Didrik Wold

MAT100C – Gruppe 20

Essay

The Golden Ratio

The Golden Ratio (and related names and properties: phi ratio, sacred cut, golden mean, divine proportion, golden number, golden section) is an irrational number (expressed by Φ, or Phi) that has some curious properties and is also closely connected to nature as well as mathematics. It seems to have gotten its name from the Golden Rectangle, a rectangle whose sides are in the proportion of the Golden Ratio. The Golden Ratio is found by dividing a segment into two parts so that the length of the smaller part is to the length of the larger part as the length of the larger part is to the length of the entire segment.

|----1----|-----x-----|

If we let 1 be the shorter part of the segment and x be the longer part and the whole segment be the sum (1+x) then we have the following equation:

1/x = x/(x+1)

If we substitute Φ for x then we get:

1/ Φ = Φ /(Φ+1)

Multiplying both sides be (Φ+1) we get:

Φ+1 = Φ^2

The Golden Ratio (Φ) can now be defined as that number which is equal to its own reciprocal plus one. Or:

Φ = 1/Φ+1

Φ is also, as we showed earlier, that number which when squared is equal to itself plus one.

Φ^2 = Φ+1

(Side note) This has some interesting results if we expand on it a bit. From this we can see also that:

Φ+Φ^2 = Φ^3; Φ^2+Φ^3 = Φ^4; Φ^3+Φ^4 = Φ^5; ad infinitum

This means that the power series Φ^n where n goes from 1 to infinity is in fact a Fibonacci series which I discuss a little farther down.

If we now solve Φ^2 = Φ+1 equation we end up with the quadratic equation:

Φ^2- Φ-1 = 0

The solution of this equation is:

Φ = (1+sqrt(5))/2

This is the fractional representation on the irrational number that is Φ. The decimal approximation comes out to be plus or minus 1.618033989. Usually the positive number is regarded to be the Golden Ratio itself while the negative number is referred to as the negative of it’s reciprocal. The Golden Ratio is an irrational number but not a transcendental one (like Pi), since it is the solution of a polynomial equation.

If you are sufficiently bored you can also define Φ using several different trigonometric equations. Not being a fan of trigonometry I’ll simply list them up:

Φ = 2sin(3Pi/10) = 2cos(Pi/5) and 1/ Φ = 2sin(Pi/10) = 2cos(2Pi/5)

It is important to include Fibonacci numbers and the Fibonacci series in any discussion about the Golden Ratio. The original problem Fibonacci was working with was trying to find out how fast rabbits could breed in ideal circumstances. But that’s not what we’re interested in. We’re interested in the resulting series of numbers. This series is made up of numbers in such a way that each number is the sum of the two numbers before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 …

This is closely related to the Golden Ratio because if you take the ratio of two successive numbers in Fibonacci's series divide each by the number before it, we will find the following series of numbers:

1/1=1, 2/1=2, 3/2=1.5, 5/3=1.6666…, 8/5=1.6, 13/8=1.625, 21/13=1.61583…

It becomes fairly clear what is happening once we plot the ratios on a graph:

Here we can see that the ratios are settling down to a particular value, this is of course the number we now know as Φ, The Golden Ratio.

The Fibonacci numbers can also be used to create what are known as a Fibonacci rectangle and spiral. These are similar to the Golden Rectangle and Spiral (which are discussed next) but different in some important ways.

First we draw two squares of size 1, then continue with each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, are known as the Fibonacci Rectangles. These fold outward infinitely while the Golden Rectangles tend to get smaller infinitely. But you can see the relationship.

Now we draw a spiral by putting together quarter circles. This is the Fibonacci spiral (left). This spiral increases in size by a factor of Φ in a quarter turn. A very similar spiral found in Nautilus seashells (right) takes a whole turn to increase by a factor of Φ. These spirals are equiangular, or logarithmic.

Now that we have derived the Golden Ratio and discussed Fibonacci numbers let’s see how it applies to what is known as the Golden Rectangle. The Golden Rectangle, as we said earlier, is a rectangle whose sides are in the proportion of the Golden Ratio. The Golden Rectangle is supposedly an aesthetically pleasing rectangle and the Golden Ratio an aesthetically pleasing one and can be found deliberately turning up in a great deal of art and architecture as well as in nature. For example, the front of the Parthenon can be comfortably framed up in a Golden Rectangle and Leonardo da Vinci used the Golden Ratio in extensively in his painting of the last supper. While its artistic aspects can be argued the construction of a Golden Rectangle is an interesting procedure showing the geometry of the Golden Ratio.

We begin with a unit square and then bisect the square vertically. Then we draw a diagonal in the right half of the bisected square. The length of the diagonal is easily calculated using the Pythagorean Theorem based on the triangle formed by the ½ unit base and the 1 unit side. The resulting diagonal’s length is sqrt(5)/2, which is approximately 1.118033989. If we extend the bottom side of the unit square and draw a circle with a radius of the diagonal and its center at the midpoint on the unit side, the circle will intersect the extended side at a point that will be 1.618033989 units from the corner of the square. Now we just extend the top side of the unit square the same distance and connect them to form a rectangle. This gives us an equation for the Golden Ratio:

Φ = ½ + sqrt(5)/2

Which can be easily combined to make the equation we derived earlier:

Φ = (1+sqrt(5))/2

But the really interesting part comes when you realize that the new rectangle formed when you remove the unit square and leave just the bit to the right is also a Golden Rectangle. And if you divide that rectangle up in the same was as the first one you get another Golden Rectangle, ad infinitum.

Now we have an infinite number of nested Golden Rectangles with side lengths in proportion Φ. This frames what is known as a logarithmic spiral. This is a form that pops up in natural patterns very often.

As I mentioned, this spiral formed of Φ ratios appears naturally quite often because it is in a sense the best pattern that you can arrange something in and still allow for easy growth. For example, let’s say you were deciding how to pack circular seeds in a flower head. Most people would agree that a hexagonal pattern is the most efficient for circles. But that is only taking into consideration static sized seeds. Once the seeds start growing there’s no room for expansion. That’s where the spiral comes into play. It packs thing efficiently and doesn’t lose its efficiency at all when the plant grows. Once a seed is positioned on a seed head, the seed continues out in a straight line pushed out by other new seeds, but retaining the original angle on the seed head. No matter how large the seed head, the seeds will always be packed uniformly on the seed head.

There is an enormous amount of information on this subject and to list up everything I’ve read about would take at least a hundred pages. This is meant as an introduction to show off some of the basic elements involved in the Golden Ratio. The most obvious connection to calculus is through power series but it is relevant in many ways. At any rate, Φ is an important constant and probably one of the more interesting ones. Φ appears all over the world. Sure Pi goes on forever but Φ creates beautiful natural artwork and also is involved in fractals and all sorts of manmade art. And in fact you can derive a formula for calculating Pi by using Φ. I guess I just like the idea that a ratio can be connected to so many strange and wonderful natural phenomenon.