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Monday, June 6th 2016


Direct from the Pentagon


This entry comes to you from a state park in northern Indiana, where we are spending a few days. In order to make sure I get this installment out and start to establish a regular pattern again, I’ll save our adventures here so far for a later time.

Some Quick Catching Up

With regard to our health in general it’s been up and down for both of us, slightly down for June due to her fibromyalgia, slightly up for me because the expected neurological symptoms are not progressing as expected, and I’m extremely thankful for that fact.

If you’ve been seeing my various entries on Facebook, you know that I’ve been putting in a lot of time making music on StreetJelly.com as well as honing my drawing skills a little bit, the latter mostly for the sake of various websites. Long-time readers know that some of the websites I work on require use of pseudonyms to protect our readers (I’ll be happy to explain that idea if it doesn’t make sense to you), so I’m afraid that I can’t just provide a link from here under my name in order to show off my creativity, such as it is.  

I might mention that, during this blog hiatus I have made one new video, namely the Buddhist story of “The Death of Kisa Gotami’s Son.” 

I’m in the process of producing a Christian commentary video to go with that narrative. The graphics for it are already pretty much in place, and I just need to put together its narration.

After writing the entries on militant Islam, I was really beginning to feel burned out with the blog. I had no intention of dropping it then, and I still don’t, but I felt that I really needed a lengthy hiatus. Contributing to that sentiment was the fact that the various issues in the air, e.g., the presidential races and other social hot buttons have been matters on which I’d just as soon not get too caught up in the never-ending discussions, except for the occasional zinger on FB. (Please note: I didn’t say “mindless chatter,” because not all of it is, but much of the current discussions run in vicious circles that seem to do little more than to generate a lot of unwanted centrifugal force.)

On Thursday our air conditioning went out. Apparently, the motor fused, the wires burned out, and there’s nothing much to salvage on the outside unit, which is more than 20 years old. Since the temperatures have been around 90, this was not a good thing at all. D. of Riteway Heating and Cooling has informed us that replacing the outside unit (which is unavoidable) involves replacing the entire system, due to the prevailing rules of the EPA.  So, as mentioned, at this moment we are in a state park in Indiana, this time a couple of hours north of Smalltown, USA, waiting for the good folks of RHC to find a good replacement at a do-able price.


And now back to phi. Wow! That was a long time ago. Yes, indeed, but I had never intended to leave off where I did. I collected the previous entries on this topic, going all the way back to last November, in one single website, entitled PHI—Let’s Get It Right­­­­, and I highly recommend that you at least look very quickly at what I said earlier on the topic, so that this new addition will make more sense to you. I left off with this question: It’s all very well to stipulate a “golden triangle,” viz. one in which the proportion of one of its sides to its base is equal to the proportion of the sum of the base and the side to the side. 

golden triangle proportion

And it’s great that this proportion is precisely phi with all of the properties that phi has to offer. But is this geometric figure anything other than something concocted by an ancient mathematician for the entertainment of his guests on long November evenings? Can we find the Golden Triangle somewhere where it is right in place, playing a significant role in geometry? Let's go to the Pentagon to find out!

Let me issue an insincere apology to anyone who may have been misled by the term “Pentagon.” I was, of course, not referring to the building that houses the U.S. Department of Defense, and so I sup­­pose I should not have capitalized “pentagon.” Could it be that I was deliberately creating an ambiguity to get people to click on my blog?


Not this Pentagon!

As long as I’m pretending to issue apologies, let me hand out another one together with a rain check for the Fibonacci series, which figures so prominently in discussions on this topic. It has its own formula, Fn = Fn-1 + Fn-2, which is just a compact definition of the Fibonacci numbers and will tell us nothing new once I’ve explained the series. There also is a formula that we can use to determine directly the value of a specific Fibonacci number at any given point in the series, but to do so, we already need to know the value of phi and its negative reciprocal. As I continue to insist, as amazing as the Fibonacci series it, it yields phi only by convergence, not directly. Nothing wrong with convergence, but it’s not what I’m after at this time.

So, instead, let’s look at a pentagon in geometry. It’s easy to draw one with modern software programs; let’s remember, though, that in geometry anything that we draw will always be approximations. On the one hand, that means that you can’t just solve a geometric problem by measuring the lines. On the other hand, we need not attempt to go beyond normal human abilities in drawing our figures. And if­­­­ they turn out a little lop-sided, that’s okay, they’re only attempts to illustrate something that some aspects of reality cannot be reproduced by a sketch with paper & pencil sketch or a computer program.

So, let us think of a regular pentagon, viz. a two-dimensional figure with five equal sides.

pentagon/golden triangle

Now let us bisect two side-by-side angles. For convenience, we’ll go with the bottom two in the drawing, those that are located at points C and D. They converge at point A.

pentagon/golden triangle

Surprise! We have found the triangle ADC, and it turns out to be a golden triangle. (I’m skipping how that fact is derived from calculating the angles involved. See Livio, pp.  if you’d like more on that topic.)

pentagon/golden triangle

golden ratio

which comes out to phi on both sides of the “equals” sign. But that’s not all. Let’s take one of the triangle’s base angles, bisect it and run a straight line to the edge of the new triangle. We get a new triangle, DGC, and it, too, has the coveted “golden” proportions, as indicated by the number phi.

pentagon/golden triangle


No reason to stop there. Go ahead and bisect another base angle of the latest triangle, and welcome another golden triangle (GHC) into the family.

pentagon/golden triangle

We could go on and continue the process, but let’s go no further and celebrate our new discoveries. 1) We have found that a golden triangle is not just something created ad hoc for the sake of accommodating phi, but it is a direct property of a regular pentagon. 2) A golden triangle gives rise to further, but smaller, golden triangles each time one draws a straight line bisecting one of the base angles.

It’s not just the golden triangle that has the property of reproducing similar smaller versions of itself. Next time, we’ll look at the golden rectangle and take our first outing into the physical world by considering the chambered nautilus.


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