Last month was the annual celebration of Pi Day, an important holiday that’s enjoyed worldwide by math enthusiasts. Interestingly, people who develop severe allergic reactions to a simple 2+2=4 math problem also celebrate Pi Day because of free food.
The Pi Day of 2015 is uniquely significant, since 3/14/15, 9:26:53 corresponds to the first 10 digits of pi. Several studies have reported that hungry math devotees consumed over 10 billion tons of pie at that time.
Hardly a social person, I celebrated Pi Day by treating myself with a simple cherry pie and a glass of cherry wine. Surrounded by a gigantic library of esoteric math books I gathered over the course of my studies, I deeply contemplated the history behind pi.
At 3/14/15 9:26:53, I was struck by a troubling epiphany. The pie I was eating was circular. Circles, like triangles and rectangles, are simple and elegant shapes. So much aesthetic value can be realized just by looking at a circle.
What do circles have to do with pi? By definition, pi is defined as the ratio of circumference to diameter. Who cares about diameters? Usually, pies are cut only up to the center, so radii are important. Hence, the ratio of circumference to radius should be used, and this ratio is simply two times pi.
Moving once around a circle is called a revolution. Thanks to an arbitrary system invented by Babylonian astronomers, a revolution is equal to 360 degrees. A degree is equal to pi divided by 180 radians, which implies that a revolution is equal to two times pi radians.
That doesn’t make sense! Why should pie be called pi if a revolution around pie is only two pi radians? Why is a quadrant marked by pi divided by two radians, instead of pi divided by 4 radians? Both problems constitute the so-called Pi-Pie Paradox, indicating a rather uncomfortable coexistence between the two great wonders.
The Pi-Pie Paradox can be resolved in two ways. The first method is to offer pies as semicircle-shaped desserts. However, semicircles are aesthetically unsuitable, and there have been cases whereby sharp edges and pointy corners were used as lethal weapons.
Semicircles aren’t the only option. Pies can be triangular as well, since the sum of the angles inside a triangle is 180 degrees, or pi radians. Triangles are always convex, and can be constructed from any three noncollinear points. Many important solids, like tetrahedrons and octahedrons, contain triangular faces.
Yet, there’s a caveat: Triangles aren’t aesthetically pleasing. The only beautiful shape as declared by Euclid is the rectangle. After all, four is the only number that has the same number of characters as its value. The most important solid is the cube, which consists of square faces.
The sum of the angles inside a rectangle is 360 degrees, or two pi radians. Curiously, the sum of the angles outside any convex polygon is also 360 degrees, or two pi radians.
Consequently, the Pi-Pie Paradox can be resolved geometrically only by shaping pies as circles or rectangles. Naturally, this is a problem since both shapes claim two pi to be a superior value, which brings the centuries-old tradition of pi into question.
The second method resolves the paradox by replacing pi with tau, which is equal to two pi. In addition to its geometric significance, tau is used popularly in factorial approximations, normal distributions, error functions, coordinate conversions, Fourier transforms, Planck’s constant, and Maxwell’s equations.
Understandably, there is a popular opposition to banning the lovable pi, declaring that the area of a circle will always be elegantly expressed as A=π×r^2, instead of the ugly A=½×τ×r^2. On the contrary, the ½ term is fascinating, since it connects between integrals and areas of triangles.
From a non-mathematical viewpoint, hungry critics argue that shifting from Pi Day to Tau Day will eliminate the serving of pies. Clearly, this is misleading, since everyone will be offered twice as much pie on June 28!
Besides, who has time to celebrate Pi Day on the weekend before final exams? Instead, we can relax and enjoy the warm sunny weather on Tau Day, while eating twice as much pie.