The Numbers Game

People who think that ridding humanity of religion and such will remove the weirdness and episodes of random oddball wackiness from the world will probably continue to be disappointed as long as there's math around. Case in point is an irrational number that goes by the name phi or the Greek letter Φ. Like its more famous cousin π, Φ comes from a geometric relationship. It's a way to divide a line in such a way that the ratio of the two unequal pieces added together to the longer piece is the same as the larger piece to the smaller one. And like the other one, it never repeats and never ends, starting out as 1.618033 and going on from there.

The ancient Greeks calculated the number, called the Golden Ratio because of its aesthetically pleasing quality. But things started to get weird when Φ started showing up in nature. Such as the spiral shell of  a nautilus, which spiraled inward along the same ratio as the line. Botanists found it showing up in the distribution of leaves on a tree branch -- if not exactly, close enough often enough to be significant. Modern researchers have found it in different qualities on the molecular level.

Astrophysicist Mario Livio, in 2003's The Golden Ratio, reviews some of the places Φ is supposed to have shown up across history. He finds that in a lot of those cases, Φ's either not really there or the similarity to it is something of a coincidence. It probably didn't influence the construction of the pyramids or the way Da Vinci painted Mona Lisa, for example. But it does show up in enough different places to be weird enough.

The chapters get a little repetitive and it's possible that Livio could have dropped one or two suspected appearances of the Ratio that turned out to be incorrect. But he's a gifted science writer with a real knack for moving complicated concepts into the realm of lay understanding, and he leaves plenty of room for a readers to figure out for themselves what they think about the prevalence of Φ in the universe and why this particular mathematical expression shows up as often as it does in the real world.

-----

Irrationality in math is not the same as in other arenas. In fact, if the old definition of insanity is doing the same thing over and over again while expecting different results, mathematical irrationality is its opposite. Irrational numbers are ratios that can't be expressed as a fraction of integers -- they never repeat, no matter how many times the division in the ratio is carried out. Which is one wrinkle with them already -- how do we know a decimal never ends and never repeats? Simple math alone can't get us there; irrationality requires a mathematical proof of its existence.

Some irrationals are common and famous -- π as the ratio of a circle's circumference to its diameter is one. Some showed up when mathematicians got curious about what might happen when they played this or that game with numbers or equations, like the mathematical constant called e. In his 2017 book The Irrationals, mathematician Julian Havil offers some of the history of irrationals, first discovered by ancient Greek and some Hindu mathematicians. He explains how some of the better-known were first discovered and how new ones appear even in math today. The ability of computers and their ability to calculate immense strings of digits mean mathematicians are less sure than they used to be about the non-repeating aspect of irrationals -- they probably don't repeat, but there may be some wiggle room.

As in some of his other books Havil is not shy about using mathematical formulas and equations, many of which are blank space to people who didn't progress much beyond pre-calculus and have forgotten large swaths of that. It may be unavoidable but it's an unfortunate feature of what is a really interesting set of ideas about our weird ol' universe.