Let's have fun first, for a change. The applet below plots Lamé curves, also known as superellipses. Play with the parameters and see how many interesting shapes you can get.
|Force A = B|
Next time you need to create an icon, a logo or anything that needs a geometric form as start point, these curves may be a source of inspiration. In particular, the "rounded-edge squares" generated by exponentes above 2 are very nice. Use non-integer exponents to fine-tune the roundness.
The original Lamé curves are based on the following parametric equation:
x^a + y^a = 1
If you remember the high-school trigonometry classes, the Lamé curve with exponent 2 is simply the circle equation. The applet offers some extensions e.g. two different exponents, aspect and rotation.
Lamé was a French matematician, among many that tried to prove Fermat's Last Theorem. Certainly the Lamé Curves were conceived as a tool in that endeavor. The Last Theorem states that, in equation
a^n + b^n = c^n
there is no triplet of non-zero integer numbers (a, b, c) that satisfy the equation for integer exponents higher than 2. For the exponent=2, every solution is known as "Pithagorean triple" (e.g. 3, 4, 5) and there are infinitely many of them.
A simple manipulation of the Fermat's Last Theorem equation makes it equal to the Lame's curve equation:
(a/c)^n + (b/c)^n = 1
Accordingly to this equation, the last Fermat's theorem says: there are no pairs of non-zero rational numbers that satisfy this equation when the exponent is integer and greater than 2. So, except for the "four cardinal points" where either x or y are zero, no point of a Lamé curve is a tuple of rational numbers. Either x or y must be irrational.
Something similar happens with Euler's number: every rational exponent of "e" is irrational, and every rational number must have an irrational natural logarithm. (The opposite is not true: the log of an irrational number is most probably irrational.)
In the case of exponent 2, the Lamé curve becomes a circle and then we have infinite rational pairs that are x/y points on the circle, based on Pithagorean triples. Most points over the unit circle are irrational, given that irrational numbers are more abundant. Also, some relevant angles like 45º are irrational (x=y=√2).
Even so, given the infinite number of Pithagorean triples, there are infinite rational points on the unit circle as well. We could trace a densely-dotted, if theoretically discontinuous, circle using only rational coordinates.
Anyway, Lamé could not prove the Fermat's Last Theorem, but the unique properties of the exponent-2 Lamé curve (the circle) hints the theorem is true. Among all Lamé curves, only the circle
The Fermat theorem was originally stated for integer exponents higher than 2. It is noteworthy that Lamé curves with exponents above 2 (integer or not) are perfectly "smooth" while exponents between 0 and 2 generate kinked curves. This fact suggests they are very different species of curve.