13363: : This Week in Haiti 20:30 10/9/2002

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"This Week in Haiti" is the English section of HAITI PROGRES
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and Creole, please contact the paper at (tel) 718-434-8100,
(fax) 718-434-5551 or e-mail at <editor@haitiprogres.com>.
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                           HAITI PROGRES
              "Le journal qui offre une alternative"

                      * THIS WEEK IN HAITI *

                       October 9 - 15, 2002
                          Vol. 20, No. 30

HAITIAN MATH WHIZ MAY HAVE UNRAVELED AGE-OLD GEOMETRY MYSTERY
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 Haïti Progrès. "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 Médard (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."

Médard'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 Collège Fernand Prospère 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 Haïti
Progrès, "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 Carréga 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 Haïti Progrès, Carréga 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

All articles copyrighted Haiti Progres, Inc. REPRINTS ENCOURAGED.
Please credit Haiti Progres.

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