One Sunday morning, following the visiting speaker’s sermon that dealt in some way with the beautiful, the service leader asked members of the congregation to put forth the things they found beautiful. Finally, a retired professor of physics stood and asked whether anyone found mathematics beautiful. No one admitted to it.
As someone who finds beauty in a lot of different things - music, art, nature – I was made to ask myself whether I had ever found beauty in any mathematical formula and decided that it had never happened. It turns out that for mathematicians there are beautiful formulas and ugly formulas, and a beautiful formula causes the same area of the brain to be stimulated as listening to Bach or viewing a Van Gogh does in many of the rest of us.
Mystified, I went in search of explanations of mathematical beauty, and this article in BBC News by James Gallagher explains it as well as it can be made understandable to those of us who are mathematically inert.
On an intellectual basis I get it. Certain seemingly unrelated mathematical concepts like, pi, imaginary numbers and prime numbers may be shown to be related in mathematical equations, and the more concise that demonstration the more beautiful. Thus Euler’s equation,
reveals an unexpected relationship between all of these conceptual numbers. The value e, like pi, is an irrational number that is approximately 2.71828, and like pi it is a mathematical constant. The value I is the unit imaginary number and is equal to the square root of (-1). As one mathematician explained the beauty of Euler’s equation, it in the most concise manner explains the relationship between the constants.
Another mathematician when discussing the equation that expresses the fact that any prime number divisible by four with a remainder of one is the sum of two squares was beautiful, but not especially for the equation. In his words that equation is like the final chord in a symphony. The derivation is the symphony.
I understand that on an intellectual basis, but not on the emotional basis that occurs with art or music.
Last evening we were watching television and Lynn paused the recording while she answered a phone call. The image on the screen was of a beautiful, rocky coastline, and it reminded me that when I first read, in the mid-1970s, about fractal geometry and Mandelbrot Sets they were beautiful, not for the equations, which just look like equations to my eye, but for what they represent when expressed visually.
The equations of fractal geometry - which were impossible to perform before computers because they involve continuous equations in which the result of the first equation is entered into the second, the result of which is entered into the third out to infinity - are beautiful because they express the real world.
“"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."