When a Falcon attacks its prey, it swoops in along a path that is mathematically related to the golden ratio.
In his book Elements, Euclid explains the golden ratio and how to divide a straight line running between points A and B into two by a point P so that the ratio of the longer segment (AP) to the shorter one (PB) is exactly the same as the ratio of the entire line (AB) to the longer segment (AP).
A P B
x 1
In symbols:
AB = AP
AP PB
It doesn’t matter if the actual length of the line AB is one foot, one meter, or one shoelace length. It’s the ratios that count. So let’s say the length of PB is 1 to simplify our discussion.
With PB = 1, the length (x) of AP is what we now call the golden ratio. To calculate its value, we have to do a bit of algebra. The length of AB will be x + 1. This means we can rewrite the above geometric identity as the equation
x + 1 = x
x 1
This can be rearranged by cross multiplying to give 1 (x + 1) = (x) x, which becomes x + 1 = x^{2}. We can then rearrange this to give the quadratic equation x^{2} – x – 1 = 0.
If you think back to your college algebra class, quadratic equations have two solutions, and there is a formula to give you those solutions. When you apply this formula to the above equation, you get the two answers:
x = 1 + √5 and x = 1 - √5
2 2
Using a calculator to three decimal places, the answers are 1.618 and –0.618, respectively. The golden ratio, φ, is the first of these two solutions—the positive number.
You start to suspect there’s more to φ than meets the eye when you ask what happened to the negative solution to the quadratic equation, –0.618, which also goes on forever as a decimal. Apart from the minus sign, it looks the same as the first solution (φ) but with the initial 1 missing. But that turns out to be a false lead. Calculate a few more decimals and you will see that the two numbers are not the same. But if you dig a bit deeper still, you will find a surprising identity. The negative solution is equal to 1 – 1/φ. Hmmm. That doesn’t usually happen with quadratic equations.
So in the forthcoming UAAP season..beware of the Adamson Soaring Falcon’s formidable attack! Unawakanahimo!
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