: Categories. One time the director of a major museum explained to me, "You know, we like to keep these things separate."

H.vB.: Really?

K.S.: Another reason I'm not a scientist is because I'm not good at math. I have a kind of block and that's the way life is. So I get around it by making things in space instead. But going back to the question of quantum mechanics and visual urges ... I bet a survey would turn up few visual people among quantum physicists. By natural selection it has weeded out visual types so it's no wonder you come across such warnings not to try visualizing atomic structure. In introductory chemistry and physics books, authors urge students to forget about pictures, to avoid thinking about that familiar logo atom - the Bohr model - a nucleus with orbiting electrons. But what if some very visual people had gotten in early on? Might there not be more visual models, might there not be more interest in finding one? Right now, there's little interest in that direction. The immediate reaction is we don't need them. We deal with these things through mathematics - and mathematics is quite enough.

H.vB.: Pauling is a very good counter-example: He did think of molecules in visual terms.

K.S.: There couldn't have been a more abstract, a more non-visual and mathematical group, though, than the men who caused the quantum revolution - the second wave - the new quantum group.

H.vB.: Right. Heisenberg.

K.S.: So we know fashions happen in all parts ... I want to bring visual models back. I started thinking about this sort of thing in 1960 when I discovered these magnetic phenomena. More than anything I'm trying to make them find their way into the models which exist ... or, I should say, the evidence which exists. [Brings out magnet models, above]

H.vB.: I'm sure you've checked out these magnetic fields, that they all really do fit in the right way.

K.S.: The gearing is just part of their binariness: They reverse-rotate in a checkerboard pattern, and alternate magnetically. I first discovered the group of eight magnets. I thought this must be the octet of the Periodic Table (I hadn't taken chemistry in high school, only physics). In reading chemistry books, though, I saw that shells and subshells of electrons exist in certain numerical patterns: There are not only eight - there are two, there are six, ten, 14, 18 and 32. Then, by playing more with the magnets, I began to identify the rest of the possible patterns. Two is obvious. But there was a group Of 14 and 18 and then 32. Every time I'd say: wrong, it couldn't exist, but then I'd come up with evidence that it did exist. Then I realized there was a ten-magnet group as well which completed the entire set of shells and subshells with the one exception: Six magnets (to represent the subshell from chemistry) wouldn't allow for alternating polarities. The nearest to six was this form with five. But then I constantly doubted the whole proposition, saying, as any reasonable person would: No, an electron can't move around the nucleus, off-center, like an astronaut circling just around the north pole. insane idea ...

H.vB.: ... unless something was holding it UP.

K.S.: OK, something else must be working. Then I read about de Broglie. The leap I take - and it is a leap - has to consider the obvious question: What would make it possible for the electron's orbit to move off-center like an astronaut circling only around the north pole? I say it can and does happen because the electron is committed to a marriage between its de Broglie wavelength - to lock that wavelength like a homing signal - into a quantized shell, that is, a nuclear electrical sphere. No matter then whether its path is around the equator. All it must do is maintain a keyed-in wavelength at whatever level it finds itself To move away from a great circle route, then, it needs to include fewer whole waves while keeping the same flight speed.

H.vB.: What's different about your thinking and de Broglie's thinking is that de Broglie still had very firmly in his mind a belief in classical mechanics. And classical mechanics makes the atom a ...

K.S.: Planetary system.

H.vB.: Yes, a rigid structure. De Broglie believed very firmly in equatorial planetary orbits, which have nothing to do with quantum mechanics or atoms, or anything like that. Then he superimposed the waves, but you're doing away with half of that underlying thinking and keeping the other half.

