Chapter 5

UCLA (1955-1985), Anharmonic Shaker, Analytical Symmetry Books, Grothendieck Correspondence & Biography, Geometry Competition

Overview
Recall that at Rockefeller, Nils Jernberg and I invented and built a programmable anharmonic shaker. New Brunswick Scientific took on its commercial production but abandoned their efforts after 7 years, having exploited some of its novelties in their own line of shakers but having been unable to dynamically balance it.

Thoughts concerning symmetry were not in my mind then, nor did I suspect that the curves traced by the movements of the shaker platform would be unknown, or that at my fingertips lay keys to vast, unexplored fields of geometry. The heart of the discovery was a simple transformation wherein the rotation of an initial curve (the shaker drive cam) about any point in its plane gave rise to another curve (traced out by any point on the shaker platform).

The above keys revealed fascinating new areas of geometrical beauty; established powerful analytical foundations for studying the symmetry of curves; greatly enlarged bases for classifying curves; led to new insights regarding relationships between curves and reference elements (coordinate systems), and the concept of symmetry; and opened up innumerable new fields of Euclidean geometry.

Though my curiosity was aroused initially, I did not carry out a thorough analysis, as there was no reason to believe that the curves or the transformation were unknown. One aspect that intrigued me ultimately proved to be of the greatest significance, namely, how the properties of the new curve obtained are influenced by the location of the point about which the initial curve is rotated. In time I came to realize that the symmetry of an initial curve about a specific point of rotation can be defined in terms of the properties of the new curves.

In the years following my discovery of the transformation I consulted with knowledgeable mathematicians, but the transformation was unknown. I also wrote to Martin Gardner, the famous math puzzle specialist who contributed a monthly column for Scientific American. He replied, in Aug., 1966:

I've delayed answering your letter.....because I wanted to mull over the possibility of devoting a column to that family of curves you have discovered. I'm still fascinated by the thing, but I believe it's just a bit over the borderline into an area that is too specialized and technical for most of my readers.

In 1976, I decided to use the transformation as an exercise in my mathematical modeling course. But first I undertook a detailed analysis. In my first half hour of setting down a few equations I was 'thunderstruck' by the transformation's significance. In my estimation, it was the long sought after 'most general' analytical tool for characterizing the properties of plane curves.

The simplest procedure to carry out the transformation is to take the equations of an initial curve in polar coordinates, letting r = x for the angle Θ and r = y for the angle (Θ+90°), and eliminate . This yields the 90°-circumpolar transform of the curve about the origin (the pole) in terms of the rectangular coordinates x and y. Using any angle , instead of 90°, yields the transform.

Early on, I discovered the fundamental property of traditional foci: A curve has greater circumpolar symmetry (lower degree transforms) about a traditional point focus than it has about neighboring points. It then became obvious that previous definitions of foci depended on relationships that merely accompanied this basic, previously unknown, property. From the point of view of symmetry, the traditional foci of curves were not their only foci. From that time on I had no doubt that the transforms of curves about points (poles) in the plane were previously unknown. Had they been known, they would have been common knowledge, because of their fundamental significance.

New findings repeatedly shifted my studies' emphasis. Eventually, I studied the most symmetrical curves relative to the reference elements of different coordinate systems, leading to the conclusion that they are the curves with the simplest and lowest degree equations. It then became mandatory to examine the literature. But even a cursory search revealed no hint of a connection between concepts of symmetry and coordinate reference elements. In fact, indices of most geometry texts do not even contain an entry of "symmetry."

In the later phases, I was much occupied with the significance of other elementary transformations of curves about points in the plane, in relation to the curves' symmetry and classification. One of these transformations, inversion about such points was most fascinating. [In cases where the initial curve "self-inverts," the new curve is identical to, and coincident with, it.]

At about this time my son, Chris, was taking a math course from Prof. Basil Gordon at UCLA. Gordon had a stellar reputation in the Mathematics Dept., regarded by all as a genius. Chris frequently was greatly impressed by Gordon's ability to answer any math question students posed. I suggested that he ask Gordon a question in the field of my symmetry studies. To make a long story short, at the next class meeting Chris asked Gordon, "what relationship does the Devil's Curve have to conic sections?" Well, of course, Gordon didn't have an answer. But no other mathematician would have had one, either.

Even today, no geometer could answer that question unless he had read one of my books. For background, the Devil's curve is famous, with several applications. But no one ever suspected that it could be derived by a simple transformation involving two best known conic sections. Even with all the new relationships between curves I was discovering daily, I was utterly amazed at the relationship.

After Chris told Gordon I was working in the field of symmetry, I joined Gordon in his office one afternoon and spent two hours introducing him to my transformation and its ramifications. Suffice it to say that, no sooner had I defined the transformation, than Gordon immediately grasped its significance and far-reaching implications. He congratulated me profusely and asked to be placed on my mailing list.

Later, Gordon characterized my first book on symmetry as, "one of the most original treatments of plane curves to appear in modern times. The author's new and deeper studies reveal a great number of beautiful and heretofore hidden properties of algebraic plane curves." Subsequently he also commented on my second book, as "replete with fascinating, provocative new findings; accompanied by a wealth of beautiful and instructive illustrations."

My first symmetry book, "Symmetry, an Analytical Treatment," was published in 1980. My second and third books followed shortly thereafter; "Curves and Symmetry" in 1982 and "Structural Equation Geometry" in 1983. Both my sons Warren and Christopher and my student, Roozbeh Sadeghiyan, gave valuable assistance in some proofs, derivations, and proofreading. Aaron White proofread all three books and my student Dr. Donald Perry assisted with the Introduction to the third book.

During this time I began corresponding with the legendary Alexander Grothendieck. Many mathematicians would agree that he was the greatest, most prolific, living mathematician, if not the greatest of all time, but more about him later. I had only a small window of opportunity for this. After attaining all honors in the field of mathematics, he withdrew from the 'rat race' in 1991, secluding himself in a remote hamlet in the Pyrenées. In Feb., 1982, I dared to send him my second book and a fairly detailed brochure covering the contents of my first book, saying that I would greatly appreciate his comments. He replied on Feb. 27 with words of encouragement and some questions.

I replied in detail on March 17. In the meantime he had had time to examine the material, responding on April 2,

Thanks a lot for your letter....The outline of your approach that you sent me this time seems to me quite adequate for giving an overall picture of your new approach to symmetry and of some of your findings, as well as of the seemingly inexhaustible wealth opening up for inquiry."

By the time I sent him a copy of my third book, in 1984, he had warmed up more to my approach and provided some outstanding evaluations, as well as advice and philosophical observations. Grothendieck's evaluations were to the effect that I had "discovered innumerable new worlds in that innocuous looking Euclidean plane," and that the book contained "a wealth of geometrical reflection and insight."

Inasmuch as my books followed relatively closely upon one another, comments on them sometimes pertained to the first two volumes, or even all three. Besides the comments of Gordon and Grothendieck, there were those of the physicist and mathematician, Richard Fowler of the U. of Oklahoma, He commented, concerning the first book, "provides sharp new tools for studying the properties of general algebraic curves." The review in the American Mathematical Monthly for 1981 stated:

.....striking new results on symmetry and classification of curves....read this book for more in symmetry than meets the eye.

Concerning my second book, Fowler commented:

....casts much new light on inversion and its generalization, the linear fractional (Möbius) transformation, with promise of increasing their utility by an order of magnitude.

Prof. Morris Newman of the UC Santa Barbara commented:

 .....represents tremendous amounts of new information.

A lengthy review in the German journal, Zamp, in 1984, concludes with:

....one is astonished by the forms that are obtained by the reported transformations.....whoever....delves into this voluminous work will be richly rewarded.

Prof. Newman also remarked,

Kavanau was born a century too late, for if the books had appeared 100 years earlier, they would have caused a sensation.

On the third book, in addition to the comments from Grothendieck, Dick Tata, in Mathematics Teaching, 1984, stated:

 ....turns the whole subject in a new and original direction....His claims demand and deserve attention....I thought the chapter on the bipolar system was stimulating and I began to agree with the author that this system could well be used as the one to which students are first introduced. It certainly offers a fruitful area for investigations of various sorts and at various levels....I would certainly recommend this book strongly as a stimulating source of ideas and highly original points of view....

A.S. Posamentier of CUNY, in Mathematics Teacher, 1984, stated:

....appropriate at the graduate level where a student can study the topic of analytic geometry in a concentrated fashion. The author takes great pains to make the book's development as intelligible as possible. He succeeds nicely, largely because of his fine command of the language. The book is well written and ought to be seriously considered by college professors interested in geometry for its own sake!

I feel there can be little doubt that the techniques and approaches of what I referred to as "structural equation geometry" will be found to have significant practical and heuristic applications. However, I've had little contact with such matters since the early 1980s, and can say little even about their use in the classroom. I contributed an article to New Scientist (March 14, 1985) titled, "Curves Cut through the Mathematical Jungle."

In 1981 my publisher Science Software Systems, Los Angeles, sponsored a geometry competition for a prize of $2,500 to further promote interest in the first book. The competition was to derive unpublished material from my studies. We received 3 meritorious entries. Although none solved the assigned problem, all were detailed, erudite, and accompanied by beautiful illustrations, particularly that of Prof. W. Wunderlich of Vienna. In the eventuality we declared a 3-way tie, and awarded $833.34 to each entry. Dr. J. B. Wilker of Cardiff subsequently published his entry in The American Mathematical Monthly, April, 1984.

End Overview

More about symmetry and the anharmonic shaker - 1955-1985

Recall that during my stay at Rockefeller, in 1955, Nils Jernberg and I invented and built a programmable laboratory shaker whose motion's amplitude and path could be controlled with interchangeable cams, and with a gearshift allowing either reciprocation or gyrorotation. As described in Chap. 2, New Brunswick Scientific took on the commercial production of this instrument but abandoned their efforts after 7 years when they were unable to balance it dynamically.

Thoughts concerning the symmetry of curves were not in my mind at that time, nor did I anticipate that the curves traced out by the movements of the shaker platform, using only simple circular and elliptical drive cams (or gears) would be unknown. The heart of the discovery was a very simple transformation wherein the rotation of one curve or figure (the shaker drive cam) about any internal (or even peripheral) point gave rise to another curve or figure (the movement traced out by the shaker platform).

