Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
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 gi 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 0o (mod 360o), the product becomes a translation.
At first, Connes looked at gi'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 gi'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
the multiplicative subgroup of non zero elements of
k. d is a morphism from G to
k*. Define also
Let g1, g2, g3G be such that
g1g2g3 are not in
Let gi = aix + b,
i = 1, 2, 3. The equality
Using j = a1a2a3, we transform this into
where the fixed points have been expressed explicitly as
Now, aij since by hypothesis the pairwise products of gi'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.
= 1 because each gi3 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,
Connes published his proof in a collection of papers on the occasion of the 40th 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:
tools that suit our problems
problems that suit our tools.
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 20th 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 1st and 2nd 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 7th 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, IC and OC 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.)
In the applet, by checking the Show calculations you may obtain the (approximate) sum of the Distances of the circumcenters OA, OB, OC 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.
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 firstname.lastname@example.org
Copyright © 1996-2000 Alexander Bogomolny
Let there be given two rotations gP and gQ around
points P and Q, respectively. Assume gP is executed first,
gQ second. gP leaves P in place. Denote
Now prove that
A plane isometry is a transformation of the plane that preserves distances between points. By SSS, any isometry preserves also angles. Isometries may or may not preserve orientation. An orientation preserving isometry is either a translation or a rotation.