K.S.: Yes, I've been selective about what parts ... [laughter] I've tried to take the blind-men's elephant (from the folktale) and reconstruct it. One day I hope to find the right computer program to get me around the math (which I don't know) to tell me how much the electrons' magnetism contributes. I can't find anywhere in the books where orbital magnetism is believed to contribute to the electronic structure - only to spin magnetism. But one ought to look for a use in the atom for those strong orbital magnetic forces. it's that part of my - what I'll call the speculations of an artist - I've not had much success in talking to people about. The argument always is that orbital magnetism is so puny compared with the electrostatic repulsion.

H.vB.: You tried to get around that by saying that the electrostatics is sort of saturated by virtue of the nucleus being positive.

K.S.: Well, I see it as a good possibility that the electrons don't see one another as so repulsive in the atom as people assume since they're all swimming in a neutralized mush. I've been told that the magnets I'm interested in are ineffectual because, compared to the electrostatic forces, they are minuscule. I'm looking for a rational connection whereby ... Look, it's simply remarkable that those magnetic groups exist, and that only certain numbers of them exist. Intuition, romantic as it may be, says: "Nature wouldn't let this principle go to waste."

H.vB.: I would counter that by saying this is a very beautiful structure, but there are an infinite number of other structures that nature does waste.

K.S.: I've had some nice compliments over time and one of them was from a scientist who didn't believe in my atom at all. He said: "Just because it's beautiful doesn't make it right."

H.vB.: That's a flat-out contradiction to Dirac who said exactly the opposite: "If it's beautiful, it's got to be right." Yes, both the artist and the scientist seek beauty in their way. I see it more as a problem of process. As I said before, they both start exactly, identically, the same way. A crude example is Kekule's famous dream, but another example is the way Einstein thought. He really imagined shapes and things like that floating around. Then, it seen is to me, the next step is that the artist pursues one direction, which is visual expression of those realizations, and the physicist has to - and here comes that awful word - pursue a mathematical expression. That is absolutely necessary. And I know you've asked if that is the only way. Mathematics is the only way for physics.

K.S.: I understand that - that's not surprising to me. I think there's where I'm constantly at odds. For instance, my sculptures certainly have properties, many of them, which could be dealt with mathematically. I don't even know how many could be imagined. I wouldn't know how to calculate them, at any rate. But I can tell you my sculpture is likely to stand successfully because I've made lots of them and they do stand in many places. Tension is always in a contest with compression. The compression is pushing outward like the gas in a balloon, and the tension members surround it like the skin of a balloon. in this analogy, just like over-inflating a balloon, you could expand the compression system until finally a wire would break. Now, I confess I do take care, but any kid can blow up a balloon without knowing what mathematics may be involved.

H.vB.: OK, what you're saying is that mathematics is in everything. You're right. But the very precise question that the physicist asks after he makes the same model that you make is: What is the mathematical prediction I can make in the next experiment? And in your case, the next experiment for those atoms would be: What happens when 1, for example, shoot an electron into the thing? That's classically how we examine the structure of an atom, and that's where your model would fail and my model would not fail because I can tell that, for hydrogen in its ground state, there would not be any trace of a disk in the spray pattern. It would come out a nice sphere. It would come out beautifully spherical.

K.S.: Uh huh.

H.vB.: And then what do you say?

K.S.: I say there has to be something that works as a mechanical
principle to describe what goes on under the Schrödinger equation. The heart of that question has to do with how electrons actually move around in the atom - how, specifically, they interact with one another, not Merely statistically. The consensus at the 5th Solvay Congress - (the year of my birth, as it happened) was that mechanical models were to be abandoned. Maybe it's harsh to say in this way - but I have the feeling the questions which were being asked before 1926 and '27 were swept under the rug.

H.vB.: I agree with that, I completely agree with that. I think mechanical models are a wonderful thing, and an important thing and an instructive thing. I happen to think yours doesn't work very well, but I would love to see ... what you're talking about is what we call the physics. What's the physics of it? That's what we always ask. And in the case of the atom, I think your questions are right on the money. The atom is a structure. it holds up - it's the same question as you've asked in the macroscopic world. It's a structure. Structures usually (most structures we know of) are composed of two opposing forces - in the case of the balloon, you mentioned it. In the case of a star, it's gas pressure outward and gravity inward. in the case of your sculptures, it's tension and compression. What are the analogous balances and trade-offs inside an atom? Inward, there's certainly electrical attraction; outward, there's partly centrifugal force, and there's repulsion between electrons, and what else is there? One of these mysterious things is the Pauli exclusion principle ...