Only when highly symmetrical curves are rotated about certain specific points-their highest ranking foci-are the new curves sometimes well known. When I say "highly symmetrical curves," I mean highly symmetrical in a new sense that can be defined in terms of simple equations. But even though defined in a new sense, these highly symmetrical curves include the curves or figures that always have been recognized in the past as being highly symmetrical-a circle, a square, an ellipse, etc.

Since I hardly was prepared to entertain the notion that unknown properties of the circle or ellipse would be disclosed through such a simple procedure, I little suspected that at my fingertips lay keys to vast, unexplored fields of geometry-keys that would: reveal fascinating new areas of geometrical beauty; establish powerful analytical (by "analytical" I means through the use of equations) foundations for the study of the symmetry of curves and surfaces; greatly enlarge the bases for the classification of curves; lead to new insights regarding the relationships between curves, and between reference elements (coordinate systems) and the concept of symmetry; lead to the discovery of, and ability to design at will, remarkable new curves; and open up innumerable new fields of euclidean geometry.

Though my curiosity was aroused by my initial observations of the new curves, which I call circumpolar intercept transforms, for reasons which will become apparent, I was not compelled to carry out a thorough analysis, as there was no reason to believe that the transforms or the transformation (generated by the cam rotation) that produced them was new, that is, unknown. The circumpolar intercept transformation is so simple and straightforward that I felt it would have been a logical object of investigation for the first ancient geometer to study the rotation of an off-center wheel. In fact, any student of geometry since classical Grecian times could have chanced upon and explored the potential of this transformation to generate curves with only the most primitive tools of the geometer.

From the time of its initial discovery, that is, the time when I first attached a pencil to the shaker platform and let its movements trace out a curve, I was intrigued by an aspect of the transformation that ultimately proved to be of the greatest significance. This was the matter of how the properties of the circumpolar intercept transform of an initial curve (in this case, the cam) are influenced by the location of the point about which the transformation is carried out, that is, the point about which the initial curve or cam is rotated. In time, I came to realize that the symmetry of an initial curve about a specific point of rotation could be characterized completely in terms of the properties of its transforms about that point, that is, in terms of the equations of the curves formed by rotating the initial curve about that point.

It will help to focus the reader's attention to repeat some simple examples of intercept transforms for a circular cam, as I did in Chap. 2 for both a circle and an ellipse (Figs. 2-4 & 2-5). If we rotate this cam about its center, then the two drive shafts, at 90° to each other, will not move, because the perimeter of the circle (or the circular channel in the cam) will not be displaced at all. The platform, driven by the drive shafts, will not move either, so the pencil will simply make a point. Now as the point of rotation is moved out from the center of the circle toward its circumference, the cam will 'wobble' more and more, and the platform will undergo greater and greater displacements (Fig. 2-4). When the point of rotation departs slightly from the center, an almost circular curve is traced. As the departure increases, the curve takes on an increasingly triangular appearing aspect until, at the greatest feasible distances from the center (close to the channel in which the ball bearing rides), the curve looks like a triangle with slightly curved side and rounded corners (vertices). Except for the 'first curve,' the point, which is a curve of zero degree (x = a; y = b), all of the curves traced out by the platform were previously unknown and are of 6th degree.

If it were physically possible to rotate the cam about a point on it circumference, and have the drive shafts displaced accordingly, the limiting curve traced out by the pencil on the platform would be a arc of a circle. Accordingly, we see that the center of the circle is its most symmetrical point, because the circumpolar intercept transform about it is only of zero degree. Any point on the circumference is the next most symmetrical point, because its circumpolar intercept transform is of 2nd degree, while any other point in the plane of the circle is next most symmetrical, as well as least symmetrical, because the degree of its transform is six, that is, all other curves are of 6th degree .

For further illustration, consider that the two drive shafts do not have to be perpendicular to each other, that is, at 90°, giving 90°-circumpolar intercept transforms. Suppose we align them to each other at an angle of only 30°. Then, the first curves seen look like ellipses and also are of 6th degree. As the point of rotation is moved out further and further, the 30° -intercept transformsstill look like ellipses, and are of 6th degree, but the vertex at one end becomes pinched and rounded (Fig. 2-4, curves for 30°). At the periphery, instead of getting a arc of a circle, as for 90° transforms, we get a segment of an ellipse. Hence we get the same answers for the symmetry of a circle about a point in its plane for both 30° and 90°-circumpolar intercept transforms-most symmetrical about the center (zero degree), next most symmetrical about any point on the circumference (2nd degree), least symmetrical about any other point in the plane (6th degree).

Contact with Martin Gardner

In the years following my discovery of the intercept transformation I consulted with knowledgeable colleagues-mathematicians or those with a mathematical bent-on various occasions, but the transformation was totally unknown to them. It was suggested that I write to Martin Gardner, the famous math puzzle specialist who wrote a monthly column for Scientific American. If anyone knew the answer, it would likely be he. My letter to Martin Gardner produced the following answer, dated August 10, 1966.

I've delayed answering your letter of July 20, partly because of pressure of work, and partly because I wanted to mull over the possibility of devoting a column to that family of curves you have discovered.

I'm still fascinated by the thing, but I believe it's just a bit over the borderline into an area that is too specialized and technical for most of my readers. The editors of SA always want me to be even less technical than I sometimes am, and although some readers know enough about graphing curves on the xy plane, I think most of my readers would know too little analytic geometry to get into the topic.

I may be wrong about this. But I'd prefer to pass it up as a topic for an entire column. I admit that the column in Piet Hein's superellipse got into analytic geometry of curves, but the superellipse was so tied up with other things-the new center in Stockholm, the balancing eggs, Columbus' balancing egg story, and so on-that readers who didn't understand the mathematics of the curve could still find the column interesting.

I begin an analysis of the intercept transformation

It was not until 12 more years had passed, when I decided to use the circumpolar intercept transformation as an exercise in a course I was giving in mathematical modeling, beginning in 1976, that I undertook a detailed analysis of it. In my first half hour of setting down a few equations I was 'thunderstruck' by the significance of what I had found. The intercept transformation was nothing less than the long sought after most general analytical tool for characterizing the properties of curves.

To clarify this assertion for the reader I now give a simple explanation of how to plot the circumpolar intercept transform of any curve (circumpolar, meaning about a pole or point). We could begin with any initial curve, whatsoever, in the xy plane, that is, an infinitely extending plane about two perpendicular intersecting axes, the x axis being taken as horizontal and the y axis being taken as vertical. Now we take any point in the same plane, no matter whether it be inside of the initial curve, on the curve, or outside of the curve, and draw an infinitely extending line through that point. For convenience, let the line be parallel to the x axis. Let us also pass a line through the point parallel to the y axis, that is, at 90° to the first line.

For simplicity, without losing any potential for generality, let's just say, for now, that our initial curve is plotted about the rectangular x and y axes in the xy plane, and includes the origin as an interior point. We lose no generality by doing this because our initial curve can always be shifted or plotted in such a way as to include the origin. Now, let us start rotating the initial curve counterclockwise about the origin. For each angle of rotation of the curve, let the point of its intersection with the positive x axis be called X, and with the positive y axis be called Y. Now, we simply plot successive pairs of values obtained for X and Y against each other on a new set of xy coordinate axes, yielding the 90°-circumpolar intercept transform of the curve about the origin (Fig. 5-1, for an initial ellipse and the first four pairs of values).

Consider that this procedure is the equivalent of taking the equations of the initial curve in polar coordinates, letting X = r for the angle α and Y = r for the angle (Θ+90°), and eliminating , to yield the 90°-circumpolar intercept transform in X and Y. With this realization, it becomes clear that any beginning college freshman, even some high school students, studying analytical geometry would have the potential to obtain the 90°-circumpolar intercept transform of any initial curve, for which he or she could obtain the equation in polar coordinates, plotted about any point in the plane. In fact, he or she also could obtain the equation for any angle, α, of transformation, that is, the α transform. In some cases, it might even be easier to accomplish the same derivation in rectangular coordinates.

In my early analyses, I first concentrated on the fascinating geometrical relationships to be found between common initial curves and their occasionally common, but usually unknown, transform curves (see Fig. 2-5). My first mapping of the exponential degrees of the transforms (meaning 90°-circumpolar intercept transform, unless otherwise stated) about points in the planes of conic sections led to a key finding. This was the discovery of the fundamental property of the traditional foci of curves:

A point focus of a curve is a point about which a curve has greater symmetry (lower degree circumpolar transforms) than it has about neighboring points

It then became obvious that the various previous definitions of the foci of curves depended on relationships of these points to the curves that merely accompanied this basic, previously unknown, property. Mapping the degrees of the transforms, that is, finding the degrees about all points in the plane, also revealed that, from the point of view of symmetry, the traditional foci of curves were not their only foci; while some of the other foci frequently were known, many of them were unsuspected. From that time on I had no doubt that circumpolar intercept transforms and their properties had been, until then, unknown. Had they been known, they would have been common knowledge, because of their fundamental properties.

These findings shifted the emphasis of my studies; I began to investigate the symmetry of well known curves about their foci, as revealed by the degrees of their circumpolar intercept transform equations. The approach was to select a highly symmetrical curve (judged by visual criteria), such as, having a center, the number of lines of symmetry, etc., and then locate its foci by mapping the degree of its transforms onto points in the plane considered as poles for the transformation (see Fig. 2-5).

Very shortly, even this goal was changed. The new emphasis was on synthesizing curves for which the highest ranking foci automatically would be known. But once I began this new program, the conclusion became inescapable that the concepts of the symmetry of a curve could not be separated from the elements of reference. A curve possessing the highest symmetry relative to a single point pole might not occupy the position of highest symmetry relative to an ensemble of two points, including the point in question, or relative to a point and a line, a point and a circle, two lines, two circles, etc.

Accordingly, the emphasis of my studies took yet another turn. I ceased studying exclusively the properties of the foci of curves, as revealed by their transforms about points, and took up the study of the most symmetrical curves relative to the reference elements of different coordinate systems, that is:

The most symmetrical curves relative to the reference elements of different coordinate systems are the curves with the simplest and lowest degree equations in the respective elements of the coordinate system.

Up to his juncture my studies had been carried out independently of the original literature of geometry. After all, if the transformation was unknown before I chanced upon it, I couldn't be missing much relevant material by not looking back into history. But from the time my perspective broadened, coming to realize that concepts of symmetry and coordinate reference elements are inseparably linked, it did become desirable to look into any pertinent original literature, including the studies of the great geometers of the past: Descartes, Newton, Chasles, Cayley, Moutard, Darboux, Crofton, Salmon, Williamson, Hilton, Yates, and Zwikker.

