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Tuesday, June 7th 2016


Golden Rectangles


I’ll never be able to maintain a regular blog as well as take care of other things if I keep making myself include all kinds of diagrams and drawings. Still, it is fun, and for most of us a picture paints a thousand words—which is not to say that people who draw a picture necessarily forego an additional thousand words (ca. 1050 for this post). I started this entry yesterday (Monday), while still in Pokagon State Park. It was a good day, too. I had my first trail ride on horseback of the year, and June and I spent lots of time outside and swimming in the pool. Now we just got back to Smalltown, USA, and we need to see what we can do to get the new a/c system installed. Also, of course, I’m trying to finish this entry.

Pokagon Swan

Before returning to phi allow me to mention that my next StreetJelly date will be my regular set on Thursday evening at 9 pm EDT. I had to cancel last week due to circumstances that weren’t entirely beyond my control. After all, I was not forced to go to the dentist. Regardless, I see no impedance for this week. I’m planning on doing a show of some of my original songs. That’s at StreetJelly.com.

Please let me remind you that all of the earlier discussions on phi (except the last one—for now) are collected in a single site so that you can read the previous sections in a sequence that makes sense.

I left off last night by showing that phi manifests itself when you bisect two adjacent angles of a pentagon and, thereby, create a “golden triangle.” Then we were able to give birth to more and more golden triangles of diminishing size by bisecting one of their base angles each time. There is also a process that gives rise to new generations of golden rectangles, as we shall see below.

golden rectangle

By now I’m sure you have figured out why this rectangle should be golden: the ratio of the longer side AB to the shorter side AD is the same as the combined sides AB+AD to the larger side AB.

Now, we can lop of a square of length AD from the one of the ends rectangle, and we have a smaller rectangle left. I have placed the square on the left side of the rectangle. There is no rule governing that placement, nor can there be, since one can always flip the figure without doing it any damage. I’m placing my squares so that I can use the ongoing generation of golden rectangles to make a specific point in a short while.  

golden rectangle

The remaining rectangle (EBCF) now has the golden proportions. Let’s continue the process and remove another square designated by EBHG, and we have produced yet another golden rectangle, answering to the name of GHFC.

golden rectangle

Are we done now? Only if you want to be. We can remove another square and enjoy the sight of golden rectangle GIJF.

golden rectangle

And let’s do one more and call it GILK.

golden rectangle

And so forth … This is another unique treat that phi brings to us: We can go on and on bringing out golden rectangles by removing squares from one of its side.

Let us now reverse this process by starting out with the smallest golden rectangle and adding squares to it so as to create a newer, larger one, which will yield another golden rectangle by means of the same procedure.

golden rectangle

It is at this point that the placement of the square takes on significance. If I were to continue the enlargement procedure indefinitely according to my pattern, my arrangement will give us a spiral. In order to turn the tiniest of our rectangles into the next largest size, we’ll put a square underneath it. To reach the next size, we can place our square to its right. Moving on the next larger one, we can place the square on top of the one we have. Finally, to reach the largest size with which we began, we can expand it by means of a square on the left. Again, there is no point at which we have to stop, except for intrusions into our mathematical world, such as lack of available bandwidth, old age, or boredom. What you see is the beginning of a spiral. If we were to continue the process, the sequence can continue with the pattern of adding squares: down, right, up, left. Each time we get a new rectangle, it’s a golden one, and each one stands in proportion to all of the other by multiples of phi, and the spiral, that we call a "Golden Spiral" grows.

Here is a golden spiral from Wolfram Alpha.  I chopped it up and turned it into an animation.

growth of golden spiral

Are we still in math or, more specifically, geometry? Yes, we are. However, having shown you these constructions, I cannot forego mentioning one example of where we some people believe that they can see such a building process in action in nature, viz. an increase by a factor of phi in each particular stage.

The item in question is an animal belonging to a group of mollusks called "cephalopods"  [“feet on the head”], known among his friends and family (e.g. the octopus) as the chambered nautilus. This animal sets out its life in a rather small shell, but as it grows older and bigger, it manufactures larger chambers for its comfort and convenience—modern living for mollusks! A common belief is that this growth occurs according to the golden ratio. Unfortunately for ardent enthusiasts of the golden ratio, this is not so. It’s really a shame because once upon a time even your bloggist, who holds a B.S. in zoology, had been a victim of the same misconception. Gary B. Meisner, who calls himself the “phi-guy” and maintains a quite sizable website devoted to the golden ratio, explores a number of ways in which phi could be found as part of the proportions in the chambered nautilus, and leaves us with the ambivalent answer that, as yet, no nautilus that fits the pattern has been discovered, though it could not be ruled out that it will. For my purposes (and Meisner’s, if I read him correctly), that’s not enough to marvel at the golden ratio in nautili, though it leaves plenty of other things about which to marvel.

Chambered Nautilus

There are other, much better, examples in nature, but I’m once again postponing writing about those. There’s still too much I want to share with you about phi itself and where it shows up in its own world, the cosmos of numbers and formulas.


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