K.S.: ... and magnetism.

H.vB.: Magnetism, and so on. And then, based on that physics (not mathematics), one should be able to build a model. But that model then has to accord not only with the structural information but with such questions as: How do you excite the next state? Or a prediction: What happens when you shoot two atoms together? Or when you shoot an electron at an atom, at what angle will it come off? All that kind of stuff.

K.S.: The fact is, my model is an eclectic one ... de Broglie's model was a flat pancake model ...

H.vB.: Before you get into that ... Even now, without any waves at all, the electrons don't have to be in a pancake configuration: one can be above the equator and one below, so long as all the forces are in equilibrium. That's what's so beautiful about this kind of model - it's exactly like your sculptures, your structures. The forces though are explicit in that model: attraction to the nucleus, mutual repulsion which is electrical, and centrifugal force that you get whenever something is going around. And built into that is the tiny, tiny magnetism that is properly accounted for. But when you talk about your atom, you never talk about the centrifugal force, because you don't really have something going around the nucleus.

K.S.: I think there is a first principles phenomenon which is truly not known, unimagined, unrecognized, and there's no way to identify it. That doesn't mean it doesn't exist, however. Some other factor has to be at work for the electronic structure ... something that's not on the usual list. I think that Heisenberg* was quite right: We're lost when we try to see if there's a real orbit or not. Anything you use to look for it is going to wipe away the stuff you're trying to look at. But I think it's quite reasonable to imagine that it isn't a planetary system in a strict sense. The electron has a profession all its own and when it's around the nucleus, it must form a standing wave. It can form that standing wave only at a quantized energy level. Though its preference would be an equatorial orbit, it can't always find its way there because other electrons' orbits (according to my view of Pauli's principle) aren't allowed to be in the same piece of space at the same time. So when more than just a single electron is on a shell, then they've got to move over to make room. They're free to do that since it's not a simple planetary system - but a wave attaching itself to an energy surface.

H.vB.: OK, OK, you're essentially ignoring centrifugal force because you I re really rejecting orbits.

K.S.: I think it's not an orbit in that strict sense. I think it's a phenomenon that governs the electron and that's all that it can do, obey its own quantized rules. As I see it, if it can't form a standing wave on a shell, it isn't going to be in the atom. It's allowed in the atom only if it inhabits a standing wave, either equatorial, or small-circle and non-planetary - but always lying on a spherical electrical shell. H.vB.: Well, fair enough. I agree with you that we cannot follow those orbits so we shouldn't talk about the centrifugal force. The reason I was talking about it is because it turns out that these classical orbit calculations, the way I'm describing them with the centrifugal force, mock up the quantum mechanical calculations to a high degree of accuracy. The two questions are: What other a priori and unconventional kinds of forces do you have to build in, and, secondly, what are the predictions that your model makes? For example, one prediction your model of hydrogen makes is that the hydrogen atom is not spherically symmetrical. Is that part OK?

K.S.: No, no, no. No, no, no. I say the hydrogen atom is spherically symmetrical, that it has to do with gimbling ...

H.vB.: Precession.

K.S.: In other words, I don't think it turns out to be a flat pancake, although I think there is a circular motion. But it's a question of gimbling. You've got it moving like this, and the moment you have it moving in a second direction, you have a sphere. Now, considering how rapidly gas atoms move about in a chamber, it wouldn't be any marvel to find out that a hydrogen atom in a volume of gas seldom, if ever, appears as anything but a sphere.