But even a cursory examination of the works of these men was sufficient to reveal that the literature contains no hint of interdependence between concepts of symmetry and coordinate reference elements. In fact most specialized geometry texts do not even contain and entry of "symmetry" in their index. Nor is there mention or discussion of a significant property of bipolar coordinates that was evident even from my preliminary studies, namely, that a great deal of information about both the foci and the symmetry of a curve generally can be obtained through mere inspection of its bipolar equations. This applies to a much lesser degree to the equations of curves in polar and rectangular coordinates.

Coordinate systems with more than two poles were mentioned only infrequently. There was little to be found relative to polar-linear constructions beyond the focus-directrix constructions for conics, for example, a point moving so as to maintain equal distances from a point-pole and a line-pole traces a parabola; if the distances are unequal, one obtains either ellipses or hyperbolas (Fig. 5-2). Only very infrequently was there mention of the polar-circular construction for central conics. The feasibility of basing a coordinate system on the reference elements for these known constructions did not receive serious consideration. Nor was there mention of linear-circular (Fig. 5-3) or bicircular coordinates. The only non-rectangular bilinear system treated was the "oblique" system, and that with non-orthogonal distances, that is, distances not measured at 90° from the axes, and exclusively within a directed-distance framework (that is, including negative distances).

There seemed to be little or no realization that the well known focus-directrix (Fig. 5-2) and focus-focus construction rules for conics (Fig. 5-4), and the lesser known focus-directive circle construction, are only highly exceptional cases among numerous undirected-distance (all distances positive) constructions that exist for these and many other curves relative to the same pairs of reference elements.

It is well known that it is possible to derive many rectangular equations for the same curve, each depending upon the curve's location and orientation. All these rectangular equations are of the same degree in the variables (x and y), for example, 2nd degree for parabolas, circles, ellipses, and hyperbolas. But in coordinate systems in which no more than one reference element consists of a line (but not necessarily including a line), different equations for the same curves, not only can have vastly different complexity, they also can be of different degrees in the variables. These differences are highly significant, because they relate directly to the concept of the symmetry of a curve relative to the coordinate reference elements.

In view of these observations and certain other impressions from the literature, I felt that there was little to be gained at that time from further study of the literature. My studies to that point already had provided sufficient insight and perspective to point the way both for a broader approach and in greater depth. Accordingly in the next months I concentrated on studies of the types of curves defined by various degree classes of equations in undirected-distance coordinate systems.

In the last phases of the studies at that time, I was much occupied with the significance of elementary transformations about a point, particularly inversion (Fig. 5-5, inversions of parabola about points in the plane), in relation to the symmetry and classification of curves.

[To invert a curve about a point at 0°, one draws many lines from the point to the curve and measures the distances from the point along the lines to the points of intersection with the curve. One then takes the inverse of a given distance (for example, the inverse of 4 is ¼) along a given line, and measures off that inverse value along the same line in the same direction. This establishes a point on the inverse curve. Doing this for every point of intersection traces out the inverse curve. If one were inverting at 90° or 180°, one would measure off the inverse value at a right angle to the line, or in the opposite direction along it, respectively. If the curve "self inverts," the new curve will be identical to and coincident with the initial curve (Fig. 5-6, self-inversion of an equilateral strophoid about its loop vertex). The term "self-inversion" also is used loosely to refer to an inversion locus which is merely congruent (or similar) to the initial curve. Strictly speaking this should be called a "congruent inversion" locus (see Appendix I).]

My inversion analyses were carried out independently of the methods employed by previous workers. For that reason, it's not surprising that my approach was different. Considerations of symmetry and questions relating to the classification of curves directed my concerns to very specific questions - framed analytically (that is, as equations) - about the relationships between the inverse curves of conic sections about points on their lines of symmetry (Fig. 5-5, the upper ensembles of curves), and the inverse curves of these inverse curves, etc. Accordingly, I was at least as interested in the properties of each of the curves in an inversion chain relative to one another and the initial conic, as in the properties of the transformation itself. On the other hand, in the classical work, the emphasis was chiefly on the most general properties of the transformation and on the circumstances under which self-inversion occurs, and in how many different ways.

I carried out a further search of the literature on the inversion transformation, and located three old works, the latest dating to 1902, that were by far the most extensive prior works on self-inversion in existence. But very few of my specific results concerning the inversion transformation and , particularly, the ability to organize curves through their relatedness by inversion, were to be found. The latter finding relates to the most beautiful and unique property of the inversion transformation, among all geometrical transformations, a property I call "immediate closure" (see Appendix I).

Discussions of symmetry with Basil Gordon

At about this time, early in 1980, my son, Chris, was taking a course in Multivariate Calculus from Basil Gordon at UCLA. Gordon had a stellar reputation in the Mathematics Department, regarded by all as a genius (recall I had been a teaching assistant and University Fellow in The Department in 1946-1949). Chris frequently was greatly impressed by Gordon's ability to answer any math question, any student posed. I suggested to Chris that we give him a question in the field of my analytical studies of symmetry. Neither Chris nor I remember the precise question he asked. However, it could have been any one of hundreds, even thousands, that were available. But it certainly would have been one phrased in conventional terms, readily comprehended by Chris, the other students, and Gordon, as suggested in the following.

To make a long story short, at the next class meeting Chris asked Gordon a question such as, what relationship the Devil's Curve had to conic sections, a topic treated at greater length below. Well, of course, Gordon didn't have an answer. But nor would have any other mathematician in the world. Even today, no geometer could answer that question unless he had read one of my books. For background, the Devil's curve (Fig.5-7) is a famous curve with several applications in various fields of mathematics. But no one ever suspected that it was closely related to, and could be derived by transforming, conic sections. Even with all the new relationships between curves I was discovering daily, I was utterly amazed when I discovered that relationship. More amazing, it is the result of a unique, one of a kind, transformation between conic sections. The derivation even can be used to find previously unknown families of Devil's curves.

After Chris told Gordon I was working in the field of symmetry, I phoned Gordon and arranged for a meeting to tell him about my work. Early one afternoon I joined Gordon in his office and stood at the blackboard for 2 hours introducing him to my circumpolar intercept transformation and all its known ramifications. Suffice it to say that, no sooner had I outlined the transformation, Gordon immediately grasped its significance and far-reaching implications. When I finished he congratulated me profusely and asked to be placed on my mailing list for future findings.
As it happens, at that time I was completing the first rough draft of my first symmetry book, Symmetry, An Analytical Treatment, 1980. As an afterthought, I decided to ask Gordon if he would be willing to give me a characterization of my studies for future use in promotional material for the book. He gladly did so, characterizing it as:

One of the most original treatments of plane curves to appear in modern times. The author's new and deeper studies reveal a great number of beautiful and heretofore hidden properties of algebraic plane curves. [His first suggested quote included "to appear in this century" but he later suggested the change to "modern times." I actually prefer the latter, since in the field of geometry, "modern times" would extend back into the latter years of the nineteenth century.]

Later, he also commented on my second symmetry book, Curves and Symmetry, Vol. 1, 1982, as:

Replete with fascinating provocative new findings; accompanied by a wealth of beautiful and instructive illustrations.

Paper on symmetry submitted to Science

With my first book out, my inclination was to submit a preliminary paper for journal publication. As usual, Science was my first choice. Accordingly, early in January 1981 I prepared a rather lengthy review of my most interesting findings to that point, Symmetry and the Analysis of Curves, and, as a long shot, sent it off to Science. The first two reviewers split, so a 3rd had to be called on. Unfortunately. it came out with only one strong "for" versus two weak "againsts," so the paper was rejected. The favorable review was from H. S. Coxeter, whom I had suggested as a referee, and whom many regard as the greatest classical geometer of the 20th century. The reviews follow.

Coxeter. The author deals effectively with a very interesting topic, the quantization of symmetries..(I had better say quantification, since the other has a peculiar meaning). He begins by showing that some of the classical construction rules for curves are themselves analytic in form, and may be used as a basis for classification. He then extends these construction expressions hierarchically to discover new and largely unsuspected symmetry relations which cast new light on old curves and to develop new "classical" curves.

These ideas definitely deserve publication in a widely read national journal. I could have wished that the article were much expanded rather than shortened. And because of this it may not be suitable for Science. In many places it reads like newspaper headlines merely announcing work done, with no clear indication of what was accomplished.

I found the last paragraph on p14 might be improved in clarity.

I thought that on p16 the author had missed the implied possibility of discussing the fact that the reference curves for symmetries are themselves part of the scheme, and so a hierarchy can be constructed.

On p17 I suspected that there must be a systematic approach possible for finding points of highest symmetry, but the text left me with a cut and try sensation of method. I refer to the last paragraph.

And, as remarked above, I would dearly like to have heard more about systematizing the use of the inversion transformation via a knowledge of its symmetry properties.

But nonetheless it is clearly a rich paper in an ancient field.

Reviewer 2.The author is developing analytic geometry in an interesting and unorthodox direction. Science would not seem the appropriate place to publish his results however, unless his concepts have some clear relevance outside pure mathematics.

Reviewer 3. The paper is technically sound but its importance is highly reader dependent. The paper lacks general appeal.

I would suggest the author shorten the paper by deleting some types of coordinate systems and kinds of symmetry and then submit the paper to a more specialized journal such as Mathematics Magazine or the American Mathematical Monthly.

First symmetry book

Symmetry, An Analytical Treatment was published in July, 1980. I introduce it with the first description I wrote about it, followed by material from the brochure written for it. This is followed largely by excerpting from the book's Preface and Foreword, which deal more with the nitty gritty of my work and its goals.

First description: SYMMETRY is a concept of common interest to artists, architects, engineers, and scientists, particularly mathematicians, crystallographers, and physicists. In fact, considerations of symmetry in a different context occupy a central position in particle physics, one of the most active research areas in "big physics." [The context is sufficiently different that Julian Schwinger, quantum electrodynamicist and co Nobelist in physics in 1965, and also a colleague at UCLA, never replied to my note asking if I might discuss my concept of symmetry with him (I had called one evening earlier but he was not home yet. His wife hoped my business with him would not interfere with their plans to go to a movie).]

This book presents the first and only existing analytical treatments of symmetry. It provides analytical definitions of symmetry, introduces the concept of the relativity of symmetry with respect to reference elements, and expresses various aspects of the symmetry of curves in analytical terms. These developments transcend the existing treatments of symmetry in terms of group theory, wherein symmetry is conceived solely within the narrowly circumscribed framework of rotations and reflections of figures that result in coincidences.

