Category Archives: math

On Fermat’s Last Theorem (Conjecture)

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Fermat, a French mathematician of the late 17th century, came up with a conjecture that baffled other mathematicians for three and half centuries until Andrew Wiles published a proof in the mid-nineties. Most of you are familiar from high school geometry with the Pythagorean theorem, that the sum of two integers squared may be equal to another integer squared: a2 + b2 = c2, but Fermat imagined a more general problem for integers where an + bn ≠ cn where n>2: Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. That last bit is the mystery—that the margin was too small for his proof. Many mathematicians believe he did not have a proof, but all the same, he did throw down the gauntlet by making the conjecture. He just wrote the conjecture, that an + bn = cn, is not possible. Wiles’ proof is so complex and convoluted, however, that you have to be a brilliant mathematician to even begin to understand his arguments. For as simple as the Pythagorean theorem looks, Fermat’s conjecture is inversely complex, and complex in ways that not even a great mathematician can dream. The conjecture looks simple, but the answer seems to be one of the most complex ever proved in the history of mathematics. The proof, almost as elusive as the Holy Grail, is unintelligible to the average lay person, and difficult for even the gifted. What kind of mind does it take to fathom the dark and profound reaches of Fermat’s conjecture? This conjecture, according to a French academy of math, has the dubious honor of having the highest number of incorrect proofs written about it. In other words, many mathematicians have tried to conquer the proof, but died ignominiously on the battlefield without having succeeded. That fact that Wiles did his work in secret suggests that even he thought the little problem might be paradoxically unsolvable—a no-win scenario, as it were, and a career-ending catastrophe. That there is, after all, a solution to Fermat’s last theorem is of little consolation to all of that failure. (Sorry mathematicians,formatting limitations don’t allow for the little raised numbers in the equations.)

On Fermat’s Last Theorem (Conjecture)

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Fermat, a French mathematician of the late 17th century, came up with a conjecture that baffled other mathematicians for three and half centuries until Andrew Wiles published a proof in the mid-nineties. Most of you are familiar from high school geometry with the Pythagorean theorem, that the sum of two integers squared may be equal to another integer squared: a2 + b2 = c2, but Fermat imagined a more general problem for integers where an + bn ≠ cn where n>2: Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. That last bit is the mystery—that the margin was too small for his proof. Many mathematicians believe he did not have a proof, but all the same, he did throw down the gauntlet by making the conjecture. He just wrote the conjecture, that an + bn = cn, is not possible. Wiles’ proof is so complex and convoluted, however, that you have to be a brilliant mathematician to even begin to understand his arguments. For as simple as the Pythagorean theorem looks, Fermat’s conjecture is inversely complex, and complex in ways that not even a great mathematician can dream. The conjecture looks simple, but the answer seems to be one of the most complex ever proved in the history of mathematics. The proof, almost as elusive as the Holy Grail, is unintelligible to the average lay person, and difficult for even the gifted. What kind of mind does it take to fathom the dark and profound reaches of Fermat’s conjecture? This conjecture, according to a French academy of math, has the dubious honor of having the highest number of incorrect proofs written about it. In other words, many mathematicians have tried to conquer the proof, but died ignominiously on the battlefield without having succeeded. That fact that Wiles did his work in secret suggests that even he thought the little problem might be paradoxically unsolvable—a no-win scenario, as it were, and a career-ending catastrophe. That there is, after all, a solution to Fermat’s last theorem is of little consolation to all of that failure. (Sorry mathematicians,formatting limitations don’t allow for the little raised numbers in the equations.)

On fractals

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Though they are complex to describe, you have seen them many times in snow flakes, branching river deltas, the branches of pine trees, ancient ferns, and the florettes of a cauliflower. Fractals, though difficult to define, seem to be repeating self-similar patterns that repeat until they are infinitesimly small, but always the same. Fractals, if you were to analyse them from a mathematical standpoint, are non-linear functions that form all sorts of beautiful loops, and swirls that go on and on into a vanishing point somewhere off of the graph paper. We see fractals that occur in nature all the time. They are so common that we would miss if they weren’t there, but we ignore them because they are ubiquitous. Fractals are imprinted in our subconscious to the point that a nautilous shell can only have one design–a spiral of ever increasing size. If the fractal weren’t there, it wouldn’t be a nautilous shell, or pine tree branch, frost on a window, branching lightening, or the Mississippi River delta. Ever look at the way medieval architects imprint a fractal design on the front of Gothic cathedrals? Fractals are pleasing to the eye and soothing for the soul. Part of the universes harmony is wrapped up in fractals, including the designs of galaxies. Now, in the Oscar winning song of the year, “Let it Go,” the word fractal is included in the lyrics, and the main character creates an ice palace out of macro-fractal snow flake. Fascinating.

On fractals

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Though they are complex to describe, you have seen them many times in snow flakes, branching river deltas, the branches of pine trees, ancient ferns, and the florettes of a cauliflower. Fractals, though difficult to define, seem to be repeating self-similar patterns that repeat until they are infinitesimly small, but always the same. Fractals, if you were to analyse them from a mathematical standpoint, are non-linear functions that form all sorts of beautiful loops, and swirls that go on and on into a vanishing point somewhere off of the graph paper. We see fractals that occur in nature all the time. They are so common that we would miss if they weren’t there, but we ignore them because they are ubiquitous. Fractals are imprinted in our subconscious to the point that a nautilous shell can only have one design–a spiral of ever increasing size. If the fractal weren’t there, it wouldn’t be a nautilous shell, or pine tree branch, frost on a window, branching lightening, or the Mississippi River delta. Ever look at the way medieval architects imprint a fractal design on the front of Gothic cathedrals? Fractals are pleasing to the eye and soothing for the soul. Part of the universes harmony is wrapped up in fractals, including the designs of galaxies. Now, in the Oscar winning song of the year, “Let it Go,” the word fractal is included in the lyrics, and the main character creates an ice palace out of macro-fractal snow flake. Fascinating.