H.vB.: It looks like a sphere, but you're not willing to admit that a ring is really an electron going around so fast that it looks like a ring. Why do you say in one dimension you want it to be fairly solid and you're allowing the other dimension to be taken care of by gimbling, but you're not willing to say: OK, it's only a particle going around so fast it looks like a ring.

K.S.: Oh, no. I do say that, but I say the particle's quite lost in there. But the difference, the big difference is that in its orbit the electron is constantly regenerating its de Broglie matter wave. In the gimbling motion, like flipping a coin, it's more analogous to an airplane propeller. Yes, the propeller does become a disk, in effect, just as the gimbling orbit becomes a sphere. But there's no standing wave involved in these cases. With the orbiting electron there is.

H.vB.: Fine.

K.S.: In the same way, each electron orbit fills space excluding its neighbors. It does what every hydrogen atom's electron is supposed to do, according to the standard explanation: It moves rapidly and carves out the space. The illusion of solidity starts with the individual electron's orbit.

H.vB.: Let's go to one of the things you said at the beginning: What about the spectrum? The spectrum of hydrogen was what got the whole shebang going. Do you worry about that at all when you're thinking about the structure?

K.S.: My hydrogen picture can't be too much different from the normal one. Mine has energy levels for transitions also. It offers the entire assortment. The excited states in my model portray the electron stretching to a new energy level with a wavelength that's correct for that higher shell. As with the original Bohr model, those changes relate to the distance from the nucleus - the atomic radius. So, presumably, I would end up with the same energy changes.

H.vB.: The same spectrum?

K.S.: The same spectrum. But, obviously, the spectral information is absolute. We're stuck with it. There's all the evidence, the hard evidence. And so whatever model you would make ...

H.vB.: That's true for a host of other data, too. Scattering information, for example. And that's the part that I find hard to believe, that even your helium model could agree in any way with the scattering evidence.

K.S.: I don't know [sigh]. I can't declare that it does and I can't assume it doesn't. There ought to be a single physical reality which is identifiable as a rabbit, or whatever. A rabbit is a rabbit, and any model you make of a rabbit would have to end up reflecting rabbitness.

H.vB.: What's the role in your mind of the intermediate thing, an abstract model of a rabbit? What's the role of the Schrödinger equation, which does describe the atom and which does in fact make exquisitely accurate predictions for a huge fat catalogue of observations? Did you just now admit that?

K.S.: Sure. Anything that's one-to-one with the experimental information, must, I guess, relate to the real atom. The pictures of the charge clouds that we're shown don't seem to me to be able to relate to the real atom.

H.vB.: No, but I'm talking about the intermediate thing now, the Schrödinger equation. No picture, just the mathematics.

K.S.: Well, I don't pretend to understand the Schrodinger equation. But from what I do understand, the Schrödinger equation has been able to give information about very simple atoms. The assumption is that it would be applicable to all atoms if we had big enough computers, and so forth. I understand, too, in regard to the charge cloud model, that the hydrogen model is the example and when other electrons are added, they are simply added together as if you added up hydrogen atoms. Am I correct?

H.vB.: Well ...

K.S.: So I think that hydrogen, as magnificently wonderful as it is, is awe inspiring and you'd think somehow that all atoms would imitate it, only with more complexity. But just don't believe it! I think there are systems which go to work as soon as you've got helium.

H.vB.: But back to your belief that there is some kind of an underlying model. Let's call it the rabbit - that it's really there. I believe that, too: There is an underlying reality. But what that rabbit looks like depends very much on how you look at it. Is it a rabbit as seen by x-rays, which is a very different thing from a rabbit seen with ordinary light, or a rabbit seen by neutrinos shining through, which is very ghostly and hardly seen at all?

K.S.: I understand what you're saying: It's true that it depends upon how you look at it.

H.vB.: And it would be very difficult for you to produce a model which is going to simultaneously produce all that information.