The contents of this work complement the existing subject matter of analytical geometry. Whereas the latter deals almost exclusively with the properties of curves and figures that are readily apparent to the eye, the material introduced in the book provides fascinating new insights into intrinsic and largely inconspicuous symmetry properties of curves--properties that provide broad foundations for comparative studies of relationships between curves. Knowledge of these relationships has made it possible to "custom design" curves with desired symmetry properties, a new field to which a full 50 pages of the book are devoted.

It is anticipated that the new findings will find their way into geometry courses at all levels. Their greatest impact probably will be in elementary courses, because these findings include much provocative material for restructuring the presentation of elementary geometry along lines more likely to stimulate and maintain student interest.

The book also opens possibilities for undergraduate mathematics students to make original contributions in the field of geometry, since the analytical treatments of symmetry have revealed numerous intriguing but almost totally unexplored avenues for geometrical investigations that do not require advanced mathematical training.

From the brochure: Is the circle the most symmetrical curve? If not, what curve is most symmetrical? Or can any curve be regarded as the most symmetrical? Can analytical comparisons be made between the symmetry of the parabola, ellipses, and hyperbolas? Do spirals have definable symmetry. What is the nature of the symmetry of a straight line? How is the symmetry of curves related to points, axes, and other characteristic loci? Can curves be ranked hierarchically according to their symmetry? Are there analytical criteria for classifying curves, as opposed to heretofore employed structural criteria--namely their projective properties and branches at infinity? Is the possession of two branches by hyperbolas and one branch by the parabola a fundamental property of these conics?

The answers to the above questions often are not simple or straightforward, and most questions have to be qualified precisely. Within the present framework of analytical geometry, answers to most of them would lack a firm theoretical foundation. The results and method of analysis introduced in this book make possible a rigorous analytical approach to these topics, as well as to numerous others, many of which lie outside the domain of current concepts in geometry. The material presented reveals fascinating new areas of geometrical beauty, establishes powerful analytical foundations for the study of curves and surfaces, greatly enlarges the bases for the analysis and classification of curves, leads to new insights regarding relationships between curves, and between reference elements and the concept of symmetry. It has resulted in the discovery of remarkable new curves and new relationships between curves, including extraordinary homologies between the circle and the equilateral hyperbola.

From the Preface: If this were a conventional work addressed only to mathematicians, I would shorten some of the treatments. There are two principal reasons why I have not done so. First, because I desire the book to be understandable to college students who have had a first course in analytical geometry. Although certain analyses require differential calculus, a background in it is not needed to follow the principal results. However, even in its present form, great dedication will be required for an undergraduate college student to comprehend the entire book. Symmetry analyses are not for the faint of heart. One sometimes works with equations of staggering complexity--in some cases containing scores to hundreds of terms. But equations that I regarded as being impractical to work with in the early stages of my studies were handled routinely in later stages. My assessment, then, of the degree of complexity that constituted impracticality rose several levels.

The 2nd reason for not abbreviating the treatment is that the topics of the book are by no means conventional. Although I expect the basic tenets to be accepted after due consideration, they are not directly intuitive. In the course of arriving at them, it was necessary for me to unburden myself of numerous misconceptions. Nor were errors uncommon. In fact, the self-checking feature provided by the requirement for dimensional balance of equations (that is, with all terms having the same dimensions = sums of exponents) seems essential in dealing by hand with equations of such complexity.

For these reasons I not only have not condensed the work, I have introduced a certain amount of redundancy and let stand some unintentionally repetitious material. I have preferred to leave it unmodified, in the belief that many readers will find it easier to follow in its present form. On the other hand, for mathematicians, only a few hints often suffice to lay bare the essential features of complicated relationships [as was the case with Profs. Gordon and Grothendieck (see below)]. To them, some parts of the work are bound to be excessively detailed and repetitious, and to them I apologize for not shortening some of the treatments.

I have been more concerned to explore the many and far-reaching geometrical applications of symmetry and inversion analyses than to construct a formal mathematical system. Consequently, I have presented certain results as Maxims, rather than theorems, and have supported many of them by heuristic discussions rather than formal proofs, though I see no obstacle, but time, that prevents these proofs.

It remains to explain why I wish the book to be understandable to college students. I believe that the findings of this work will lead to greatly increased interest in analytical geometry and, ultimately, in all areas of mathematics, particularly among high school and college students. Because of the new areas of geometry that have been disclosed, a situation has arisen in which it is possible for the first time in many years, perhaps since the time of the Ancients, for beginning students in mathematics to make significant original contributions to the discipline of geometry.

For example, the intriguing topic of curcumcuvilinear symmetry, that is, the symmetry of curves with respect to other curves (of which the Devil's curve is a special and unique example), rather than merely to points or lines, is one of vast extent, heretofore unexplored. While the derivation of circumcurvilinear transforms of curves of higher than 2nd degree involve complex calculations, those of quadratics (curves of 2nd degree) are readily accessible to beginning students. Aside from the circumconical symmetry of curves of 2nd degree (involved in the hypothetical question asked of Basil Gordon), only the circumcurvilinear symmetry of the parabola about the cissoid of Diocles is treated here (the cissoid of Diocles is the inversion of the parabola about its vertex).

In the fascinating area of inversion taxonomy (classification by inversion), there are hundreds of questions that remain unanswered even unexplored--merely for the lack of time. There are scores of curves of unknown affinity that remain to be explored systematically. For example, I do not know whether even such a well known curve as the folium of Descartes is related by inversion to any other previously known curve. And the taxonomy of other elementary point transformations (classification by point transformations) of conic sections is barely in its infancy.

Some of these areas of inquiry are within the competence of any analytical geometry student with a programmable calculator or computer. Of course, that was true of certain areas before the appearance of this work, but there was no prior appreciation of the significance of point transformations about all points in the plane for the classification of curves (although it goes without saying that a transformation of an initial curve always gives a curve related in some way to the initial curve) or of their importance for providing a systematic organizational framework for symmetry analyses.

Similarly, in the areas of undirected-distance (all distances positive) coordinate systems, my limited survey only whets the appetite for a systematic study of the comparative symmetry rank (ranking being by exponential degree) of curves in and between various coordinate systems. For this, only the bare rudiments of algebra and analytical geometry are required. With few exceptions, undirected-distance coordinate systems have been neglected by geometers of the past; they have several features that may account for this neglect. But some of these very featured that may have made them unattractive are at the heart of their exceptional utility for certain aspects of the study of symmetry (for example, in synthesizing curves for which the highest ranking foci automatically would be known, as mentioned earlier).

Almost all aspects of symmetry analyses using intercept transforms about points and lines are within the competence of students with a background in analytic geometry. Here, also, the field is wide open. Even just a characterization of the appearance of transforms about different point poles of an initial curve has yet to be accomplished. I have explored this matter only for conics, the cardioid, and the spiral of Archimedes. Aside from conics, conic axial vertex cubics (inversions of conics about axial vertices), limaçons, and Cartesians, no curve has received more than an exploratory circumpolar symmetry analysis, and even for the curves already studied, the gaps are many and wide. But this latter area is one that sometimes demands great fortitude and a calculator or computer that will compute zeros of functions (roots of equations).

Another fascinating area is that of non-focal self-inversion loci (loci = points in the plane or points on curves, that do not have circumpolar rank). Here, also, many curves remain to be discovered. At one and the same time, this is both the easiest and the most difficult of the areas I have mentioned. For deriving the plots of the loci, anyone with a ruler and dividing calculator could do the work. But for the equational analyses, the difficulty for initial curves of higher than 2nd degree are great. Some special cases, however, are certain to yield readily to solution.  Even with the experience gained from my studies, I could spend years working in any on of the 5 areas mentioned above and cover only a small fraction of the unknown ground.

A comforting aspect of carrying out basic symmetry analyses is that one is fortified with the knowledge that, no matter how formidable a derivation may appear to be at first sight, it very likely will work out beautifully in the final analysis. Time and again I have been awed at the beauty of the combinations, reductions, cancellations, and factoring that occurred in derivatives or eliminants of expressions that appeared, at face value, to be most intractable and unpromising. But because of the relatively high degree of symmetry that underlies almost all of the problems dealt with, one expects the final form of the equations to be relatively simple and esthetically pleasing.

Whenever a result of unexpected complexity was obtained, the likelihood of an error was great. The dogged checking and double checking of such results lies behind the achievement of many a difficult derivation--and errors were almost as often based on misconceptions as upon faulty manipulations. Once the likelihood of a procedural error has been eliminated, I repeatedly reconsidered the basic tenets until the misconception was discovered.

In fact, having corrected many misconceptions in the course of this work, it is hardly likely that none remains; I will be grateful to readers who detect and inform me of additional ones. The reader should not be surprised if he discovers relationships that I have overlooked or failed to mention. Virtually every time I took a fresh look at a topic, or merely in the course of retyping or preparing illustrations, I found relationships that had previously been overlooked, new paths that invited exploration, or shortcomings of the treatment that that warrant eventual revision. To have followed all these leads would have delayed the appearance of the book indefinitely.

Aside from classical analytical geometry, all of the material presented is the result of original derivations carried out independently of the literature. In cases where results subsequently were found to be known, appropriate references are made or the fact is otherwise acknowledged and, in fact, other of my results may exist somewhere in the literature in connections unrelated to symmetry or the classification of curves.

From the Foreword: Regarding implications of the treatments of symmetry in this book, it will be evident that the analytical approaches also can be applied in 3 dimensions, though this topic is not touched on here. But concrete assessments can be made of their implications within the field of plane geometry; in essence they greatly expand the theoretical foundations for the study of curves. This is achieved, moreover, by complementing, rather than by supplementing, conventional methods. To clarify this statement, attention is called to the fact that studies of the properties and classification of curves have heretofore been based heavily on overt structure, chiefly the "exceptional points" of curves (their double points, cusps, points of inflection, etc.), the projective properties of curves, and the branches of curves at infinity.

In other words, both classical and modern studies have been concerned almost exclusively with properties of curves that can be seen and measured directly or described in terms of branches at infinity. The basic tools of analytical geometry are applied to find such things as intersections, slopes, tangents, double tangents, normals, asymptotes, and inflection points, to calculate distances, angles, and areas, to find the bisectors of angles, the feet of perpendiculars, inscribed and circumscribed circles, etc.

