The Golden Ratio (Math Lair)

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The golden ratio is a value often found in arithmetic and geometric proportions. This proportion is often found in art and architecture as well, this ratio is believed to be aesthetically pleasing. It is often denoted by the Greek letter φ (phi).

To see what the golden ratio is, imagine that a line has been divided into two lengths A and B. The ratio of A to B is in the golden ratio if

A + B

To determine what

is, we'll solve the equation for

. We'll start by cross-multiplying:

AB + B² = A²

and then divide both sides by B²

+ 1 = (

)²

Letting φ =

, we get:

φ + 1 = φ²

Solving this equation, we get φ = 1 +

5
2
, which is approximately equal to 1.6180339887... .

Here are some other interesting ways of representing φ:

1 +

1
1 +

1
1 +

1
1 +

1
1 + ...

and

1 + 1 + 1 + 1 + ...

It is easy to verify that the golden mean has the property

1 + φ = φ²

If you take this equation and multiply both sides by φ, you get:

φ + &phi² = φ³

In general, we could multiply the equation by any power of φ and get

φ^{n - 2} + φ^{n - 1} = φⁿ

Have a look at the following geometric sequence where, to get from one term to the next, multiply by φ:

1, φ, φ², φ³, ...

Based on the equations above, it can be seen that each term of this sequence is the sum of the two preceding terms in the sequence. Another sequence like this is the Fibonacci sequence. The Fibonacci sequence has the property that the ratio of successive terms approaches φ in the limit.

φ is also the ratio of a diagonal of a regular pentagon to the side length.