Happy is the man who knows to enjoy himself in things big and small. Here's the news: Alain Connes — a French Fields medalist — has come up with a new proof of Morley's theorem. Indeed the proof was published in 1998, and the news may be stale for some readers, but I was pleasantly surprised to learn about it as recently as last month. A message with the proof was posted to the sci.math.fr newsgroup and translated for the benefit of the Hyacinthos discussion group by its French member, Jean-Pierre Ehrmann. The group, thought up by John Conway and Antreas Hatzipolakis, was named after the French geometer Emile Michel Hyacinthe Lemoine (1840-1912). Its interests are appropriately centered around triangle geometry.

Connes was introduced to the theorem at lunch time. The theorem was mistakenly attributed to Napoleon Bonaparte (1769-1821) — another French geometer. Which, besides curiosity, served as an impetus to pick up the challenge. Recall that Morley's theorem states that

Connes began with an observation that the points of intersection "are the fixed points of pairwise products of rotations g_i around vertices of the triangle (with angles two thirds of the corresponding angles of the triangle)." Rotation is a transformation of the plane which is defined by the center of rotation — its fixed point — and the angle of rotation. The plane, and with it every figure in the plane, rotates around the center through the angle of rotation. The product of two rotations (which is often also called the sum) is the result of the successive execution of the two transformations. The product of two rotations is another rotation through an angle which is the sum of two rotation angles. If the angles add up to 0^o (mod 360^o), the product becomes a translation.

At first, Connes looked at g_i's as plane isometries, with the intention to express in their terms the threefold symmetry of the equilateral triangle. This attempt failed and eventually he established impossibility of such a presentation. The success came with the interpretation of g_i's as affine transformations of the (complex) line.

The proof is carried out for a field k and the affine group G over k. G consists of affine transformations g(x) = ax + b, xk, where ak, a0 and bk. For gG, define

d(g) = d(ax + b) = ak^*,

the multiplicative subgroup of non zero elements of k. d is a morphism from G to k^*. Define also T = Ker(d), which is the group of translations, i.e. the additive subgroup of k. For gT, a1, and g admits exactly one fixed point:

Theorem

Let g₁, g₂, g₃G be such that g₁g₂, g₂g₃, g₃g₁ and g₁g₂g₃ are not in T. Let j = d(g₁g₂g₃). The following two conditions are equivalent,

1. g₁³g₂³g₃³ = 1.
2. j³ = 1 and A + jB + j²C = 0, where A = fix(g₁g₂), B = fix(g₂g₃), C = fix(g₃g₁).

Proof

Let g_i = a_ix + b, i = 1, 2, 3. The equality g₁³g₂³g₃³ = 1 is equivalent to d(g₁³g₂³g₃³) = 1, and b = 0, where b is the translational part of g₁³g₂³g₃³. The first condition is exactly j³ = 1. Also, by hypothesis j1. b is computed to be

Using j = a₁a₂a₃, we transform this into

where the fixed points have been expressed explicitly as

Now, a_ij since by hypothesis the pairwise products of g_i's are not translations. The theorem thus follows.

Corollary (Morley's theorem)

Let there be given PQR. Take k to be the field of complex numbers C and let g₁ be the rotation with center P through two thirds of QPR and similarly for g₂ and g₃.

g₁³g₂³g₃³ = 1 because each g_i³ is the product of reflections in the two sides forming the corresponding angle. Due to the definition of rotation angles, A, B, and C are the intersections of trisectors. This establishes condition a) of the theorem, which implies b). In particular, A + jB + j²C = 0. The latter is a classical characterization of equilateral triangles.

Remarks

1. I forgot to mention that there is no limitation on the characteristic of k, except that some complications arise when characteristic equals 3.

2. The expression g₁³g₂³g₃³ has a built-in invariance with respect to multiplication of any of g_i's by a cubic root of 1. The theorem thus implies existence of the total of 18 equilateral triangles. As I mentioned on another occasion, these are often designated as Morley triangles. The fact is however that Morley has discovered 27 equilateral triangles which is clearly not reflected in the current nomenclature.

Connes published his proof in a collection of papers on the occasion of the 40^th anniversary of l'Institut des Hautes Études Scientifiques. The volume addresses various aspects of the evolution of the relationship between mathematics and physics. Most of the papers are historical surveys with emphasis on the last few decades. Symmetry groups are of course only one element in the multifaceted relationship. Topology, cohomology, Lie algebras, knot theory have all become valuable items in the physicist's tool chest. More than that, with the emergence of the superstring theory, a theory that holds a promise of becoming a TOE — a Theory Of Everything — the attitude of physicists towards mathematics underwent a definite metamorphosis.