As an example of the markedly different emphasis of analyses of the symmetry of curves presented in this book, I take the main topic of circumpolar symmetry, that is, the symmetry of curves about a point or pole. The heart of such an analysis consists of finding, ranking, and determining the properties, of the focal loci of curves, all of which objectives involve derivation of the intercept transforms of the curves about specific points or representative points of point arrays, that is, points on curves or at other defined locations. The basic feature of a circumpolar intercept transform is that it expresses the algebraic relation between the lengths of the two radii or lines that extend to the curve from a given point, and at a given angle to one another, as these radii rotate about the point and "sweep the curve" (see Fig. 5-1).

A circumpolar intercept transform gives a complete characterization of the symmetry of a curve about a point at a given angle. Transforms differ from slopes, tangents, etc., in that they only rarely can be "seen," that is, only very infrequently can they be perceived by mere inspection of the curve, as can slopes, tangents, etc. Correspondingly, the properties of the curves that they disclose usually are initially hidden.

As a specific example of a circumpolar intercept transform of a conic, I take the "harmonic mean" definition of the traditional foci of conics, known for well over 100 years, in its classical form. This definition is that "the harmonic mean [ twice the product divided by the sum, that is, 2XY/(X+Y)] between the lengths X and Y of the segments of a chord passing through the axis of a conic is constant and equal to of the length of the latus rectum (the chord passing vertically through the focus). In other words, 2XY/(X+Y) = (latus rectum length)/2 = a constant.

This 2nd degree equation, the harmonic mean relationship, is precisely the transform of a conic section about its traditional focus for a reference angle of 180°, that is, for the two lines, of length X and Y, extending from the traditional focus to the curve in opposite directions. In other words, the harmonic mean relationship characterizes the 180° symmetry of a conic about a traditional focus. But it is only one of many such characterizations (transforms), virtually all previously unknown, that define aspects of the symmetry of conics about different specific points and representative points of point arrays.

Even for the traditional focus, the 180° transform is only the most simple of several transforms, which is another way of saying that the symmetry of the curve about the traditional focus is greatest for a reference angle of 180°; three other transforms, for example, are those for reference angles of 0°, 90°, and between the radii, the latter being the α transform (the general transform for any angle, α). Transforms about the traditional foci for these other angles are of 2nd, 4th , and 4th-degree respectively, while the 180° transform about other points on the line of symmetry of the parabola (say, adjacent to the focus) is of 4th-degree.

Because the harmonic mean relationship has such a simple form, it happens to be one of the few non-trivial transforms that was discovered empirically, Once discovered, however, it was a signpost that pointed directly toward the core relations of the circumpolar symmetry of conics. Thus, given that this remarkable relationship exists for radii through the traditional focus at 180° to one another, it is natural to inquire as to the form of the corresponding relationships for other specific angles, and as to the general form of the relationship for any unspecified reference angle. The degrees of some of these relationships are given above. But the relations for the traditional focus are only the cases for a point on the x axis. What are the corresponding relations for any point on the x axis. And what are the relations for points on the other line of symmetry of hyperbolas and ellipses, for the vertices, the center, and the general point in the plane, with its location unspecified?

Most of the approaches to the study of curves and their interrelationships that are provided in this book previously were unknown. They fall into four principal categories: (1) relating to the symmetry properties of curves; (2) relating to the design and synthesis of highly symmetrical curves; (3) relating to the classification of curves; and (4) relating to undirected-distance coordinate systems.

In the first category are: (a) ranking of the symmetry of curves relative to two or more fixed reference elements; (b) ranking of the symmetry of curves about specific points (circumpolar symmetry), specific lines (circumlinear symmetry), and specific curves (circumcurvilinear symmetry); and (c) characterization of the symmetry of curves on the basis of the properties of their intercept transforms about their focal and sub-focal loci (points or curves that have focal status or less than focal status).

The second category deals with the manner in which knowledge of characteristic algebraic relationships for known curves of high symmetry can be employed in the synthesis of new highly symmetrical curves possessing specific symmetry properties. The third category includes: (a) new application of the inversion transformation about points in the plane in which the primary emphases are on its utility for the classification of curves and for providing a systematic organizational framework for circumpolar and circumlinear symmetry analyses; (b) classification of curves according to criteria of circumpolar symmetry; and (c) comparative distribution of 1st-generation tangent pedals and inversion loci of given groups of initial curves. Fourth category considerations relate to the equations and properties of highly symmetrical curves in undirected-distance coordinate systems.

I wish to thank my teen aged sons, Warren and Christopher, for aid with certain calculations and curve plotting. Among other things, Christopher assisted with the derivation of general equations for the normal pedals of conics, and Warren with those for the tangent pedals. I also thank my student, Michael Suzuki (in the mathematical modeling course), for assistance with two derivations.

Crossing paths with the legendary Alexander Grothendieck

By way of introduction, some observations of Allyn Jackson (Notices Amer.Math. Soc., 2004;51:1038, 1056, 1196, 1212) follow.

Grothendieck is a mathematician of immense sensitivity to things mathematical, of profound perception of the intricate and elegant lines of their architecture....changed the landscape of mathematics with a viewpoint that is "cosmically general"....left his deepest mark on algebraic geometry....had an extremely powerful, almost other worldly ability of abstraction that allowed him to see problems in a highly general context, and he used this ability with exquisite precision. Indeed, the trend toward increasing generality and abstraction, which can be seen across the whole field since the middle of the twentieth century, is due in no small part to Grothendieck's influence.

The work of Alexander Grothendieck has had a profound influence on modern mathematics and, more broadly, ranks among the most important advances in human knowledge during the twentieth century. The stature of Grothendieck can be compared to that of for example, Albert Einstein. Each of them opened revolutionary new perspectives that transformed the terrain of exploration, and each sought fundamental unifying connections among phenomena.

Some excerpts from Steve Landsburg's posting of Oct. 26, 2004 in Science Permalink follow:

Alexandre Grothendieck....was the greatest mathematician of the twentieth century and arguably the greatest of all time. Between 1958 and 1972 he reformulated the fundamental concepts of geometry ---concepts like point, space and covering....so completely that it is no longer possible to imagine what geometry would be about if Grothendieck had never lived....In this endeavor, he collaborated with several of the world's finest mathematicians who put their own research agendas on hold for the privilege of attending Grothendieck's daily seminars, fleshing out his ideas, and committing them to paper. The resulting documents, totaling over 10,000 pages, revolutionized geometry, arithmetic and algebra by viewing all of mathematics from a height of abstraction at which subjects blend together, every unnecessary detail is stripped away, and essential truths are almost automatically revealed.....This is, after all, a compelling story. Its hero is a brilliant eccentric described by everyone who's known him as a man of indescribable charisma. (It was this legendary charisma, no less than the brilliance and clarity of the Grothendieck vision that lured so many first-rate mathematicians away from their own research for the sake of the grand collaboration.)....

Many mathematicians would agree that Alexander Grothendieck was the greatest, most prolific, and most influential living mathematician. I had the great good fortune to correspond with him in the period, Feb., 1982 to March, 1984, exchanging 5 or 6 letters, each, about my studies in geometry. Inasmuch as his assessments of my books were highly favorable, and he stands almost alone in his fields, I devote some space to his biography and also quote from some of his letters

Grothendieck was born on March 28, 1928 in Berlin. He entered Montpellier University, France, one of the most backward in the teaching of mathematics, in 1941. There, he was mostly self taught. After graduating he spent the year 1948-1949 at the Ecole Normale Suprieure in Paris attending in the legendary seminar conducted by the great algebraist, Henri Cartan, the theme being simplicial algebraic topology and sheaf theory--his first real contact with the world of mathematical research.

Thence, on the advice of Cartan and André Weill, he went to the U. of Nancy, a more suitable milieu for his developing interest in topological vector spaces, to study with Laurent Schwartz and Jean Dieudonné. To their astonishment he solved a series of problems in locally compact spaces, any one of which could have qualified for his dissertation.  He became one of the Bourbaki Group, along with Weill, Cartan, Chevalley, Delsarte, and Dieudonné. His is a classical example of a student far exceeding his professors. In this connection, the Bourbaki were proponents of formal, algebraic mathematics devoid of geometrical reasoning. They made it a point of honor to publish no diagram in their books and papers

After obtaining his doctorate he spent two years as a visiting professor at the U. of Sao Paulo and 1955 at U. of Kansas where he began to immerse himself in homological algebra. He left there in 1956, returning to the Centre National de la Recherche Scientifique. Afterward, he accepted a chair at the Institut des Haute Etudes Scientifiques, of which he was a founding professor. He shortly developed his version of the Riemann-Roche theorem, characterized by Armand Borel as "a fantastic theorem....really a masterpiece of mathematics." His unique capabilities are best conveyed by excerpts from his students and other contemporaries.

Grothendieck over and over again completely transformed what people thought a subject was about (Nicholas Katz of Princeton U.).

He was not only solving outstanding problems but reworking the very questions they posed. His creation of a new viewpoint on mathematics is his greatest legacy (Allyn Jackson).

He would aim at finding and creating the home which was the problem's natural habitat. (Pierre Deligne, the most brilliant of all his students, and the one with whom he had the closest mathematical affinity).

Grothendieck had....this ability to make an absolutely startling leap into something an order of magnitude more abstract (David Mumford of Harvard U.).

Before Grothendieck I don't think people really believed you could do that. It was too radical. No one had had the courage to even think this may be the way to work, to work in complete generality. That was quite remarkable....another remarkable thing, he had complete control of the field, which was not inhabited by slouches, for a period of about 12 years (Michael Artin).

Grothendieck had a real talent for matching people with open problems. He would size you up and pose a problem that would be just the thing to illuminate the world for you. It's a mode of perception that's quite wonderful and rare (Barry Mazur).

His technical superiority was crushing (René Thom).

I had had in Paris some of the great mathematicians of the day, from Schwartz to Cartan, but Grothendieck was completely different, an extra terrestrial. Rather than translating things into another language, he thought and spoke directly in the language of structural mathematics, to whose creation he had contributed greatly....It was fascinating to work with a genius. I don't like the word but for Grothendieck there is no other word possible....It was fascinating but it was also frightening, because the man was not ordinary....the greatest memories of my life as a mathematician (Yves Ladegaillerie).

