Notes on the Golden Ratio & the Golden Rectangle What is the golden ratio? Imagine that we have cut a whole line randomly into a bigger and smaller piece.
whole bigger
smaller
We could express the ratio of the whole line to the bigger piece as a fraction:
And we could express the ratio r atio of the bigger piece to the smaller piece as a fraction as well:
We could do this wherever we randomly cut the line. However, there is one special place where we could cut the line that would create a unique situation:
When this special relationship happens—when the ratio of the whole to the bigger piece is equal to the ratio of the bigger piece to the smaller piece—we call this ratio the golden ratio. Now, instead of using words like "whole" and "bigger," let's express this as an algebraic equation with variables. Let's call the whole "a," the bigger part "b," and the remaining smaller part "a‐b". So then, we have a golden ratio when:
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What is the golden rectangle? Let's take our a and b above, which are in the golden ratio, a nd construct a rectangle contained by them. a
b
This is a golden rectangle. It has often been said that the golden rectangle was deliberately used in a lot of our art from the Greeks on. (It is also debatable how much that is true.) Now let's look at an interesting property of golden rectangles. Let's construct a square and a rectangle inside this rectangle: a
b
b
b
a-b
Notice that the resulting rectangle has sides b and a‐b. But remember that we started with a golden rectangle, which means that this relationship is true:
So the ratio of the sides of the new, smaller rectangle is the same ratio as the bigger ratio; in other words, both rectangles are golden rectangles. This is always true; whenever you mark off a square on a golden rectangle, the leftover rectangle is a new golden rectangle.
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Let's imagine now that we mark off a square on the new rectangle:
a
b
b
a-b
b
Notice that we have a new golden rectangle in the top right. We can mark off a square in that, and on and on…
a
b
b
a-b
b
If we put quarter‐circle arcs into the squares, we would have a sort of spiral:
a
b
b
b
a-b
Calculating the golden ratio (This is some real math stuff; I want you to read it, but only some of you will remember this kind of algebra, and I will not ask you to repeat it.) Consider the number we get when we take a circle and Page 3 of 5
. We call that number π, and when we calculate the number we find it is approximately 3.1415… As it happens, π is an irrational number, and so we cannot write it out calculate the fraction
exactly as a decimal. (How do we calculate π? Well, in the spring we will see how Archimedes did it.) So what about the golden ratio?
by a Greek letter; we call it φ (phi). How do we calculate the number? Well, we could make it easy on ourselves if we choose to make b in In this situation, we also call the number we get when we calculate that equation the number 1:
1 1 1 φ (the golden ratio) and , if we solve for a, we will have found φ, the number that results from the golden ratio. So that we remember that clearly, let’s not call the variable a; let’s call it φ . Since
φ
1 So how do we solve the equation
φ
1 φ 1
φ ? First of all we cross multiply: φ φ 1
1
φ φ 1 Uh‐oh, we have a squared term. That means we will need to use the quadratic formula to solve it. You remember the quadratic formula, right?
If
√ 4 0 then 2
So we need to take our equation and put it into the right form by subtracting 1 from each side:
φ φ 1
0
Hmmm… What are a, b, and c? Well, since any number is equal to 1 times itself, therefore:
φ
1 φ , so 1,
φ 1 φ , so 1, 1 Putting that into the quadratic formula gives us:
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φ
1 1 411 21 φ
1 1 4 2 φ
1 √ 5 2
Because of the ±, this would give us two solutions. But one of the solutions is negative, which is irrelevant to us, so we will only calculate with the +. On my calculator that comes out to 1.61803.
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