K.S.: Well, I know that. I would have to say right away that I'm going to have to set aside the neutrinos. I would figure most of all this is the kind of model one would find in a book of chemistry or crystallography - in other words, a picture of the atom which sits in the site and relates to the objects around it, the other atoms. The charge cloud model portrayed in chemistry books is also static. I know there are a lot of different ways of looking at the atom, that what I'm saying is simplistic. It's got to be. Of course, it can't possibly be the actual rabbit. But it might be the rabbit that would be projected on the wall if you cast a light on it and let it sit still - then saw its ears wiggling. I suppose that insofar as the chemistry book describes a three-dimensional object (not to my satisfaction), this is the kind of limited model I would have.
What use is my atom? I really don't know what use it is, and in fact the more it is useless, the more it is identifiable as art!

H.vB.: I think it is very constructive to place what you're doing between the Bohr model that we all ...

K.S.: Love and hate ...

H.vB.: Love and hate, and let's call it the Schrödinger equation, just as a shorthand - the mathematical description. I've said it several times: I think the idea of making those models is important, it's useful and not only pedagogically. When three-dimensional geometry gets so difficult, it's very helpful.
What I have to ask you (and I have to say this delicately!): I'd love to engage you to do what you're doing but just coming from the other direction - or, rather, why are you doing what you're doing from the direction you are doing it? You're not doing it from the direction of science: What can the scientists tell me from their many measurements that I can put into my model so as to make as accurate a reflection as possible to be helpful to them because I'm an artist and have a good understanding of stress and all that. No, you're not coming from that direction. You're coming from another direction of having invented a model (which is a fine thing to do) but pushing from the direction of the model up to the physics, rather than from the physics down to the model.

K.S.: I know, it's an ass-backward way of doing it but ...

H.vB.: I'm glad you said that! I don't understand.

K.S.: The reason I'm doing it is because I discovered the curious fact that magnets do what they do - circle magnets, current loop magnets. That provoked my interest. It's true, it's completely assbackward and I might be totally wrong. But I don't believe I'm wrong. If this idea has no merit, I'd at least appreciate the credit due me for having made a marvelous invention, totally artificial thoughts about the atom. To me there is enough of a correspondence. You say I'm not taking proper instruction from physicists ...

H.vB.: ... from the physical evidence ...

K.S.: OK, but the only people who give me that are physicists ... live human beings ... who write books. In fact, we've all read the same books. I was born just about the time the uncertainty principle was invented. I see what to me look like foot-faults, some mistakes. Since I'm not a scientist, yet I believe I've earned my spurs in the realm of structure, I see that the atom's set of forces could be described as I describe them-as a force-diagram in space that ought to be scale-independent, as far as I can tell. I think the atom as it's held together has got to be finally a force system in space.

H.vB.: I agree with that. I agree with that.

K.S.: So what I'm aiming for is a structural conception which must develop from one's imagination, but also, I confess, biased by this magnetic information, because I see as part of all of it a most extraordinarily flexible geometry. It's a universal Tinker-Toy for building spatial systems out of a hypothetical electronic structure.

H.vB.: OK. Let me say one thing you said earlier: It would be surprising or a pity if God had not made use of this nice structure and this nice geometry. And I completely agree with that. I can tell you, I would not at all be surprised someday to learn that these very structures exist in geology or chemistry or biology or at the molecular level. I would not be surprised. These are very wonderful structures and they have magical properties and they do balance in a very coherent fashion - stresses and strains, and so on. So I, too, would not be surprised if ...

K.S.: If nature didn't exploit them somehow.

H.vB.: Yes, the same way as Penrose's tiling patterns. He invented those and they were then found in nature (and it was very surprising because they violated everything that had been said until that time). But the one thing he did not do was he didn't pick the place where he said nature should repeat them. But you are picking the place!

K.S.: That's true. But that's what makes me an artist and not a scientist ...