Grothendieck's Esquisse d'un Programme of 1984 was completely different. It was a wild expression of mathematical imagination. I loved it. I was bowled over, and wanted to start work on it right away. Some of it doesn't even seem to make sense at first but then you work for two years, and you go back and look at it, and you say, 'He knew this' (Leila Schneps of the U. de Paris)

It is no exaggeration to speak of his 1959-1970 years at the IHES as the 'Golden Age'. During this period a whole new school of mathematics flourished under his charismatic chairmanship, establishing the IHES as a world class center of algebraic geometry, with him as its driving force. In those glory years it is said that he was a babbling font of mathematical creativity, throwing off startling insights, deep conjectures, and brilliant results nonstop. It was said that he had so many ideas that he essentially kept all the serious people working in algebraic geometry in the world busy during that time.

He was awarded the Fields Medal, the Nobel of mathematics in 1966 but would not travel to Moscow to attend the International Congress of Mathematicians to receive it because he objected to the militaristic policies of the Soviet Union. Similarly, in 1988, at the age of 60, though awarded the Craaford Prize of $200,000 by the Royal Swedish Academy of Sciences he declined it, railing against the dismal ethical standards of mathematicians and scientists.

During this period at IHES Grothendieck's work provided unifying themes in geometry, number theory, topology, and complex analysis. He introduced the theory of schemes in the 1960s, which allowed certain of Weill's number theory conjectures to be solved. He worked on the theory of topoi, which are highly relevant to mathematical logic. He provided an algebraic definition of the fundamental group of a curve.

The mere enumeration of Grothendieck's best known contributions is overwhelming. They include topological tensor products and nuclear spaces, sheaf cohomology as derived functors, schemes, K theory (the Grothendieck Groups and Rings), and Grothendieck Riemann Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck topologies (sites) and topoi, derived categories, formalisms of local and global duality (the 'six operations'), étale cohomology and the cohomological interpretation of L functions, crystalline cohomology, 'standard conjectures', revolutionized homological algebra in his celebrated "tohoku" paper, motives and the 'yoga of weights', tensor categories, and the motivic Galois groups. It is said to be difficult to imagine that they all sprang from a single mind, and that those who understand the mathematics of his Golden Years speak of it with awe and wonder.

Beginning in 1969 Grothendieck came in conflict with the founder and director of IHWS over military funding of the Institute. As a result he resigned in 1970, abandoning mathematics as his chief focus. In 1970-1972 he held a Visiting Professorship at the Collège de France, then a similar appointment at the U. de Paris at Orsay for 1972-1973. After that he became professeur a titre personnel, which is attached to a single individual and can be taken to any university in France.

 Grothendieck took his title to the U. of Montpellier, where he remained until his retirement in 1988. There he had a regular faculty position, taught at all levels, and poured a great deal of energy, enthusiasm, and patience into his teaching. He did not have a seminar that met consistently but formed a small working group that, according to Ladegaillerir, never really got off the ground. He took leave during 1984-1988 to direct research at the Centre National de la Recherche Scientifique. He retired at age 60 in 1988, in which year, as noted above, he declined the Crafoord prize. He felt that the work, itself, was its own reward, cheapened by awarding prizes in competition. During his years at Montpellier he wrote his autobiography, Récoltes et Semailles (Reapings and Sowings), which was never published but widely circulated in manuscript form (see, also, <www.grothendieck circle.org> and many web sites).

Needless to say, I was awed by his resumé when, on Feb. 10, 1982, I dared to send him my second book, Curves and Symmetry and a fairly detailed brochure covering the contents of my first book (together with figures), saying that I would greatly appreciate his comments on the works. He replied on Feb. 27 with words of encouragement and some questions. I replied in detail on March 17. In the meantime he had had time to examine the material, responding on April 2 that:

Thanks a lot for your letter and your candid description of your current practical motivations. The outline of your approach that you sent me this time seems to me quite adequate for giving an overall picture of your new approach to symmetry and of some of your findings, as well as of the seemingly inexhaustible wealth opening up for inquiry (emphasis added). I am not surprised however to hear about your difficulty for finding audience and recognition among a reasonably wide circle of the so called "mathematical community".

One point surely is that there is among mathematicians, as well as in any other type of congregation, an in built inertia against the appearance of any substantially (or even radically) new approach to things, even against so childishly simple and natural steps such as introducing the 0 cipher for use in [illegible], or groups of symmetries of figures, or thinking in terms of sets, or accepting the empty set, or negative numbers and the like. Such resistance is of course the greater, when the person proposing some new approach is not (or not yet) widely known and backed by prestige won through endeavours along more traditional lines.

This however is not the only point. Another one, equally important I believe, lies in the fact that, for anyone (like yourself) for whom "the ice was broken" in his relationship to mathematics, namely who has come to like looking into "mathematical" relationships between "mathematical" objects, it becomes gradually clear that wherever our eye chooses to start looking somewhat closely at things, from whatever angle we choose, there opens up a world of mystery that calls out for being dived into, seems to get ever deeper and richer and ever more inexhaustible as we are yielding to the call (This by the way, you surely do know yourself is by no means something restricted to mathematics or to scientific curiosity alone...),

This implies that every "genuine mathematician" is as much absorbed by his own endeavours, as you are absorbed by your interest for the aspect of "structural geometry of curves" you discovered a few years ago. Of course he shares most of his mathematical interests with a circle of colleagues, more or less wide or narrow but always quite limited. In any case not only through in built inertia and "blocks" against innovation, but also through evident limitation in available energy, everyone has to make drastic choices (mostly implicit) as to what he is willing to devote any attention to.

If he is backed by prestige, what he devotes attention to will be deemed worthwhile of attention by a corresponding more or less wide and more or less passive "audience" of colleagues, students, etc. In any case, he'll have plenty of pleasure and excitement discovering things and doing the work that goes with it. The pleasure and joy is something independent of recognition, by its essence - but I noticed it often wears out with the years through the vanity that goes with recognition (and sometimes, too, through the frustration that goes with lack of recognition).

To come back to the "practical matter", your work--with the entire emphasis of metrical properties of curves and figures--is very far from my own main themes of interest. Therefore I am afraid I have no substantial comments to offer on the mathematics you have been engaging in - I can but acknowledge the impression of imaginativity and momentum. Nor am I in a good position, for the same reason, for making a meaningful recommendation to a mathematical journal, all the more as I have more or less lost contact with the world of journals, meetings and the like for the last ten or twelve years.

I am sorry I am not being of any help really, and wish you every success in both your practical as in your scientific endeavours. Yours very sincerely,

Grothendieck was said to be very generous in the willingness with which he shared his ideas with colleagues and students. Certainly, he was generous with his time with me. Others of his letters also were lengthy. By the time I sent him a copy of my 3rd geometry book, in 1984, he had warmed up more to my approach and provided some outstanding evaluations. I replied on April 16, among other things asking if he would allow me to quote his favorable comment, above. The answer, on May 3, together with a discussion of other matters, was in the affirmative.

Inasmuch as my books followed relatively closely upon one another - July, 1980, January 1982, and December, 1983, comments on them sometimes pertained to the first two volumes or even all three. Besides the comments of Gordon and Grothendieck on the 1st book, there were those of the physicist and mathematician, Richard Fowler (with whom I worked on a voltage stabilizer at war's beginning; see Chap. 1), who was visiting Los Angeles at the time, and the reviewer for the American Mathematical Monthly (suggested as a better medium by the 3rd reviewer of my paper submitted to Science), as follows.

Striking new results on symmetry and classification of curves....Read this book for more in symmetry than meets the eye.

American Mathematical Monthly, 1981

Provides sharp new tools for studying the properties of general algebraic curves.

Professor Richard Fowler

Second symmetry book

My second book on symmetry, Curves and Symmetry, Vol. 1, appeared in January, 1982. I quote excerpts from its Preface.

The present work is the first of two volumes to illustrate and elaborate upon the topics of Symmetry, An Analytical Treatment (Kavanau, 1980) in which only the first five of its chapters were accompanied by figures. This volume illustrates and amplifies the material of Chaps. VI-VIII. Among the topics treated are: circumpolar intercept transforms of conic sections; inversions of quadratics (2nd degree curves), cubics, quartics, quintics, and sextics; central Cartesians and their inversions--including serpentine circular cubics, bicircular quartics, and corresponding monovular representatives; curves of demarcation; non-focal self-inversion loci; some general and specific mapping properties of the inversion transformation; and Inversion Taxonomy--the classification of curves on the basis of their relatedness by inversion.

The emphasis of the contents upon inversion reflects the fact that the inversion transformation occupies a unique position in relation to classifying curves and organizing curves for systematic symmetry analyses. This is a consequence of its recently discovered fundamental property of immediate closure (Kavanau, 1980). [Since this property is discussed in more detail in Appendix I, I do not treat it here. JLK]

Although the inversion transformation was discovered independently several times in the early years of the 19th century and has been studied intensively for over 140 years, its most fertile and intriguing properties, namely, the modes of mapping equivalent inversion poles (points in the plane about which inversions yield similar curves) have heretofore escaped notice. The oversight would appear to be related to the fact that inversions of lines and circles, and ensembles thereof (and planes and the surfaces of spheres in the solid domain) are the most readily visualized of all inversion relationships and lead to very useful theorems. Because of this utility in geometrical proofs and constructions, they have dominated completely almost all previous treatments of inversion. But if one deals exclusively with the inverse loci of lines and circles (which invert solely into one another), one encounters solely lines and circles, which are of practically no heuristic value in relation to inversion mapping properties.

Inverse loci of other curves are mentioned and tabulated repeatedly in the classical literature, but these are few in number and consist almost entirely of inversions about exceptional points on the lines of symmetry of well known curves, and about the poles of spirals. In the few instances in which inverse loci about non-axial points of curves were studied in depth, the chief concern was with such topics as the relationships between points of contact or intersection of tangents and bitangents (for example, whether these points lay on a conic), the envelopes of circles related in a certain manner, etc., rather than with questions related to the fundamental and much more fruitful topic of the mapping properties of the transformation, itself.

If one deviates from the conventional approaches mentioned above, one is confronted forthwith with a plethora of provocative questions. Consequently, my original intention to present all the illustrations for Chaps. VI-XIV in a single volume was short lived. I had barely begun the figures for Chap. VI, when basic questions concerning the inversion transformation intruded. Answering these questions led to the discovery of the fundamental mapping properties of equivalent inversion poles, which comprise universal and remarkably simple relationships. In keeping with the unique suitability of the transformation for classifying and organizing curves, the answer to virtually every inverse mapping query is one of elegant simplicity. For example:

Question....what is the inverse locus of a given curve about a point on the ideal line at infinity?