L. Michel, who joined l'IHES in 1962 writes in Physique et Mathématique: "Mais je n'y ai été qu'un utilisateur, très gourmand et un peu gourmet, des mathématiques..." G. Veneziano recollects how in the early 1960s Prof. Gatto used to say: "A good theorist should not study maths, should just know it!" But he ends his article Physics And Mathematics: A Happily Evolving Marriage? with a remark that "... mathematical rigour and elegance are becoming a driving force, a true guiding principle for theoretical research" and adds a dictum of his own:

Yuval Ne'eman of Tel-Aviv University, Israel makes a reference to the Erlangen Program announced by Felix Klein in 1872. Klein suggested to classify different geometries by means of groups of symmetries — transformations that leave invariant essential properties of geometric figures. In his paper De l'Autogéométrisation de la Physique, Ne'eman writes: "Le Programme d'Erlangen se réalise à travers la physique!" (Well, at times symmetry gets spontaneously lost and a definitive reference about this phenomenon is Noncommutative Geometry by Alain Connes.)

So physicists turn into mathematicians. In fact, this may not be at all surprising. V.I. Arnold once categorized mathematics as that part of physics in which experiments are inexpensive. Given the growing costs of experimentation in particle physics, the evolution of theoretical physics into a part of mathematics might appear as a natural development.

Arnold made his remark during a discussion on mathematics education that took place at Palais de Découverte in 1997. The thrust of his argument was directed against what is known in the United States as the new math, although his outlook was much broader. He lamented an attempt that originated in France in the middle of the 20^th century to separate mathematics from physics, in mathematics instruction in particular. The attempt is associated with a group of (then) young mathematicians that published volumes of mathematical texts in a rigorous axiomatic manner under the assumed name of N. Bourbaki.

Incidentally, J. Dieudonné — one of the more active contributors in the group — was one of the first two mathematics professors (the other one was Alexander Grothendieck) who received a permanent appointment at IHES in 1959. Bourbaki's treatise was extremely influential on the development of mathematics and physics — and not necessarily in a negative sense. Here's just one short affirmation from the aforementioned collection of papers: "Bourbaki traite l'intégrale de Wiener d'une manière que je pouvais généraliser facilement à l'intégrale de Feynman." (C. Dewitt-Morette, Des Ponts)

As regards mathematics education, Arnold holds a twofold view. First, he classifies mathematics as a natural science. Elsewhere (Polymathematics: is mathematics a single science or a set of arts?) Arnold writes:

Second, like every natural science, mathematics is developed by trial and error, modeling and experimentation, with proofs coming last as a method of hypothesis verification. Both aspects of mathematics must be reflected in mathematics education.

In fact the trend of melding of theoretical physics with mathematics was reciprocated on the part of mathematics by fostering its experimental nature. Technology (besides traditional paper and pencil) permeates today's mathematics. Triangle geometry serves as a good example.

Clark Kimberling — the man largely responsible for a renewed interest in triangle geometry — writes

Kimberling's book Triangle Centers and Central Triangles, published in 1998, describes nearly 400 centers of various kinds and his online encyclopedia currently lists in excess of 500 remarkable points. (Points X356 and X357 are associated with Morley's triangle and are known as the 1^st and 2^nd Morley centers.) Most of these were first discovered with the help of the computer.

Computers open new horizons for experimentation while still keeping experiments inexpensive. As tools in mathematics education, computers promise to get more students involved in mathematical experimentation. It remains to be seen whether many will and with what results. Experimentation leads to uncertainty. No doubt many students will find this difficult. New pedagogy will be required to teach students how to use computers successfully. Meanwhile, I wish to offer a curious example. I borrowed the example from Paul Yiu's lecture at the 7^th Statewide Conference organized by Florida Higher Education Consortium, November 12, 1998.

Let I be the incenter of ABC. Consider three triangles AIB, BIC, and CIA. In each of the three triangles pass a straight line through its incenter and circumcenter. (In the applet, I_C and O_C stand for the incenter and circumcenter of AIB, and similarly for the other 4 points.) Doodling with the applet one may form (at least) two conjectures. (A working geometer would probably dismiss one of them as a known result. I still carry on.)

1. The circumcenters of triangles AIB, BIC, and CIA all lie on the circumcircle of triangle ABC.
2. The three lines through the incircle and the circumcircle of triangles AIB, BIC, and CIA meet at a point.

In the applet, by checking the Show calculations you may obtain the (approximate) sum of the Distances of the circumcenters O_A, O_B, O_C to the circumcircle of ABC and the (approximate) Area of the triangle formed by the above three lines.

You may see that one conjecture is probably right whereas another conjecture is probably wrong. You may want to formally prove the former. There is no disagreement between mathematicians that a proof should crown any kind of purposeful experimentation. According to Floor van Lamoen — a Dutch member of the Hyacinthos group — "computer exploration is nice. Proving is even nicer!"

And just one more note. Paul Yiu is one of the organizers of a new online publication: Forum Geometricorum. This is a peer reviewed journal devoted to the Euclidean geometry. The first issue planned for January 2001.

References

1. Les Relations Entre Les Mathématiques et La Physique Théorique, Institut des Hautes Études Scientifiques, 1998
2. I.M. Yaglom, Geometric Transformations, v 1, MAA, 1962

Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at alexb@cut-the-knot.com

Copyright © 1996-2000 Alexander Bogomolny