When the word “pie” is spoken aloud, or when we chuckle (or groan) at the above image, we will probably think of a tasty baked concoction, not its homonym, the Greek letter pi or π. Most people learn what pi means in math class as part of their secondary education. You may recall that, in geometry, it’s the constant ratio of a circle’s circumference to its diameter. We learn that this number can never be expressed as an integer, or as a digital value. Although pi can be expressed as an exact fraction (3 1/7), its decimal equivalent is 3.1416… followed by an infinite sequence of base-10 digits, making it an irrational number – a quantity incapable of being expressed as an integer. (I won’t attempt to summarize here the long history of pi’s use in mathematics, which is readily found in Wikipedia, with appropriate citations.)
A similar phenomenon occurs when we examine another well-known measurement that is assumed to be a constant, namely the speed of light. In my previous post (http://kenlesure.com/Blog/What_“Now”_Really_Means) , I mentioned what has been long accepted as a scientific fact: the speed of light has a constant value of approximately 186,000 miles per second. Notice that I said, “approximately”; in fact, the exact value of this famous constant has never been determined. Because if any unit of measure, whether it be a mile, meter, or second, is completely arbitrary, then it follows that any measure of the speed of light must also be arbitrary. To solve this conundrum, in 1983 the scientific community formally defined a meter (the world’s standard measure of distance) as the distance light travels in a second, namely the fraction 1/299,792,458. By solidifying c, a core element of Einstein’s famous equation, E = mc ², as nature’s most fundamental constant, its basic unit of measure became, paradoxically, a measurement that can be expressed only as an irrational number, like pi.
I won’t get into the intricacies of how properties of light were discovered and measured over the course of centuries. That has been well done by Brian Clegg in his readable and entertaining book, Light Years: The Extraordinary Story of Mankind’s Fascination with Light (Icon, 2015).
The similarity in the dilemmas of measuring the two most fundamental constants in physics and geometry has both intrigued and puzzled me for years. Lacking much formal training in either discipline. I questioned my own ability to solve it. This endeavor would be daunting, if not presumptuous, to say the least. Nonetheless, I couldn’t stop myself from engaging in thought experiments.
I had read in many places that thought experiments were Einstein’s m.o. – his modus operandi. His use of mathematical equations to express relativity and other natural phenomena was secondary; after all, an equation is nothing more than a symbolic expression of equivalence. In fact, he often got assistance from others, including his first wife, to help him “do the math.”
Einstein said that he arrived at the special theory of relativity on one of his daily streetcar rides to and from his work at the Zurich patent office. He began to imagine himself riding on a beam of light at its unimaginable speed. What would happen came to him as an epiphany that changed our thinking about everything – space, time, matter, and energy. Science had not even fully accepted the existence of atoms until Einstein’s theory of special relativity helped prove the equivalence of energy and matter, as expressed in the famous and elegantly simple equation E = mc ².
When I first read this story back in graduate school in the 1970’s, I tried numerous times to replicate the famous thought experiment for myself. One day I finally did it, I later convinced myself, although I didn’t understand it from a mathematical perspective. It was, however, an epiphany that carried me, in my mind, to another dimension.
Later, perhaps while staring at a vinyl phonograph record spinning as it played a now-forgotten song, I wondered where its exact center was. I thought, it has to be in the turntable’s spindle, but where is the center of that cylinder, around which the record revolves within its hole? There had to be center point around which the spindle revolved. How small a particle could this non-revolving center be? A nucleus of an atom? How could such a sub-atomic particle not revolve with the rest of the disc? Maybe the indeterminable nature of the center could explain why the exact value of pi is also indeterminable.
This is finally getting to the point, literally. What is a point but a “place” in space-time without a dimension? And with no starting point, this “place” is therefore immeasurable. Does that which is immeasurable even exist? Ergo, the value of pi, although constant, cannot be determined because the central point of a circle cannot be determined.
Besides the coincidence that both pi and c are irrational numbers, what else do these mathematical constants have in common? The answer is that they both take us to a new dimension, each in an opposite direction. If I were able to travel at the speed of light, the mass of my body would transform into pure energy. I would become light itself, stretching out into infinity until a force of gravity changed my direction or until I struck an object and was reflected elsewhere or was absorbed back into matter. In E = mc ², the numeral 2 represents the next power or dimension; geometrically speaking, the measure of one dimension times itself, as two feet times two feet are four square feet. Now imagine 186,000 miles per second squared! That’s an unimaginable number, especially considering that its exact value can’t be determined. It’s like trying to imagine our physical selves suddenly turning into pure energy. Some might say that’s like imagining entering another world, which is just what another dimension would be. I can’t think of anything more enigmatic than that, except maybe finding the true center of a circle or a cylinder or a disc. In a way. this is what experimental physicists are trying to find as they seek the smallest particles in existence at the Large Hadron Collider (LHC) near Geneva, Switzerland.
For me, then, pi is much more than a simple mathematical constant that we use in grade school calculations, only to become almost forgotten for the rest of our lives. Pi really is an enigma – a mystery that takes my mind to the very center of the universe, the same way that thinking about light takes me to its opposite end (as long as nothing gets in its way on its potential journey to oblivion).
[To be continued…]