Answer....a reflection of itself in a line.

Question....what locus is obtained by inverting an inverse curve about a given point in its plane?

Answer....the same locus obtained by inverting the initial curve about the point inverse to the given point.

A notable consequence of the discovery of the above mentioned mapping properties of the inversion transformation relates to the continuum of ideal points of the line at infinity. One cannot regard these points as equivalent to one another from the point of view of inversion geometry, and unify them under the designation "the point at infinity." Not only do the inverse loci of a curve about different points of the ideal line at infinity differ in their orientation, the points of this line also may differ in their circumpolar focal status. Thus, each ideal point of the line at infinity in a direction at 90° to a line of symmetry of a given initial curve is a focus of self-inversion, whereas the other points are merely poles of congruent inversion (see below).

In the same vein, inversion mapping identifies the line at infinity as an asymptote of certain curves, whereas classical approaches generally exclude this line from consideration. The asymptote of an inverse curve obtain by inverting a given initial curve about an incident point is the inverse of the circle of curvature of the initial curve at that point. As the curvature of such circles becomes infinite, the corresponding asymptotes approach the line at infinity. Thus, from the point of view of inversion geometry, there is no more reason to exclude the ideal line at infinity as an asymptote of an inverse curve than to exclude a point of infinite curvature from an initial curve.

Perhaps the most significant new property to emerge from inversion mapping studies is that of congruent inversion, that is, the existence of previously unknown poles (and inversions) about which certain curves invert to congruent loci differing from the initial curves only in their orientations and locations. Non-axial inverse loci of central conics, for example, possess one pole of congruent inversion in the finite plane, while the corresponding inverse loci of biovular (possessing 2 ovals) central Cartesians possess 3 such poles. Thus, foci of self-inversion can be regarded merely as special cases of poles of congruent inversion; about the former poles the inverse loci are both congruent and coincident, as opposed to being merely congruent about the latter poles.

The term identical subspecific inversion denotes both self-inversion and congruent inversion (any similar inverse locus qualifies as a congruent inversion, since the unit of linear dimension is at one's disposal, and this determines the size of the inverse locus). The inverse of a curve with 4 lines of symmetry may have as many as 15 poles of identical subspecific inversion, not more than 6 of which can be foci of self-inversion. On the other hand, the inverse loci of some periodic functions have an infinite number of poles of identical subspecific inversion, including infinite numbers of both poles of congruent inversion and foci of self-inversion; the inverse loci of other periodic functions have no focus of self-inversion, but an infinite number of poles of congruent inversion.

Since topics related to inversion are dealt with in both Chaps. VI and VIII of Symmetric, An Analytical Treatment, the length of this book was much increased on that account. A 2nd principal factor that led to its expansion was the treatment of non-focal self-inversion loci in Chap. VIII. In the original text only the non-focal self-inversion loci of conic sections were derived and described. A very limited extension of the treatment to limaçons (inversions of conic sections about their traditional foci) and parabolic Cartesians revealed so rich and compelling a new  area, that the 15 pages of figures devoted to this topic amount to no more than an exploratory treatment.

Both the topic of non-focal self-inversion loci and the topic of curves of demarcation of Chap. VIII led to considerations of Cartesian self-inverters (including curves known classically as Cartesians). When the affinities of these curves are viewed in the perspective of Inversion Taxonomy, one is led to a consideration of the central initial curves, Central Cartesians. The latter loci possess the distinction of being the 4th-degree equivalents of central conics outside the quadratic based inversion superfamily (QBI superfamily; where QBI curves = quadratics and their inversion loci). In fact, central Cartesians, central conics (ellipses and hyperbolas), and central quartics (4th-degree curves) the inverse loci of central conics about their centers--are the only central curves whose point in the plane inverse loci have rectangular equations that do not exceed a degree of 4. Thus, these 3 groups are classified together as central curves with 4th-degree maximal inverse loci.

The fact, alone, that central Cartesians are the 4th-degree non-QBI equivalents of central conics would have justified analyses of their circumpolar symmetry and inversion properties. When it also became evident that the intriguing geometrical properties of central conics and their inverse loci (Fig. 5-8) are at least matched in interest by corresponding properties of central Cartesians and their inversion derivatives, extensive further lengthening of the projected treatment was inevitable; 19 pages of illustrations for Chap. VIII are devoted to the latter curves.

I am indebted to my son Christopher for carrying out a number of derivations for me and for developing a useful parametrization technique for plotting inverse loci and intercept transforms of high rectangular degree. Christopher also was the first to prove Maxim 6-7-5 and he originated a fruitful method for locating the asymptotes of inverse curves. Thanks also are due to my student in mathematical modeling, Roozbeh Sadeghiyan, for elegant geometrical proofs of Maxims 6-4-2, 6-4-4, and 6-7-5 by procedures that also can be employed to prove many other inverse mapping Maxims, and for calling my attention to the inversions of periodic functions with an infinite number of (parallel) lines of symmetry.

My friend Joseph Bechely, the bridge expert, was engaged in graduate studies with Professor Morris Newman in the Mathematics Department at UC Santa Barbara in 1982. I gave him copies of the first two symmetry books for Professor Newman. The latter commented that I was born a century too late, for if the books had appeared 100 years earlier, they would have caused a sensation. Additional comments and quotes about the second book, or both books, were as follows:

....represent tremendous amounts of new information.

Professor Morris Newman, Mathematics, UC Santa Barbara

Casts much new light on inversion and its generalization, the linear fractional (Möbius) transformation, with promise of increasing their utility by an order of magnitude.

Professor Richard Fowler, Physics, U. of Oklahoma

Replete with fascinating, provocative new findings....accompanied by a wealth of beautiful & instructive illustrations.

Professor Basil Gordon, Mathematics, UCLA

In recent years, the symmetry content of geometrical forms has fascinated a wider audience. These two works, comprising 1,000 pages between them, bear witness to this. To begin with, the exposition on page I of the first volume is recommended. There, the author describes how he was led into a systematic study of generated symmetrical figures through the observation of curves obtained experimentally. In connection therewith, he can lean on only a few predecessors, among others, G. Salmon, A Treatise on Conic Sections, 1848/1879, and on H. Hilton, Plane Algebraic Curves, 1920.

We can only report topicwise the contents of the first volume: general concepts of symmetry, particularly with respect to a line and a point, Inversion of conic sections and curves of the 3rd and 4th-degree, circumpolar symmetry, etc.

In addition to much new material the second volume gives the figures for Chaps. VI-VIII that are lacking in the first one. One is astonished by the forms that are obtained by the reported transformations. The curves mentioned above serve as initial curves that are transformed by inversion about appropriately selected points. Furthermore, for example, the folium of Descartes transforms to the most bizarre curves.

Whosoever has the time to plunge into this voluminous work will be richly rewarded.

Zeitschrift für angewandte Mathematik und Physik, 1984

Third symmetry book

My last book on symmetry, Structural Equation Geometry, The Inherent Properties of Curves and Coordinate Systems was published in December, 1983. Below, I excerpt first from its Foreword and Introduction. In Appendix I I present some historical material, a survey of the contents of the three symmetry books and my proposals for improving the teaching of analytical geometry. [Incidentally, two copies of the 3-volume set were purchased by the Boeing Library in Seattle/]

From the Foreword: The present work is a much simplified version of the material of Chaps. I VII, IX, and X of my work, Symmetry, An Analytical Treatment intended as a sourcebook for instructors of analytical geometry at all levels and as an advanced undergraduate text. For this purpose, treatments of plane Euclidean coordinate systems and the methods and rationale of new approaches to the study of the inherent structure of curves have been greatly elaborated. In addition, 155 Problems have been included; these range from the very simple to the very difficult. Inasmuch as nearly all of these Problems are new, detailed outlines to the solutions to most of them also are given.

I wish to thank my son Warren for proofreading the final drafts of the manuscripts for all three symmetry books. I am greatly indebted to Aaron White for reading the penultimate drafts of all the books and for numerous suggested improvements and incisive comments. Thanks also are due to my sons Christopher and Warren for many helpful suggestions and to my former student Dr. Donald Perry for numerous helpful comments concerning the Introduction.

From the Introduction: Though today's society increasingly requires mathematical, scientific, and technological skills and understanding, the competence of the average student in mathematics and the sciences has declined alarmingly in recent times. And the modern distraction responsible for this decline appear to be on the increase. As remedial measures, educators recently have recommended additional pre college courses in mathematics and the sciences, more homework, longer school years, more exacting graduation requirements, and the provision of better qualified instructors (including "guest instructors" loaned by industry).

While these measures are highly desirable, they do not address a crucial underlying problem. This is the fact that the average student finds courses in mathematics to be unappealing, boring, or downright distasteful. This situation is not a product of the space age that can be rectified with stopgap measures. It existed long before the advent of modern distractions. Accordingly, it can be suggested that our first line of attack should be directed toward redressing this lack of appeal, and that other remedial measures will not become fully effective until this has been achieved.

Increasing the appeal of mathematics will require extensive revision and revitalization of elementary and intermediate courses. The discipline most in need of attention is Euclidean geometry. This subject not only provides the cornerstones for all fields of mathematics (and physical science) to which the average student subsequently becomes exposed, but impressions gained in elementary geometry courses are the major factor that shapes future attitudes toward other mathematics courses.

A main thrust of my admonitions is that the many fascinating features of elementary mathematics, particularly plane geometry are neither entirely appreciated nor being used in their full potential in the classroom. In the final analysis these deficiencies are the products of almost a century of comparative neglect of physical (classroom) geometry by research mathematicians. The little attention this subject has received from time to time has sought to harmonize it with conceptual advances at a level of sophistication and abstraction that is superfluous to the elementary classroom, rather than to make it more meaningful and accessible to the average student.

Headings and page numbers of topics in Appendix I, where I present some historical material, survey the contents of the three symmetry books, and make proposals for improving the teaching of analytical geometry, are given below. Many, if not most, readers will have studied analytical geometry, and will be interested in the recent discoveries and proposals.

