Haitian Math Whiz May Have Unraveled Age-old Geometry Mystery HAITI PROGRES ( http://www.haiti-progres.com), October 9 - 15, 2002 Vol. 20, No. 30
by Kim Ives PHOTO:

: Leon Romain has devised a theorem for trisecting any angle, one of geometry's great puzzles. If he is right, it could change your life. So far, nobody has proved him wrong

Around 450 B.C., the Greek mathematician, Hippias of Ellis, began searching for a way to trisect an angle. Over 2000 years later, in 1837, a French mathematician named Pierre Wantzel proclaimed that it was impossible to trisect an angle using just a compass and a straightedge, the only tools allowed in geometric construction.

But now, at the dawn of the twenty-first century, a Haitian computer program designer, Leon Romain, claims he has proven, with a "missing theorem," that it is possible to trisect an angle with those simple tools, disproving Wantzel's assertion and exploding centuries of mathematical gospel.

"This discovery shows us that the notions that every mathematician has held for the past 200 years as absolute certainty are actually false," Romain told Haiti Progres. "The mathematical and even philosophical ramifications are huge."

The trisection of an angle is one of the infamous "three problems of antiquity" which have stumped mathematicians for centuries. The other two conundrums are quadrature of a circle (the process of constructing a square equal in area to a given circle) and duplication of a cube (finding a cube whose volume is twice that of a given one).

Romain lays out his case in a recently published book entitled "Angular Unity: The Case of the Missing Theorem." In it, he explains how, as a 13-year-old student in Port-au-Prince, he learned that the trisection of an angle was impossible. Skeptical, Romain immediately set about to test whether this was true. He solved the problem in several different ways, including the invention of "a device that in fact was a modified compass with two pencils whose distance could vary at will," Romain writes. But the math teacher to whom he proudly showed his invention, Yves Medard (better known as the poet, writer, and filmmaker Rassoul Labuchin), patiently explained that all Romain's methods were unacceptable. "He taught me the most important fact concerning that problem and the discipline of geometry in general," writes Romain, that only a "straightedge and a compass are allowed in the construction of any figure.

Anything else would be considered mechanical, he said, and therefore beyond the scope of simple geometry." Medard's competent grasp of the problem had a profound effect on the young teenager. When Romain would speak of these concepts to mathematicians "with advanced degrees" in later years, he discovered that "the vast majority of them are not even aware of their existence. But there I was with that high-school teacher in a third world country, and, in a Socratic manner, he introduced me to a few of the deepest concepts of Euclidian Geometry."

This launched Romain on a three-decade quest to solve the riddle. After graduating from the CollFge Fernand ProspFre in Port-au-Prince, he studied political science and computer science at Queens College in New York. All the while, he continued to work on the trisection problem until he came up with his solution.

Simply stated it is: "In a triangle, if an angle measures twice another, the square of the side opposite that angle is equal to the sum of the square of the side opposite its half and the product of that side by the third one."

This key triangle theorem -- which Romain dubbed the "Romain triangle" for brevity's sake -- and its unique properties were noted by Greek mathematicians Nicomedes and Archimedes around 250 B.C. and Ceva Tommaso in the 17th century, but no mathematician before Romain ever established its ability to trisect an angle, any angle. Hence, Romain calls it "the missing theorem." If it holds up to peer scrutiny, many mathematical assumptions will have to be overhauled. But challenging the mathematics establishment and centuries of academic dogma will not be easy. Romain has submitted his findings to the mathematics departments at prestigious schools like New York University and Columbia University but has received no response.

Unable to disprove it but fearing its ramifications, some mathematicians are simply side-stepping the challenge posed by "the missing theorem" by pretending they have no time to review it, Romain suspects.

"I do have a copy of some excerpts of Mr. Romain's work," Dr. Henry Pollak of Columbia's Mathematics Department told Haiti Progres, "but my commitments have not allowed me to look at them carefully."

Furthermore, many mathematicians may be prejudiced. "If you start with the conception that it is insolvable, then you might not devote the amount of time you should to something which you already think is not possible to solve," said Dr. Fritz Cayemite of Columbia, who remains agnostic on Romain's premise. "I submitted it to a well-known mathematician, who said that this is a closed chapter, this has been proven to be one of the insolvable problems... I did read Romain's work and I found it very interesting, a very good piece of work. I didn't see any flaws in his proof, but I'm not really an expert in geometry

The world's principle authority on algebraic geometry, Dr. Jean Claude Carrega of the University de Lyons in France, did engage in an email discourse with Romain about his proof this past spring. "But it came to a point where he could not disprove what I was saying, and then he broke off the correspondence," Romain said.

Contacted by Haiti Progres, Carrega asserted that Romain's "method will never allow the trisection problem of a general angle with only a straightedge and a compass to be solved" because "this problem was proved impossible in 1837 by the mathematician P.L. Wantzel," whose premise is precisely what Romain claims to disprove. "For some angles, the construction of the Romain triangle is as impossible as the trisection of this angle," he said.

"He has to say why," Romain responds. "The same way I showed mathematically that it is possible, he has to prove that what I am saying is wrong. But he cannot, because he accepts the Romain triangle. Then he also embraces Wantzel. But the Romain triangle disproves Wantzel. You can't have it both ways."

So what? you may ask. What relevance does any of this have on anything other than some arcane mathematical debates? A lot, according to Romain.

Mathematical models are used to set traffic lights, provision grocery stores with apples and toothpaste, electronically transfer money around the world, keep planes from crashing into each other, distribute electricity, design buildings, determine school budgets, set insurance rates, calibrate your microwave, and run your computer, cell phone, and car. "Mathematics are central to every aspect of everyday life in modern society," Romain notes. "Mathematics are so abstract that they can be and are applied to all the other sciences we have, including the social sciences."

In fact, "99% of Einstein's discoveries are based on mathematical formulas, not physical experiments," he says. "These led to the development of the atom bomb and other technologies, on which the lives of people all over the world depend."

Romain's discovery, if it cannot be rebutted, also has over- arching philosophical implications about the way we procure and test knowledge. "The scientific method is the best approach to the truth because it tries to eliminate everything that cannot be proven," Romain says. "If the methods we use are yielding certain conclusions which are not true and which so many mathematicians can be led into believing are true, there is definitely something wrong, either in the language or the form. This clearly shows that these people, by not finding those errors, did not fully understand Wantzel's presentation. Because if they had, they would have found the holes in it. They never questioned the foundations of their own knowledge."

Has Leon Romain made a discovery that will turn mathematics on its head? Has a Haitian math hobbyist out-thunk some of the greatest minds at the grandest institutions which have toyed and wrestled with these problems for centuries? The jury is still out, but nobody has been able to prove him wrong yet.

Leon Romain can be contacted at leon@kafou.com

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