There is a special constant which was studied by the Ancient Greeks, the founders of geometry. A rectangle whose sides are in the ratio of 1 to this constant is considered to be aesthetically pleasing to the eye. This number, represented by the Greek letter Phi (Φ),¹ shows up in algebra, geometry, and supposedly nature and art. Phi is equal to (1+√5)/2, or roughly 1.61803. It is also called the 'golden mean' or by the Ancient Greek name of 'extreme and mean ratio'; it has many remarkable properties.

Some mathematical properties of Phi include:

• If a line is divided in this ratio, then the ratio of the whole line to the longer part is the same as the ratio of the longer part to the shorter part.

• It is one solution to the quadratic equation x² - x - 1 = 0

• It is represented by the continued fraction 1+1/(1+1/(1+1/(1+1/(1+...

• Phi is also represented by the series √(1+√(1+√(1+√(1+√(1+...

• Φ² = Φ + 1. That is, you square it by adding 1 to it.

• 1/Φ = Φ - 1. That is, its reciprocal is got by subtracting 1 from it.

It has many other interesting properties: it shows up in pentagons and pentagrams, and in Fibonacci sequences. It also pops up in all sorts of places such as in shells, art, and the human body.

Fibonacci sequences and Phi

A mathematician by the name of Fibonacci² discovered a certain series of numbers while studying the populations of rabbits. This series is produced by starting with two numbers, commonly 0 and 1, which we'll use; each following number is the sum of the previous two. So we get 0, 1, 1, 2, 3, 5, 8, 13, 21,...

Where does Phi enter into this series? Divide each successive number by the previous one and the results converge to Phi:

1/1=1, 2/1=2, 3/2=1.5, 8/5=1.6, 13/8=1.625, 21/13=1.6153... and so on. This will also work if you start with any two numbers for the sequence.

While the individual terms of the Fibonacci sequence starting with 0 and 1 can be got by adding, you can also use Phi to find them, with this equation: Fn=(Φⁿ-(-Φ)ⁿ)/√5.

If you graph y = Φx, the line does not cross any points with whole number coordinates other than (0,0), but the points nearest the line are pairs of Fibonacci numbers, such as (2,3) and (3,5).

Fibonacci numbers are present in nature in many ways, and therefore, so is Phi.

Phi in Geometry

Golden figures

A golden rectangle is a rectangle which has sides that are in the 'golden ratio'. To create one from a square³, draw a line from the halfway point along the base to one of the top corners. Next, measure the length of this line with a ruler. Starting at the halfway point of the base, go this distance to the right and mark a point, which will be outside the square. Extend the base to meet this point and construct a rectangle on this new longer base. You now have a golden rectangle.

Besides the square you started with, you also have a smaller rectangle; this is itself a golden rectangle. It can be divided to make a smaller square and golden rectangle, and this process can be continued indefinitely, or at least until it's too small to go any further. If you've arranged all your squares right, you can draw a curve from the far corner of the original square to the diagonally-opposite corner, and continue this curve into the next square and so on, drawing a spiral. This spiral is known as a 'logarithmic spiral'; it is special, as the nautilus⁴ shell is curved in a similar way.

The same can be done with a golden triangle, a triangle with two identical sides in the ratio of Φ:1 with the third side. If you bisect one of the base angles on one of these triangles, you'll be left with a smaller golden triangle. Keep doing this, and the points of each triangle will make a logarithmic spiral just as the rectangle did.

Phi in pentagons and pentagrams

The golden ratio also shows up a lot in pentagons and pentagrams. Suppose you have a regular pentagon⁵, and you draw a pentagram⁶ inside of it, by connecting each of the points. The ratio between the length of a side of the pentagon to that of each line that connects two of the points happens to be the golden ratio. Furthermore, if you look at the pentagram, you'll notice there are two lengths of lines that make it up: the longer lines, which make up the points, and the shorter lines, which make up the sides of another pentagon inside of it. The ratio between these two lines is also 1:Φ.

The corner angle in a pentagon is 108° which is six-fifths of a right angle. The small angles in the star are 36° while the bigger ones are 72°. These are two fifths and four fifths of a right angle respectively. The cosine of 36° can be shown to be Φ/2 while the cosines of 72° and 108° are both (Φ - 1)/2.

One of the neatest applications of pentagonal geometry is the pair of Penrose tiles, discovered/invented by mathematician and physicist, Roger Penrose. This is a pair of tiles got by dissecting a rhombus (a diamond shape) with an angle of 72°. The main diagonal is divided in the golden ratio and this point is joined to the other two corners. You end up with two tiles, the 'kite' and the 'dart'. Multiple copies of these can be used to tile the plane in infinite irregular tesselations.

Other things about Phi

If you take one finger and measure from your knuckle to the second joint, and if you divide this length by the length from the first to the second joint, this value is approximately Phi.

The Greeks thought that this ratio was pretty special and beautiful, and they incorporated this into their art and buildings. The front face of the Parthenon⁷ is a rectangle in approximately the golden ratio. In many paintings certain features are in the golden ratio, particularly those of Leonardo da Vinci. Even in music, Bach, Mozart, and other composers often divide the music up into sections whose lengths are related by the golden ratio.

Like other numbers such as pi, e and √2, this number is irrational; the decimal value does not terminate or repeat.

The number Phi, written out a bit

1.6180339887498948482045868343656381177203091798057628621354...

The Reciprocal of Phi

There is another number, closely related to Phi, which has many related properties. This number is the reciprocal of Phi, that is 1/Φ. It is sometimes called phi with a lowercase 'p' and denoted by the lowercase symbol φ, although this practice is not universal. We will use this notation here. This constant, phi, is equal to Phi-1, or approximately 0.61803.

Mathematical properties of φ

• φ is equal to (-1+√5)/2
• φ = 1 / Φ = Φ - 1
• Φ + φ = √5
• In the quadratic equation x² - x - 1 = 0, the other solution, besides Phi, is -φ.

Just as dividing two Fibonacci numbers gets closer and closer to Phi, the same can be done to narrow in on phi, simply by doing the division in the opposite direction. That is, dividing each number by the one to the right of it results in 0, 1, 0.5, 0.666, 0.6, 0.625... which converges on 0.618, φ.

A Formula for Phi

The definition of Phi as a continued fraction is very striking when written out in normal mathematical notation. The x is a vanishingly small number. This formula forms a fitting conclusion for an entry on a remarkable number.