Headings and page numbers of Appendix I:

The rise to dominance of abstract methods

Revising and rejuvenating treatments of analytic geometry

The new subject matter

The equilateral hyperbola as a Cassinian oval and self-inversion at 90°

Eccentricity as a visible property
Structure rules-new analytical tools
Structure rules, inherent structure and the classification of curves
Structure rule analysis and a coherent theory of exceptional points
New applications and knowledge of polar equations
The Devil's curve as a construction from conics
Quartic equivalents of central conics
Cartesian and Cassinian ovals related by inversion
Unprecedented new types of curves
Directrices defined
The new coordinate system for deriving construction rules
Inversion, immediate closure, and a "natural" scheme of classification
Asymptotes of inverse curves

New approaches viewpoints, and paradigms

Structure simplicity-the thread of continuity
Dimensional uniformity of equations
Using more meaningful terms than "equation"
Introducing and comparing coordinate systems

The bipolar coordinate system--the introductory system of choice
The rectangular and polar coordinate systems
Other coordinate systems
Graphic depiction of eccentricity
Characterizing curves by eccentricity
Eccentricity and the exceptional lines of conics
Systematic nomenclature for position
Systematic nomenclature for curves
Algebraic calculations
Improving the style of mathematics texts

The following comments were engendered following the appearance of Structural Equation Geometry in December, 1983, including those from Grothendieck's letter of February 8, 1984.

The author has "discovered innumerable new worlds in that innocuous looking Euclidean plane."

The book "contains a wealth of geometrical reflection and insight."

Alexander Grothendieck, 1984

....turns the whole subject in a new and original direction....His claims demand and deserve attention....I thought the chapter on the bipolar system was stimulating and I began to agree with the author that this system could well be used as the one to which students are first introduced. It certainly offers a fruitful area for investigations of various sorts and at various levels....I would certainly recommend this book strongly as a stimulating source of ideas and highly original points of view....

Dick Tata, Mathematics Teaching, 1984

....appropriate at the graduate level, where a student can study the topic of analytic geometry in a concentrated fashion. The author takes great pains to make the book's development as intelligible as possible. He succeeds nicely, largely because of his fine command of the language. The book is well written and ought to be seriously considered by college professors interested in geometry for its own sake!

A. S. Posamentier, CUNY, Mathematics Teacher, 1984

Returning to my correspondence with Grothendieck, he was much impressed by the fact that my sons assisted me with the book. From his letter of February 8, 1984:

Just a few lines to thank you for the impressive volume "Structural Equation Geometry," which I got today. I spent a couple of hours having a glance through it. It struck me a lot that two of your sons have taken active interest in your work - this means a lot to me.

While I am thinking of myself as a "geometer" --namely one fascinated mainly by shapes and structures, rather than by numbers and magnitudes ("grandeurs"), this examination shows me once again that the viewpoints and approaches I am familiar with, and which I have a tendency to take for granted (just as you take for granted ambient Euclidean planes) are just a few among an infinity of others, each holding mysteries and fascination of its own. It is a part of the world's overwhelming richness, of which we can grasp throughout our life only an infinitesimal part. This is hard to accept for the mind, who has been trained to cling to the delusion that this part is all.

While from the letter of March 22, 1984:

Thanks a lot for your letter, giving me details about the fortune of your books, and the interests of your children. None of my five children displayed interest for continuing studies beyond high school, I'm afraid my example wasn't too inspiring for them! I did play some mathematical games though with two of the boys, Alexander and Matthius, who proved a lot brighter than me. I didn't try to persuade them, though, to become mathematicians. I would have liked them to learn some craft or profession though, but none of my children made a choice yet (for the last one, who is 10, it is too early to tell)

Geometry competition sponsored:

In 1981 my publisher Science Software Systems, Los Angeles, sponsored a geometry competition for a prize of $2,500 to further promote interest in the first book. The topic was unpublished material from my studies, stated as follows:

What geometrical properties of an inversion locus correspond to the property of self-inversion of a basis curve at angles other than 0° or 180°?

EXAMPLE: A Cassinian monoval self-inverts about its center at angles of 90° and 270°. No dissimilar inversion locus of a Cassinian monoval self-inverts at either of these angles. In what manner is the property of self-inversion of the basis Cassinian in 90° and 270° modes manifested in its inversion loci?

We received 3 meritorious entries: from J. B. Wilker (Möbius Equivalence and Euclidean Symmetry), U. of Toronto; J. F. Rigby (The Inversive Symmetry of Curves), U. of Cradiff; and W. Wunderlich (Congruent Inverse Curve Pairs), Vienna Technical University. All were detailed, erudite, and accompanied by beautiful illustrations, particularly that of Wunderlich. In the eventuality we declared a 3-way tie, and awarded the originators of each entry $833.34. Wilker subsequently published his entry in The American Mathematical Monthly, April, 1984. Some correspondence concerning the awards, to Science Software Systems President, Dr. R. A. Boolootian, and to me, are of interest, as well as a reply from me to Wunderlich.

From: J. F. Rigby (to SSS, July 10, 1983): I was delighted to see your letter yesterday. Having just missed me in Cardiff and Longhope, it finally caught up with me here in Singapore. One of the other award winners, Professor John Wilker, is a friend of mine; we had already exchanged copies of our respective entries, and I thought how nice it would be if we were to share the award, so everything has turned out very well! I shall write to the third winner so that we also can share our efforts.

From W. Wunderlich (to SSS, July 8,  1983): Having just received your kind letter of June 22, I am enjoyed to read that the jury has named me among three recipients of the award for your 1983 Competition in Geometry. I acknowledge also with thanks the receipt of your cheque about $833.34.

I am glad that my entry has been appreciated, since I had invested a lot of time into the investigation of the proposed problem. However, I liked the work, as the results were of interest for myself too.

Of course, I would like to see the results of the other two awardees and their methods. Following your suggestion I will write to Professor Wilker (Toronto) and Dr. Rigby (Cardiff).

Also from W. Wunderlich (to me, February 22, 1984): Thank you very much for your kind letter of November 28, 1983, and your flattering comments to my monograph on "Congruent inverse curve pairs."

I was astonished to read that you personally paid the prize, and the more I appreciate to be among the winners. Since sponsors in Geometry are very rare, you have to be highly praised for your unselfish initiative!

I confess that my investigations meant hard work, considerably hindered by my bad eyes, but as Geometry is not only my job, but also my hobby, I really enjoyed the occupation with the proposed problem.

Meanwhile I exchanged copies of my entry with those of Dr. Rigby and Professor Wilker (which is too abstract for my taste).

In December I received from Dr. Boolootian two issues concerning the next SSS Competition in Geometry. The problem proposed and aiming at relations between pedals and inverses of curve families is very stimulating again. Having some suitable ideas, I am already collecting provisional material.

My reply to Wunderlich (April 26, 1984): Thank you very much for your letter of February 22. I still marvel at your monograph and show it off to my classes. I don't believe such illustrations are to be found anywhere else in the mathematical literature.

I'm afraid I don't merit your praise for unselfish initiative for putting up the prize money personally for the geometry competitions. It is my hope in this way to stimulate additional interest in my studies. As you well know, studies in geometry do not carry high priority among today's academic mathematicians, at least not in America.

I also liked Rigby's solution better than Wilker's and for the same reason. This is not, however, to deny the beauty and power of Wilker's approach. Do you have any plans to publish any of your results? It would be a great loss not to have their essence preserved. Since the American Mathematical Monthly already has published Wilker's solution and acknowledged it to be one of the three award winning SSS Geometry Competition entries, it would seem that they would be receptive to receiving a contribution from you also.

I will look forward eagerly to your solutions to the new competition problem. To me, the pedal inverse relations for conics are the most fascinating and beautiful of those for any geometrical transformations. Although the current problem appears to me to be much more formidable than the 1983 problem, perhaps you will prove me wrong.

From J. B. Wilker (to me, September 22, 1982): [Wilker's acknowledgement letter is misplaced, but I substitute a prior pertinent letter. JLK] I have taken up your problem with considerable enthusiasm and written a paper about the subject. The enclosed copy is for your personal library and I'll send along another copy to Dr. Boolootian to enter the SSS contest.

In the paper I have substituted the word figure for curve in order to cover the various geometrical objects which occur in 2-dimensions and also be applicable in higher dimensions. It turns out that alpha-self-inversions indicate the presence of hidden Möbius symmetry in a base figure and this hidden Möbius symmetry can be brought out as Euclidean symmetry in other figures of the same species including inversion images of the base figure. The figure on the cover of the paper indicates that hidden Möbius symmetry can be present even when alpha-self-inversions are not. Consequently I have treated the slightly more general problem of base figures with arbitrary hidden Möbius symmetry rather than just those which admit alpha-self-inversions. This "in for a penny - in for a pound approach did seem to pay off in the long run and I hope you will enjoy it.

Miscellaneous Correspondence

Over the years I have received much correspondence over my books, mostly from people trying to get copies of them, which I usually provided gratis. These are still available, but only privately, as I have not yet gone to the trouble of listing them for sale with amazon.com. One chap kept after me for years, wishing to design a web site for structural equation geometry, but that happened at times when I was too occupied or stricken with sciatica to be able to help. Now he as moved on from UCLA and I have lost touch with him. [Since the above was written, he returned to Los Angeles, and contacted me again in 2003. His name is Curtis Mishiyama, and he is working for a degree at a local college after having been involved with NASA and Mars landing projects. He no longer thinks in terms of designing a web site but expressed an interest in exploring possibilities of advancing my geometry findings in some manner, for which I sent him a copy of Curves and Symmetry. As of Dec., 2003, I haven't heard from him further.]

In 1985, I received a letter from Professor R. H. Kirchhoff of the University of Massachusetts at Amherst, informing me as follows.

I enclose for your perusal a few examples of Cassinian ovals which occur in ideal fluid flow. These are from my book, Potential Flows, Computer Graphic Solutions....I only found out that these isotachs and streamlines were Cassinian ovals after paging through your book Structural Equation Geometry.

I feel there can be little doubt that the techniques and approaches of structural equation geometry will be found to have significant practical and heuristic applications. However, I've had little contact with such matters in recent years, and can say little even about their use in the classroom. I did contributed an article to New Scientist (March 14, 1985) titled, "Curves Cut through the Mathematical Jungle. In the early 1980s, Richard Atkinson became Chancellor at UC San Diego (he now is President of The University). Inasmuch as he had held a high government position in Washington, D.C. in connection with some educational activity, I send him a copy of my Structural Equation Geometry book, together with some thoughts about revising methods of teaching geometry. Nothing further came of this contact, however. The time for widespread use of structural equation geometry in the classroom may be no less delayed than that of its 'parent,' the anharmonic laboratory shaker. Fifty years on from its invention, the latter's obvious advantages for many purposes still have not been realized in scientific and engineering uses.