| Where Recherche duTemps Perdu
---- meets Kirchliche Dogmatik
Yesterday was a gorgeous day in Hoosierland! Blue sky, sun shine, no need for an air conditioner at the moment. Today not so much. It’s supposed to get very hot again, so I’m glad we were able to make the arrangements to fund a new system. D. has already removed the old unit, will purchase the new machinery tomorrow on our behalf, and have it in place, hopefully, by the end of the week. --- Well, today that didn't happen. The funding is not yet showing up in our bank account. Hopefully it will be there tomorrow, so we can get the show on the road.
I’m going to continue with my discussion of phi. If this is the first time you happen upon my blog in a while, you may be a little confused as to the topic, particularly coming into the middle of an ongoing series. So, please know that the first four entries, which go all the way back to last year, are collated into a single document, to which I will add this one and the last two as soon as I can, so that it’s all in one place, and you don’t have to find your way backwards through my blog archive in order to read what came before.
Allow me to revisit what we’re doing here. We’re examining the nature of phi (1.61803…), the golden ratio constant, and for the moment we’re doing so as far away as we can get from the Fibonacci series and its exemplifications in art or nature. My point is to try to demonstrate its inherent elegance, which stems from its Creator just as much as the arrangement of flower petals and other “fibonaccied” items in nature.
Let’s go back to the question of two days ago: How contrived is the golden ratio? For example, take the golden triangle, which includes phi as an important ratio. Is it something that has been concocted just to show off phi, or is the golden triangle something that we can uncover and discover in other places. A partial response has been that it is included in the formation of a regular pentagon, and, thus, can be discovered without being guilty of fabricating an artificial instantiation just to smuggle in phi.
Phi is also found apart from geometry in number theory, which we can combine with a little algebra. Remember that phi is just one member of the infinity of all real numbers, which, as demonstrated by Georg Cantor, has turned out to be larger than some other infinities. However, it stands out from this uncountable crowd, right next to p, e, i, √2, and a few others, due to its special properties or notoriety.
Here are two ways of finding phi without geometry. We will take a look at a couple of somewhat unusual-looking formulas and turn them into equations, which will resolve into phi. Here is the key: We came up with the formula for phi by setting the length of the base of a golden triangle as 1, and one of its sides as x. We were able to manipulate those numbers into a quadratic equation
x2 -x -1 = 0
The method for finding phi tonight is going to consist of finding several equations that can also be rearranged into the same quadratic equations. I found the following two equations in Livio.
Let’s start with a formula that consists of an endless fusion of square roots. Given:
By the way, this thing is a formula, not an equation. You can only have an equation if there are two (or more things) that are considered to be equal, as, e.g., in
It would be extremely tedious to work out a value for x from this equation if we want to continue to add the square roots of square roots, etc. But there is an easier way to do so.
We can square both sides. Squaring x gives us x2, and squaring the formula to the right resolves the first square root into a 1.
Now it is apparent that the collection of square roots that follows after “1 +” is still identical to the original given formula since both extend to infinity,
and, what’s more, we have already designated it as x above.
Then, substituting x for that nest of square roots, we get
which we can reformulate to fit the pattern we had looked for as
This is, of course, the formula which has phi as one of its solutions. Just think: we have derived it from that unwieldy collection of square roots.
So, now you feel like the Lone Ranger and want to save the world from more confusing formulas. That’s great. I’m with you all the way. Let’s try this monstrosity, which in mathematical argot is counted among a large group of “continuing fractions.” We start with a 1 and add the fraction, which has 1 for its first numerator and is followed by a never-ending, always-repeating denominator. Given:
Again, before we do anything else we need to turn this formula into an equation and label it with the variable x.
As you seek something wonderful in this equation, may I call your attention to the section underneath the topmost numerator of 1.
As in the previous equation, what you find there is actually identical to the originally given formula. There is no difference because both of them can be extended to infinity. And thus, we can safely apply the same letter variable, x to this new continuing fraction, which is also the old one. Remember now, that the area we have marked out is the denominator, and that the numerator of 1 still stands as before.
So, once we have substituted x for all of that clumsy denominator, we get a refreshingly simple equation.
Let’s get rid of the fraction 1/x by multiplying every term by x,
Then we have
and once more we can rearrange this equation into our favorite configuration:
Once again, we have turned a beast (the continuing fraction) into a beauty (phi). We have stumbled on yet another way of deriving phi without getting out our rulers or measuring tapes.