H.vB.: Well, that's NOT what makes you an artist and not a scientist. But you are an artist. Just making those structures, and saying those are magnificent structures, and really understanding those structures in detail - that's all one thing. But then you put your finger down and say: "God, you'd better make atoms this way." Why aren't you saying, "God, you'd better make molecules this way, or AIDS viruses, or? . . ."

K.S.: Because I don't see any suggestion in the ultimate forms of those things. The reason I argue this case is because atoms are spherical. Each group of those magnets resides on a spherical surface, so that's one interesting relationship. An outstanding fact of the Bohr model was that the electron was in a stable state as long as it stayed a certain distance from the nucleus. To go up or down it had to do electrical work. This was an ideal principle - and it's still worthwhile to look for a geometry that would enable electrons to do that. And here is such a geometry. Primitive magnetic fields are caused by electrons in motion - rotating electrons - so I'm drawing on a geometry which lets them do that in some useful way. I'm taking the de Broglie wave idea literally and saying maybe there are real physical electron waves, not just waves of probability ...

H.vB.: But we've been doing that, too, for the last 20 or 30 years.

K.S.: Yes, I know. So my model corresponds to that. And then the spheres within spheres - energy levels; the Pauli exclusion principle causes electrons to fill what people call shells or energy levels. The electron waves, then, in order to be related to one another in that way would have to be individual items - atoms within the atom, if you will. They mustn't infuse through one another, as I see it. Electrons couldn't read one another as separate magnetic and electrical items unless they remain in individual domains - occupying different spaces - acting item to item. So there's another correspondence. For molecules and crystals, atoms need to link together in some geometrical fashion. Here's a structure that would make that eminently possible. So, I say, why not atoms? Why not atoms?

H.vB.: You left out one of your best pieces of ammunition yourself: your numbers.

K.S.: Ah, yes.

H.vB.: But I can't get too excited about the numbers because I would not be at all surprised if the mathematical structure of what you have here coincided with the mathematical structure to be found in real atoms. I would not be in the least bit surprised if that same underlying mathematical structure could be found in real atoms.

K.S.: I think that nobody knows whether an electron shell is a succession of the subshell groups or whether if, finally in a completed shell, they all collect into a single configuration.

H.vB.: What would you like to have happen with your atom? What's your fondest wish? I'm talking operationally. What would you like to have people do?

K.S.: Some of the most effective image makers today, icon makers, are doing computer graphics. With this elaborate new computer if I can produce a really astonishing animation of this model without voice-over, just visuals, so that people could say - ah, yes, now I understand how an atom works!

H.vB.: There's a leap there! There's a tremendous leap there, because as an artist you always try to persuade people. And when people look at your sculptures, they say: "Now, I can relate to this. I can feel this. I can kinesthetically climb into this." They say that, but you don't tell them what it is. The computer images you've shown us are already very beautiful and they can become more and more persuasive, but that's a totally different thing from saying also that this is what the atom is.

K.S.: No, Hans, it's presumptuous, quite obviously. But artists are presumptuous people. On the other hand, when people look at my sculptures of steel, I hope they are saying, "Ah, yes, now I know what that kind of structure is."

H.vB.: Well, Penrose was just doodling around from his intuition, his fascination, or whatever, and then came something which essentially overturned the established doctrine of crystallography: the dogma that there shall be no five-sided symmetry. I think there's absolutely nothing in principle for it to be impossible for finding structures, or even finding the right structures, for the atom.

K.S.: But you don't think this is it?

H.vB.: No, I don't. But I'll go further than I went before: I wouldn't be surprised to find these structures in chemistry or ...

K.S.: or radiolaria?

H.vB.: Oh, yes, you could find the most wonderful shapes in those little beasts. So if you had your druthers, you'd like to show a sexy computer animation of this model that's so persuasive people will say: "This is not only beautiful and logical - it is how atoms work."

K.S.: Well, sure. Ninety-nine percent of the population could draw a picture of an atom. They'd draw the old Bohr picture.

