Saturday, March 14, 2015

Pi Day 2015

     3-14-15 9:26:53 has come and gone. I felt nary a ripple in the cosmos nor in my mathematical ethos. It was a moment in time neither momentous nor irrational.
     I thought about buying a t-shirt for a moment. It would be the logical nerdy thing to do. I might have actually done had Pi Day fallen on a day that I was actually teaching. That it fell on a Saturday and that the t-shirt I liked cost $23 yielding -2 utils for this proposition. I may, however, e-mail the company in a week or so, April 1 seems quite appropriate, and see if they would sell me one for, I don’t know, let’s say $5.
     There is a lot of friendly and fun hype about Pi Day. There are contests at various universities where folks compete to see how many digits of this most famous irrational number and be memorized and reciting. Oh my… golly gee whiz… that sounds like fun. Buying a pumpkin or blueberry pie with Pi carved in the crust and sharing it with a few people is more my speed. Even better yet, I could just blog about it.
     Why Pi? It is probably the most popular irrational number. It is probably one of the Greek letters most people are familiar with. It is fun and 3-14 is logical day to celebrate it as a prelude to Green Beer Day and the First Day of Spring. It is even cooler because 3-14 is Albert Einstein’s birthday. So, why not have a little fun with it? We love cool dates, especially like 3-14-15 9:26:53, that are once in a lifetime events. In fact, I have blogged about this phenomena already: 12-12-12 Come and Gone - Now What?
     Irrational numbers are so named because they are not rational. This has nothing to do with how easily the numbers can or cannot be reasoned with. The key part of both terms is “ratio.” A rational number is any number that can be expressed as the ratio or fraction of two whole numbers. When converted to decimals, these numbers have a finite number or digits or are repeating e.g. 1/2 = .5 or 1/3 = .33333… An irrational number cannot be expressed as a ratio of two whole numbers and when expressed at a decimal goes on infinitely and never repeats. Pi is such a number. It is the most famous irrational number.
     There is, of course, a website: They provide one million digits of the number. They also, in the Learn About Pi page, provide the following:

Pi (π) is the ratio of a circle’s circumference to its diameter. Pi is a constant number, meaning that for all circles of any size, Pi will be the same. The diameter of a circle is the distance from edge to edge, measuring straight through the center. The circumference of a circle is the distance around.
By measuring circular objects, it has always turned out that a circle is a little more than 3 times its width around. In the Old Testament of the Bible (1 Kings 7:23), a circular pool is referred to as being 30 cubits around, and 10 cubits across. The mathematician Archimedes used polygons with many sides to approximate circles and determined that Pi was approximately 22/7. The symbol (Greek letter “π”) was first used in 1706 by William Jones. A ‘p’ was chosen for ‘perimeter’ of circles, and the use of π became popular after it was adopted by the Swiss mathematician Leonhard Euler in 1737. In recent years, Pi has been calculated to over one trillion digits past its decimal. Only 39 digits past the decimal are needed to accurately calculate the spherical volume of our entire universe, but because of Pi’s infinite & patternless nature, it’s a fun challenge to memorize, and to computationally calculate more and more digits.
     There are, at the least, two other irrational numbers worth noting: The Golden Ratio, square root of 2 and the natural exponential “e.”
     The Golden ratio (= 1.618…) is the ratio of the longest side of a Golden Rectangle divided by the shorter side:

Golden Rectangle and Ratio
The Golden rectangle has been known since antiquity as one having a pleasing shape, and is frequently found in art and architecture as a rectangular shape that seems 'right' to the eye. It is mentioned in Euclid's Elements and was known to artists and philosophers such as Leonardo da Vinci. 
One of the interesting properties of the golden rectangle is that if you cut off a square section whose side is equal to the shortest side, the piece that remains is also a golden rectangle.
International Paper System
Sequence to calculate e
The Golden Ratio is approached as the ratio of two successive Fibonacci numbers.
      The European paper system is based on the basic principle that if you halve the rectangle along the longest edge, the two halves are the dimension of the next size down paper. The ratio of the longest side divided by the shorter side is the square root of 2 (= 1.414…).
     For me, e = 2.718 is the coolest of them all. The exponential function with base e is it’s own derivative. It is central to most statistical distributions notably the normal or bell shaped distribution. It is used in most life data and reliability distributions including the continuous growth formula for bacteria.
     When I think about it, irrational numbers seem crucial to the design of the universe as we perceive it.

     Happy Pi Day everyone.

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