Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

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Recognize x∧2 – x – 1 = 0?

Posted by mark schwartz on February 12, 2017


If you recognize the equation in the title then you are already familiar with the golden ratio of 1.618. There are a number of books that explore the occurrence of the golden ratio in nature, architecture, the pyramids, art, aesthetics and a variety of other suspected places. I’d like to explore something about 1.618 that I’ve not seen in any books; just something to add to the mythology of it. If you aren’t that familiar with it, just Google it and have fun.

The Story

When I was teaching a pre-service teacher’s class about math concepts, I introduced the golden ratio because it was a nice vehicle for demonstrating how a pattern can show up in a lot of places. I talked a lot about teaching students to explore for patterns because math is saturated with patterns and relationships.

I asked if anyone in the class was involved in mysticism, the occult and if they would feel comfortable talking to us about it. There was one person who spoke up. I directed the conversation to focus on the pentagram (a five pointed star inscribed in a circle) because I knew that it was a significant symbol. I asked her to talk about the significance of the pentagram and of all the stuff she talked about, the golden ratio wasn’t mentioned. I mention it because it’s a somewhat invisible yet significant characteristic of the pentagram. Let me start with the star.

From here on, you will be asked to do a lot of ‘construction’ but it doesn’t have to be elegant or perfect and hopefully my directions can be easily followed. I just believe that your participating – as I had my students do – will be entertaining and educational, not drudgery. There is some geometry and trigonometry involved so if you don’t recall stuff, I’ll provide the information needed to do the work.

You can ‘construct’ a five-pointed star, starting by sketching a circle. Don’t make it too small because some of the points and lines will need room to be labeled. When you get done, you will have constructed a pentagram.

Starting at 12 o’clock, place 5 equidistant points on the circle. Starting with the point at 12 o’clock, label that one A and then going clockwise label the others B, C, D, and E. Now, draw a straight line from A to C, then a line from B to D, and continue with C to E, D to A and E to B. You should have a somewhat respectable looking 5 point star. I want to highlight that this pentagram has a pentagon in the middle and each face of the pentagon has a triangle on it. Take the triangle at the top that has the vertex A and label the other two vertices in that triangle F and G. Still with me?

Now if your pentagram were a perfectly crafted one, the pentagon and each triangle would appear to be equilateral. The pentagon would be but not the triangles! What? – they look equilateral. How can we determine that they are not?

To do this we need to know something about the angles in triangle AFG. Allowing that AF and AG are equal to 1 (we could pick any arbitrary value and the outcome of this exercise wouldn’t change; it’s just that 1 simplifies the calculations). Geometry allows us to determine that angle A equals 36 degrees. Now drop a perpendicular from vertex A to side FG, labeling that point on FG as H. Why? Well, we just formed a right triangle AFH which allows us to use some trigonometry to determine the length of FH. The perpendicular also bisects angle A, so angle FAH is 18 degrees. The sine of this angle is FH over 1, or just FH. The sine of 18 degrees is .809 – the length of FH – so the length of FG is .618, thus demonstrating that this triangle and all the triangles in the pentagram are not equilateral.

Given this, the perimeter of triangle AFG is 1 + 1 + .618, or 2.618. If you’re not familiar with the golden ratio, let me note a few relationships. First, the square root of 2.618 is 1.618 and clearly 1.618-squared is 2.618. And, as trivial as it seems, you can see that 1 + 1.618 equals 2.618.

And now the point of this exercise! Recall that the title of this article is x2 – x – 1 = 0. If you solve this equation, you get 1.618. But substituting 2.618 for x2, 1.618 for x you now have 2.618 – 1.618 – 1 = 0, which is a true statement for the perimeter of the triangle. So, triangle AFG and all the other triangles in this pentagram contain the golden ratio. But, there’s more.

Look at the pentagon in the pentagram. I won’t ask you to do all the construction (you can do it if you like) but I’ll step through it to show that the pentagon also contains the golden ratio. Briefly, starting with point F and moving clockwise, the vertices of the pentagon are G, I, J, K. Drawing a straight line from K to G, forming triangle KFG and assigning KF and FG a value of 1, what is the value of KG? As with the star construction, geometry lets us know that angle KFG is 108 degrees, and a perpendicular dropped from F to KG (label this point M) bisects the angle, and results in a right triangle KFM with angle KFM equal to 54 degrees. Again, trigonometry lets us determine that the length of KM is .809, and thus the length of KG is 1.618.

So, what is true for each of the triangles is also true for the pentagon, although it’s not the perimeter of the pentagon, but a construct (a ‘chord’ of the circle too) within the pentagon.

And one more thing. All of this analysis came about because I was watching the National Geographic Channel and the show was all about creatures of the sea. One of the creatures shown was the star fish. If you connect the tips of the starfish arms you get a pentagon!

So, in addition to the pentagram having a powerful influence in the occult, perhaps reinforced by the presence of the golden ratio, nature again (using a little imagination) provides another possible example of the ‘hidden’ sway of 1.618.


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