Mathematical constants are built from the fabric of mathematics and can be determined a priori: Pi is based on the shape of a circle and can be calculated from numerous equations, e is from the growth of functions, and the square root of 2 is, well, a square root. Of course, mathematical constants can also be found in nature. For example, take the case of pi:
Physicists have noted the ubiquity of pi in nature. Pi is obvious in the disks of the moon and the sun. The double helix of DNA revolves around pi. Pi hides in the rainbow, and sits in the pupil of the eye, and when a raindrop falls into water pi emerges in the spreading rings. Pi can be found in waves and ripples and spectra of all kinds, and therefore pi occurs in colors and music. Pi has lately turned up in superstrings, the hypothetical loops of energy vibrating inside subatomic particles. Pi occurs naturally in tables of death, in what is known as a Gaussian distribution of deaths in a population; that is, when a person dies, the event â€œfeelsâ€ the Ludolphian number.
On the other hand, there are physical constants, which are part of the fabric of the universe, but canâ€™t be figured out simply by doing math. Instead they have to be measured and are known a posteriori. Whatâ€™s the charge of an electron? You can do an experiment to determine this. Same thing for the speed of light.
Of course, the distinction between these two types of constants is not always clear. For example, pi has long been calculated by looking at physically constructed circles. In addition, the neuroscientist David Eagleman actually announced an award a number of years ago, called the Eagleman Prize in Mathematics and Physics, for anyone who could relate various mathematical constants with physical values, such as the mass of an electron.
However, there is one physical constant that a number of scientists (including Richard Feynman), have felt might combine these two aspects: very much a part of the cosmos but also mathematically derivable. This constant is called the fine-structure constant.
The fine-structure constant combines an extraordinary number of physical properties of our universe: the speed of light, the charge of an electron, energy differences in quantum mechanics, and more. When you combine all these values by multiplying and dividing, you get a value, with no units, and one that is not too large: just about 1/137. Not 1/137 feet, not 1/137 minutes, just 1/137. This number is neither extraordinarily tiny or large, and almost exactly one over 137, though itâ€™s actually about one over 137.0359996.
Due to its lack of units, some have felt that perhaps we could use math, rather than just experimental measurement, to determine its value. The physicist Arthur Eddington produced proofs for the fine-structure constant being exactly 1/137, as well as 1/136 when it wasnâ€™t measured as precisely. And more recently, there have been attempts to explain the fine-structure constantâ€™s value using a combination of functions from trigonometry and pi.
While I donâ€™t particularly think that the fine-structure constant can be understood this way, I decided to give it a try, but with a computational twist: I used genetic programming.
Genetic programming is an evolutionary spin on function-fitting. If you have a lot of data and are trying to make sense of it, you can use a brute-force approach to figuring out what sort of formula describes what you have. Or you can evolve the solution. You start with a population of randomly generated equations that are mutated and tested for their fitness, which is simply how closely the equations describe the data. Most of the time, these equations are terrible, but evolution improves the results, in a ratchet-like manner.
So, I set out to do the same thing with the fine-structure constant. We know its value to many decimal places. So I generated some random functions that combine 1,2, pi, and e, using functions from trigonometry and hoped that after many generations of evolution, one of them would result in computing the fine-structure constant.
And that is exactly what I found. After allowing my computer program to evolve to the value of the fine-structure constant, or more precisely its inverse, which is approximately 137.0359996, I received the following function:
Unfortunately, this equation doesnâ€™t have the elegance of the type of equation that I imagined might describe a fundamental constant of the universe. So itâ€™s probably not something we should be too proud of.
Then again, it might not actually be the constant we imagined in the first place, as there are actually some scientists conducting measurements as to whether the fine-structure constant varies from place to place in the universe. So maybe we should starting evolving equations that take your cosmic position as a variable. Unfortunately, thatâ€™s a bit beyond my simple genetic programming Python script.
Top image: Mike Durkin/Flickr/CC-licensed
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