H.vB.: That's an icon.

K.S.: Right, that's why I'm talking about the computer. One fantasy is to change the icon. People would have a different icon.

H.vB.: But how would you present it? That brings us to the substrate: If you call it art, you're saying people aren't interested; if you call it science, the physicists are going to squawk at you.

K.S.: So I can't tell you it's going to help matters to invest several years of my life (which probably it will) to do this. But such is my interest.

H.vB.: But what do you care? Does it matter how it's accepted? You would like it to be seen, but ...

K.S.: Peculiar. I don't know where my interest started. A lot of people don't think there is any longer a riddle to the atom, that it's all known. Now, if there were not the familiar history of the Heisenberg affair and the debate with Bohr and Einstein and the changes which have lead people to believe that a visualization of the atom is out of the question, unnecessary, not wanted - then, I suppose I wouldn't pursue this with all this much energy.

H.vB.: Ferocity is the word!

K.S.: Ferocity, good! So, you say what is my fondest dream for this - it is built into that in some way, I'm sure.

H.vB.: Why are you stopping with the atom? Why aren't you asking the same thing about an electron?

K.S.: Because I somehow don't see a spatial order in the electron particle. Or you might ask why don't I start with the nucleus, for instance.

H.vB.: The nucleus is such a wonderful object.

K.S.: It is, but ... I don't understand it as a spatial object. I don't visualize it as a spatial object. Whereas the electronic architecture - connecting atom to atom, molecules, crystals, making these forms which are like my sculptures - pieces linked together - that I see as a spatial problem, a spatial riddle, a structural riddle.

H.vB.: You don't see the nucleus like that because you've been reading mostly atomic books? In nuclear physics books, the nucleus is so big ... big protons and neutrons, and ...

K.S.: [chuckle] There's no way to know how they order themselves in space. There's a reason for my casting about for structural principles. I think all things are made of items of one sort or another, items of form. When I see pictures of the charge cloud model with a set of six bonding orbitals, I ask: "Where are the rest of them?" You have to say: "They're in there, too, but of course we're only identifying these six for now." But I want to see the rest of them. I can't accept the notion that these six lobes, which look like individual items - two on the x-axis, two on the y-axis and two on the z - can occupy space while at the same time we know there are forty or sixty or however many electrons in there that are not in that picture. What people intuitively reach for in the charge cloud model are individual items, whether they are balloon-shaped, or whatever. A whole atom with just mush around it finally isn't aesthetically satisfying, at least it doesn't make a model builder happy. So what I'm looking for is one that satisfies my sense of structure, where separate parts come together.

H.vB.: The physicist would say that one difficulty about the inner thing is the indistinguishability of identical particles, which brings us back to the Pauli exclusion principle. But we have lots of evidence from all over physics that two electrons are indistinguishable. That's not true in our world: if you have two identical marbles, you can still distinguish them: right and left. They're identical but distinguishable. So when the electrons are inside there and you cannot distinguish them, that's so: You cannot distinguish them. If you give them labels, the wave function mixes the labels; the wave function is designed, is forced, is instructed to mix the labels in such a way that the wave function says to you: right-left or left-right.

K.S.: You can't tell one electron from the other.

H.vB.: Yes, in principle, you cannot say this is the right and this is the left. So your hope to identify the rungs of the ladder is kind of dashed by the insistence of the wave function, that you cannot tell the lines from one another. You can sort of tell the extremes because they lie outside the limits of the uncertainty principle, but you cannot - in principle - and that for us is the key (not just in practice because we're too dumb or too blind) but, in principle, you cannot tell that this is the upper rung and this is the lower rung.

K.S.: Nature might be able to tell. The electrons might be able to tell.

H.vB.: That might be so, but that desire to identify the pieces inside the atom runs counter to the experience we have from other places in physics. This indistinguishability principle really holds.

