Return to my Mathematics pages
Go to my home page
Note: If your WWW browser cannot display special symbols, like ² or 2 or ø, then click here for the alternative Golden Rectangle and Golden Ratio page.
© Copyright 1997, Jim Loy
This diagram shows a golden rectangle (roughly). I have divided the rectangle into a square and a smaller rectangle. In a golden rectangle, the smaller rectangle is the same shape as the larger rectangle, in other words, their sides are proportional. In further words, the two rectangles are similar. This can be used as the definition of a golden rectangle. The proportions give us:
a/b = (a+b)/a
This fraction, (a+b)/a, is called the golden ratio (or golden section or golden mean).
Above I have defined the golden rectangle, and then said what the golden ratio is, in terms of the rectangle. Alternatively, I could have defined the golden ratio, using the above equation. And then a golden rectangle becomes any rectangle that exhibits this ratio.
From our equation, we see that the ratio a/b=1/2+sqr(5)/2 (where sqr() is the square root function), which in turn is about 1.61803398875 . . . The symbol often used for the golden ratio is ø (phi). Sometimes, 1/ø (which is -1/2+sqr(5)/2 or 0.61803398875 . . .) is called the golden ratio. Also, other mathematical quantities are called phi. The golden ratio is also called tau. Some people call the bigger one (1.61803398875 . . .) Phi (an uppercase phi) and the smaller one (0.61803398875 . . .) phi. By the way, a more accurage value is 1.6180339887498948482045868343656 . . .
Supposedly, Pythagoras discovered this ratio. And the ancient Greeks incorporated it into their art and architecture. Apparently, many ancient buildings (including the Parthenon) use golden rectangles. It was thought to be the most pleasing of all rectangles. It was not too thick, not too thin, but just right (Baby Bear rectangles).
Because of this, sheets of paper and blank canvases are often somewhat close to being golden rectangles. 8.5x11 is not particularly close to a golden rectangle, by the way.
The golden ratio is seen in some surprising areas of mathematics. The ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13 . . ., each number being the sum of the previous two numbers) approaches the golden ratio, as the sequence gets infinitely long. The sequence is sometimes defined as starting at 0, 1, 1, 2, 3 . . . Zero is the zeroth element of the sequence. See Fibonacci Numbers.
The golden rectangle is found in some art, especially 20th Century art. But, it would seem that ancient Greek architects did not consciously use it. The Parthenon is the most famous example of the use of the golden rectangle. But, the fit is not particularly good. And, almost any rectangle can be found in pictures of the Parthenon. People find the golden rectangle in the Mona Lisa, and other Renaissance art works. But, again, almost any rectangle can be found in the Mona Lisa.
The golden rectangle and the golden ratio sometimes pop up in nature. But, this is usually far from an exact fit. To the right, we see a spiral which comes from the golden rectangle. We are told that this is very close to the shape of the shell of a chambered nautilus. This figure is self-similar, each part is similar to smaller parts and larger parts. This makes it a rudimentary fractal.
The golden ratio (and the golden triangle) shows up (as an exact fit) in mathematics in many unexpected places. The ratio shows up everywhere in the pentagram (five pointed star) and its circumscribed pentagon (shown on the left). a/b=(a+b)/a=(a+b+a)/(a+b)=ø. Here we also see the two kinds (acute and obtuse) of golden triangles (I've painted two of them green). They are any of the isosceles triangles in this diagram. These are triangles which exhibit the golden ratio. See The Regular Pentagon.
As the dimensions of a picture frame, I do not like the golden rectangle. I prefer a longer, skinnier rectangle. Other people probably prefer a rectangle closer to the golden one. But, I suspect, these preferences seldom fit the golden rectangle very well.
There are many more.
Let's look at the golden ratio, a little bit. You can verify these observations algebraically.
Here is a diagram which shows how to construct a golden rectangle. Draw the square, bisect the bottom side, and draw the arc from the upper right corner to the extension of the bottom side. Then complete the rectangle.
It is fairly easy to show that this construction produces the right lengths. If a is 1, then a+b is 1/2+sqr(5)/2.
I stumbled onto this construction while trying to figure out how to fold a square into a regular pentagon. I have since found the construction in Geometry, by Harold R. Jacobs.
Here is an infinite series which evaluates to the golden ratio:
Notice the alternating signs. Each denominator is the product of two consecutive Fibonacci numbers. Where did I get this series? I deduced it from the fact that the ratio of consecutive Fibonacci numbers approaches the golden ratio, as the numbers get larger. The sequence of ratios of consecutive Fibonacci numbers is 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,... We get the second element of this sequence with this series: 1+1/1·1. The third element is 1+1/1·1-1/1·2. The fourth is 1+1/1·1-1/1·2+1/2·3, and so on. This series was known long before I discovered it.
The golden ratio can be represented as the simplest continued fraction (as shown on the left). This fraction is 1+1/a where a is 1+1/a. You probably evaluate such a continued fraction from the right side, and work toward the left, except that it has infinitely many terms. We can get a sequence from this by starting at the left, and evaluating only part of the continued fraction: 1, 2/1, 3/2, 5/3, 8/5, 13/8... These are the ratios of consecutive Fibonacci numbers.
The golden ratio can be represented as a repeated square root, as in the diagram at the right. Evaluate this from right to left, too.
We can summarize some of the above as:
Here is an interesting sequence:
Notice the Fibonacci Numbers on the right side of each equal sign. F(n) is the nth Fibonacci number.
Also see Surprise?
I just received a catalog advertising "Golden Ratio Calipers." Here is a rough drawing of them measuring one side of a golden rectangle (gray). The calipers have a hinged rhombus at the top, and the two sides of the calipers are equal. Here are a couple of questions for you: (1) Do these calipers measure the golden ratio no matter what the top angle is? (2) And, in the diagram, what is the length of the line segment marked with the question mark?
The answers are: (1) Yes, adjust the length of the segment marked with the question mark, and the other segments that are not part of the rhombus, to get any ratio that you want. (2) If the side of the rhombus is 1, then the segment marked with the question mark should be the golden ratio. If you try to answer the second question, then the answer to the first question becomes obvious. Just use similar triangles.
These calipers come with a mystical new age kind of book, which seems to be out of print. See Health Education Alliance for Life and Longevity or Amazon.com.
I was just watching the award-winning, short documentary, Fibonacci and the Golden Mean. And they had the first diagram on the right, claiming that the builders of Khufu's (Cheops') Great Pyramid incorporated the golden mean into the pyramid. A little arithmetic shows that the diagram is wrong. What they should have drawn was the second diagram on the right. The ratios in the red triangle are very close to the golden ratio. See Pi and the Great Pyramid, where we find that the Great Pyramid also exhibits the ratio pi. If the designers of the pyramid meant to use pi, then it is merely a coincidence that the golden ratio crops up (as any pyramid that shows pi in that way will show the golden ratio, approximately). And vice versa, if they meant to use the golden ratio, then it is a coincidence that pi appears.
Return to my Mathematics pages
Go to my home page