UPDATE 8/25/2016: I have corrected an error in this post. I was careless in saying that the Golden Ratio, Phi, came from the Fibonacci Series. The Golden Ratio is the solution to the equation (a+b)/b = a/b = Phi. The Fibonacci series is one series of numbers that converges on Phi. So they are two unrelated concepts that happen to converge on the same number. I regret the error.
I also added a caveat regarding confirmation bias to the section about how Phi is found in art and nature. Some of these claims are in dispute, but I still present them anyway as food for thought. I personally find some of them quite interesting, if nothing else.
Here is the original post from 11/24/2015 with the corrections:
I was recently asked to design a Pentagram pendant from someone who found me through my Shapeways shop. I have of course seen the Pentagram many times. I was aware that the Pentagram was often used as a symbol by the occult and Satanists. So I deliberately avoided doing any designs of it or even related to it before now. It’s really unfortunate that such groups have decided to use it (although they normally use one that is upside down, not right-side up). Regardless, I was very pleased to learn the relationship of the Pentragram to the Golden Ratio, which is seen throughout nature and in many works of art. Let’s take back this amazing symbol and use it for better purposes.
What is the Golden Ratio?
First of all, what is the Golden Ratio? The Golden Ratio is derived from solving the equation (a+b)/a = a/b. If you set the ratio a/b = Phi, the you get the equation 1 + (1/Phi) = Phi. When you solve this, you get Phi = (1 + √5)/2, which is 1.61803… Phi is another irrational number like Pi that goes on forever without a pattern.
The Fibonacci Series in mathematics also converges to Phi. It’s a remarkable coincidence that the two happen to converge on the same number even though they are not directly related. The Fibonacci series starts with either 0, 1 or 1, 1. The next number in the series is the sum of the previous two. So the first numbers in the series are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…
You can see 0+1=1 and 1+1=2 and 1+2=3 and 2+3=5 all the way to 89+144=233 and so on.
As you go further and further out, the ratio of one number to the previous number starts to converge on Phi, which is 1.61803…For example, 233/144= 1.618055… It’s already starting to get pretty close. if you went further out in the series, you would get an even closer approximation of Phi.
In geometry a “Golden Rectangle” is one where the length divided by the height has the ratio of Phi.
There is also a “Golden Spiral” or “Fibonacci Spiral”. Take a Golden Rectangle and start dividing it into further Golden Rectangles. If you draw a spiral that follows the borders of these rectangles, then you get an approximation of a Fibonacci Sprial:
How is the Pentagram Related to the Golden Ratio?
The Pentagram contains several dimensions that result in a Golden Ratio. This page has a great explanation of it. Here is another way of looking at it:
In this diagram, the length of each segment is related by the Golden Ratio:
a/b = Phi
b/c = Phi
c/d = Phi
The Golden Ratio Appears Throughout Nature and Art
First I will start this section with a disclaimer. Many of the examples below are disputed. The criticism is confirmation bias: when you start looking for the Golden Ratio, you start finding evidence of it everywhere. In fact, this phenomenon of finding the Golden Ratio everywhere has been hilariously mocked by the parody Twitter account Fibonacci Perfection. At the risk of being mocked, I still personally find some of these examples compelling and interesting enough to include in this blog post.
I read that the Taj Majal used the Golden Ratio in some of its’ proportions. I had to see for myself. I overlaid a picture I found that had a Golden Spiral with a series of Golden Rectangles on top of a picture of the Taj Majal. See for yourself:
To me, it sure appears that the distance between the pillars and outer edge of the dome follow the Golden Ratio. If you look closer there appear to be other distances that follow this ratio.
Another example in architecture is the Parthenon, although that has been disputed.
Some people say that features in the human face follows the Golden Ratio, and so we may find objects or art that display the Golden Ratio to be pleasing. So even if the Greeks didn’t build the Parthenon to follow the Golden Ratio on purpose, maybe they designed it with similar dimensions because they thought it looked most appealing.
In nature, there are many examples. I already mentioned the human face. Other features such as the length of each segment of your fingers, starting at your wrist follow this ratio. This is a neat article with a lot of examples. One that I found kind of mind-blowing was that many species of flowers have a number of petals found in the Fibonacci Series.
3 petals – Lilies
5 petals – Trillium, Buttercups, Roses
8 petals – Delphinium, Bloodroot
13 petals – Marigolds, Black-Eyed Susans
21 petals – Shasta Daisy
34 petals – Field Daisies, Pyrethrum
The theory as to why so many features of plants would follow this ratio is that it has to do with the most efficient use of space. Plants have to efficiently use sunlight, so it would make sense that their leaves and other features would arrange to maximize their use of surface area.
More famously are the examples of the Golden Spiral in nature. The hurricane, spiral galaxy and Nautilus sea shells are all examples of things that are Golden Spirals, or very close.
So, what is the meaning of the Pentagram then?
The Pentagram has a direct relationship with the Golden Ratio, and the Golden Ratio has a relationship with so many things found in nature. So to me, the Pentagram symbolizes this relationship between math and nature. No one knows why this symmetry exists in nature or why so many natural phenomenon would follow the Golden Ratio. The Pentagram also symbolizes this mystery. It’s an acknowledgment that there are forces at work which we don’t fully understand but can appreciate.