K.S.: Look, it's hard to doubt Heisenberg's principle insofar as it tells about the limits - the fringe of where we can no longer measure. And of course I wouldn't think of trying to identify each electron with a colored marker. What simply doesn't follow though is that the natural workings, the structural phenomena, are therefore inoperative below there. Maybe it's OK for physics, but it's unacceptable to me. I don't believe it's at all out of the question to listen to the noises coming from behind the curtain and to try to imagine what might make those sounds. But Schrödinger also was opposed to that other view, you know, despite the cornerstone equation with his name on it. He asked why it was that such speculations which were long considered OK in other disciplines - history, for example - were suddenly excluded from atomic physics in a single stroke with the Copenhagen view.

But anyway, as far as being unable to tell one electron from the next, I didn't mean that I want to identify the parts in a particular atom caught in a trap: I want to understand how the separate parts go together, the interlocking, the structural principle. I don't really have thoughts about distinguishing electrons in my model.

H.vB.: But don't you? Each one of those rings represents an electron. It is absolutely essential in this specific structure of yours that I can distinguish all these different rings from each other, otherwise I wouldn't have anything.

K.S.: In my sculptures" if I describe what those parts do in order for the piece to exist, I'll tell you that if I cut this wire, the whole thing would deform greatly. Therefore the wire is necessary. I know I need a minimum of three wires on any end of any stick; that's what's necessary. What the particular structure is doesn't really matter. It's a general rule.

I'm looking for the principles that might make up an electronic atom, the necessary ingredients. In order to keep it from collapsing, there has to be something better than a planetary system. If it were only the planetary orbiting principle causing the electrons to stay in position, they could easily be deflected, just as a Keplerian planet can be deflected. But it doesn't happen ... they are astonishingly stable. They keep their spherical form and reconstruct themselves if they're disturbed. So what kind of structure would do that? My sculptures won't. If you cut a wire you have to put a wire back.

H.vB.: It's not robust.

K.S.: Right. But In terms of the atom, I've tried to imagine a composite model to do all the things expected of an atom, or at least as many things as possible. If I've got some of it wrong, then I've got some wrong. But some of them seem right. All these metaphysical questions - really, metaphysical questions - about how to distinguish one electron from another in order to prove this or that - it doesn't really trouble me because it's a philosophical exercise more than anything ...

H.vB.: No, no, no. You can't get away with that! If I said to you this was a speculative philosophical argument, you would turn around and say: no, I really mean this to be a real atom.

K.S.: Oh, well. I do, but as far as proving it - of course I can't. All I can do is to propose a structure which seems to me likely to fill the bill. You say I don't have the answer. And maybe the Schrödinger wave equation isn't interested in the riddle of the mechanical structure of the atom. Anyway, I urge anyone else to come up with a different mechanical picture, especially one that reflects the atom's requirements better than mine. That would be interesting, for how many examples might there be ... the numbers and shells, the geometry, the magnetic aspects, and so forth? So I want to go on record - that's my fondest wish.

H.vB.: Well, you've done that with your patents.

K.S.: Yes, maybe there's more. I still haven't gotten anyone to agree.

H.vB.: On its own merit, without labeling it as an atom, it's a hell of an invention. I can go further: It would be very interesting to think about it as a visualization of some abstract variables that really do refer to real atoms. I still don't believe the literal interpretation you're putting to it in terms of magnetic fields. The objection that magnetic fields are weak compared to electrical forces is a very real objection. What I'm saying is that maybe if you picked a different variable for describing atoms - not charge, or mass, or position of the electron, but something altogether different -and you calculated the values of this variable from the Schrodinger equation, you might come up with something like your model. The trouble is, I don't know what the shells and halos mean. But I don't know what those different variables could be, either. It would be a fascinating exercise to ask that question ...

K.S.: How about asking your students?! [chuckle]

H.vB.: [clears throat]

K.S.: Hans, let's go have a drink upstairs.