There is a special constant considered by the ancient Greeks, the founders of geometry, that when used as the ratio for the sides, creates a rectangle that was aesthetically pleasing to the eye. This number, represented by the Greek letter Phi (φ),¹ and also called the Golden Number, is remarkable in many ways.

What is Phi?

Phi is one of the solutions to the quadratic equation x²-x-1=0. This can be calculated to be equal to (1+√5)/2, or approximately 1.618.²
The other solution will be discussed later.

• Phi squared equals Phi plus one.³
• Phi's reciprocal is equal to Phi minus one.⁴
• Phi is represented by the continued fraction

  φ = 1 +

  1 1 +   1 1 + · · · +   1 1 +   1 1+x

  ⁵
• Phi is also represented by the series √(1+√(1+√(1+√(1+√(1+...⁶

But what's so great about Phi?

Fibonacci sequences and Phi

A mathematician by the name of Fibonacci discovered a certain series of numbers while studying the populations of rabbits. This series is derived by taking two numbers, commonly 0 and 1, which we'll use, and then each following number is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21,...

Where does Phi enter into this series? Divide each successive number by the previous and the results will converge on 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. A Fibonacci sequence does not necessarily start with zero and one, but this will work with any others as well.

The number Phi is used in calculating numbers in the Fibonacci sequence, with this equation: Fn=(Phi^n-(-phi)^n)/√(5).

If you graph y=φx, the line does not cross any points with pairs of integer coordinates other than the origin, but those points that are nearest the line are pairs of Fibonacci numbers, such as (2,3) and (3,5).

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 square, using a ruler, follow these steps:

1. Draw a point halfway on one of the sides.
2. Draw a line from this point to one of the opposite corners.
3. Put the ruler along this line with one corner on the 1st point.
4. Rotate the ruler down on the 1st point so that it is partially on the base, keeping track of where the corner had been on the ruler.
5. Extend the edge of the square out to the point on the ruler that had been on the corner. You should now have a rectangle that is approximately of the ratio 1:1.618: a Golden Rectangle.
⁷

The same can be done with a golden triangle, a triangle in which either one or two sides being φ times longer than its other sides. 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 curve like the rectangle did.

Supposedly, the ancient Greeks thought this was all very cool and beautiful, and incorporated them into their art and architecture. For example, it is said that the Parthenon's face was meant to be approximately a Golden Rectangle.

The length from a corner of a regular decagon to the center divided by its side length, is equal to Phi.

Phi and Pentagons

Suppose you have a regular pentagon, and you draw a pentagram inside it, by connecting each point to every other one. The ratio between the length of a side of the pentagon to that of each line that makes up the star is 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 of the star, and the shorter lines, which make up the sides of another pentagon inside of it. The ratio between these two lines is also 1:Phi.

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 Phi/2 while the cosines of 72° and 108° are both (Phi - 1)/2.

One of the neatest applications of pentagonal geometry is the pair of Penrose tiles, discovered/invented by 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.

Myths

• 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 thought to be approximately Phi.
• In many paintings certain features are supposedly in the golden ratio.
• Some say that even in music, Bach, Mozart, and other composers had sections in length that related to the golden ratio.

phi

There is another number, closely related to Phi, which has many related properties. This number, commonly called phi(p) also, but with a lowercase "p" instead of uppercase, is the other solution to x²-x-1=0. It is equal to (1-√5)/2, or approximately -0.618.⁸ However, the positive form 0.618, is usually used instead.

• φ + p = sqrt(5)
• = 1 / φ = φ - 1 = p
• p² = 1 - p
• 1 /φ = p;, so 1 / p = φ, and φ * p = 1.

Just like dividing two Fibonacci numbers gets closer and closer to Phi, the same can be done to narrow in on phi, if you extend the Fibonacci sequence the other direction, into the negatives: 0, 1, -1, 2, -3, 5, -8, 13... Dividing these 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, -p.

Further Information

• http://www.friesian.com/golden.htm
• http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html