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The Noisy Ratios That Keep Math Wild

A brief primer on irrational numbers.
November 9, 2014, 2:00pm
Image: Anders Sandberg/Flickr

Irrationality was born in a pentagram. A star.

The ancient Greek followers of the mathematician Pythagoras kept the shape as their secret symbol, a signifier of wholeness or divine blessing. Nature.

The pentagram was a symbol, but also a window of sorts. It's not a shape so much as it is a relationship. Within it we find the Golden Ratio, which is this thing:

The Ratio is noted for both its seemingly fundamental aesthetic appeal and its (seemingly) persistent recurrence in natural and manmade systems, from plant biology to financial markets to human DNA. But it's really just a way of cutting up a line.

It's a way of cutting up a line that appears to be preferred by nature, anyhow. Note that every time a line crosses another line, as in the sketch below, it results in a Golden Ratio relationship between the two resulting segments. So, follow a pentagram edge down until it hits the start of another point; that corner is a Golden joint of sorts.

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It's possible to connect the points of a pentagram to come up with a new pentagon and, from the new pentagon, a new pentagram. And this can go on forever:

Presumably, the Pythagoreans spent a whole lot of time staring deeply into pentagrams. Tradition has it that a young Pythagorean, Hippasus of Metapontum, was doing just that when he happened up the first proof of the existence of irrational numbers, a startling revelation for the "all is number" crowd.

Hippasus noted what you see above, that the inner core of a pentagram is a pentagon, and inside of that could be found a new pentagram. Clearly, this went on pretty much forever. And it had to, which is what was eventually proved by the Greek.

Imagine Hippasus gazing into the trippy arrangement of nested pentagrams above. Each one lives within a pentagon, with its dimensions limited by that pentagon. All of these shapes clearly live within some hard and fast relationship; they define each other. That relationship is what we'd normally think of as a ratio. One shape is to the other as some quantity a is to some quantity b.

So: if you were to make the pentagon bigger, the line segments running point to point grow too. We would expect that relationship to be a ratio. If a pentagon side x increases by some amount a, then we would expect the point-to-point segment y to increase by some b. It would look like this: x/y = a/b. Changing x by a means changing y by b. That seems natural.

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Given the Golden Ratio, however, this relationship doesn't exist. That is, if you change x by a, y is going to be changed by some b, but that b will always be just a little different. Some shred of line segment is always left over to be divided again.

To put another way, imagine a line cut up according to the Golden Ratio. AC becomes AB and BC. There does not exist a whole number subsegment of AC that can divide both AB and BC. We can find a whole number that's pretty close, but never close enough. We'll keep trying, as Hippasus surely did, trying to divide the segments by smaller and smaller subsegments, hoping to find, finally, a ratio of x subsegments to y subsegments that can describe what is really just an irrational number.

That irrational number, the ratio also known as phi, is this: 1.6180339887498948482… The digits, like those of pi, keep going forever randomly. The relationship between a and b that we're looking for is that a is phi times b, which will always be some other number with infinite random digits sailing off into the infinitesimal and vague.

Pi is the most famous irrational number, of course. There's no "clean" ratio between a circle's radius and its circumference, though we can get close enough with it to smash protons in a particle accelerator and bounce spacecraft around the moons of Saturn. Smashing particles, sure, dividing a circle evenly, don't even think about.

Irrational numbers are hardly special cases. There is at least one between every rational number. And there's even a good argument that there are vastly more irrational numbers than rational numbers. In decimal terms, just think about how many more possibilities there are for a number to just keep going and repeating randomly compared to stopping at a set value.

Imagine writing down some decimal number, with each new digit to the right being a random choice from 0 to 9. As determined by a 10-sided dice roll, how likely is it that you'll get a bunch of 0s (ending the number), compared to 1s and 2s and 3s, etc.? Not very. Numbers would seem to prefer uncertainty.

Contrary to the beliefs of the Pythagoreans, the world and its numbers are messy, or at least imperfect. Nature prefers harmony, but somehow the odds still suggest otherwise.