Victus Spiritus


Feel Irrational? So does Pi.

04 Dec 2009

A Final Sunset

Apples and Oranges

Neither are perfect spheres but in our rush to understand we sometimes cluster them. After all they're both fruit and somewhat round. Nor is the sun beautifully captured above a perfect sphere, but we can roughly approximate it as one (it's a complex ellipsoid like earth).

Lets take this topic 2D for the sake of simple relationships. The circumference of a circle is 2*Pi*R. Sounds reasonable, there's a "constant" relationship between the radius of a circle and the distance around it's exterior (the area of the circle is also Pi*R^2). The problem comes in nailing down the constant. We can't label it with an exact value. Calculate as many decimals as you like, and you're still no closer to completing it's numerical description. There's one heckuva backstory on the history of Pi and irrational numbers, none of which can be described by the ratio of two integers.

If a "numerical value" requires infinite precision to represent, maybe that's a hint that we're trying to describe it with the wrong language. The notion of Pi obviously resists characterization into our pleasant base 10 decimal system. It's like a ghost in our numerical language, and we're unable to pin it down.

Another concept e (natural number whos slope matches its value at any point) also defies classification numerically. At each attempt to observe or measure e with decimals we find our efforts wanting.

A perfect circle

Yes, I too love their music but we're talking geometry.

The origin of Pi is based on a perfect circle. There are no piece wise linear segments in a perfect circle. It's constantly curving, smoothly no matter how far you zoom in. Its direction is infinitely changing. Reminds me a little of a fractal, with identical recreated patterns as we zoom in.

Mathematics allows us to represent a circle as an infinite sum of linear infinetesimal segments. We can show that the features of our infinitesimal constructed circle matches that of a perfect circle. We can even write software that iteratively creates an approximation to a perfect circle at a desired resolution up to numerical limits.

For generated (physical/numerical) circles we can always zoom in far enough to make a segment appear linear. Such a simple and elegant shape resists our ability to recreate it absolutely. But our mind can envision the perfect circle based on it's features.

Identify Irrational Components Outside of Mathematics

The misconception or mixing of rational and irrational values extends far beyond mathematics. It may be worth our time to identify anomalies in any work we do, to explore the fundamental truth of their nature. Differentiation based on discriminating features allows us to better understand our experiences.

Seeing patterns and analogies is a powerful tool to begin learning, but to dig deeper we should pay careful attention to the subtle differences of each challenge.