The Wrong Circle Constant
“Mathematics possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture” – Bertrand Russell. Mathematics is elegant and beautiful, as well as logical and dignified. Take Newton’s second law of motion: F=ma. Simply stunning. It is clean, efficient, and graceful. One does not become confused at the sight of it; there is no instance where it would hinder the speed, efficiency, and progress of the person using it. Mathematics should aspire to be like this. However, there are some parts of mathematics that, for some odd reason, have remained imperfect and unchanged for decades, centuries, millennia even. To this day, there remains one that stands out more than any other; one that has persisted, despite its deficiency; one that has not only stood the test of time, but one that has gained the love of millions. A constant belonging to the most perfect of shapes: the circle. Is it not ironic that the shape we have considered to be the most flawless of shapes is one that is most directly connected a substandard geometric ratio? Perhaps you have heard of it? It goes a little like this: 3.14159265359...π – Pi. Now, I do not mean to imply that it is incorrect in any way, only that it is deeply flawed and simply inefficient to use. To me, and many others around the world, it is outrageous to think that a circle, of all things, has such an imperfect geometric constant. Pi simply does not live up to such a level of perfection; however all is not lost, as there is one constant that does: τ – tau. Firstly, let us take a look at pi’s fundamental fault: its imperfection as the circle constant. As all of us know, pi is the ratio between the circumference and diameter of a circle, thereby making the circumference equal to 2pi multiplied by the radius – or the original pi multiplied by the diameter. But think about it; is a circle made using its diameter? When you use a compass, do you not start at single point and wheel it around until it has created a circle? The making of a circle has nothing to do with the diameter. Without a doubt, it is the radius that is the most important, don’t you think? The radius is the fundamental length that determines the circumference of a circle. Defining a circle by the ratio of the radius makes much more sense. Sure enough, when you take the ratio between the circumference and radius of a circle, it gives you 2pi: 6.28318530717... otherwise known as tau. This makes the circumference the same as tau multiplied by the radius. Going back a few thousand years, looking at Archimedes’ original proof, he actually found that there were two numbers that could represent the circle constant: pi and tau. To him, they were both great candidates. He really could have chosen to use either of them. Unfortunately, he chose pi. Ever since then, people have been using pi and neglecting the much better constant, tau. I believe that it is time for that to change. I understand what you might be thinking: isn’t pi integral to other equations such as the area of a circle? A=pi r2. This is an important formula, proven by Archimedes himself. Pi is, after all, by itself. This is a good argument. Nevertheless, I am not concerned. In fact, the original proof of Archimedes shows no that the area of a circle is pi r2, as back then there was no algebraic notation, but that the area of a circle is the same as the area of a right triangle with the height equal to the radius, and base equal to the circumference. As we know, the area of a triangle is equal to half of the base multiplied by the height, ½bh. This means that the area of a circle is equal to half of the circumference multiplied by the radius, ½Cr. Recall that circumference is pi multiplied by the diameter; therefore the original area of a circle formula is A=½pi Dr. How ugly. Now, recall that circumference is tau multiplied by the radius; therefore the area of a circle is A=½tau r2. It is immediately made into a
