Golden Ratio

The Golden Ratio truely is a unique number in mathematics! It occurs often in nature, often in rivalry with the greatest irrational number of all time, p! But unlike it's rival p, the Golden Ratio is a constructable number! Officially, the Golden Ratio is denoted by the symbol Φ. By definition:

Φ = (√5+1) / 2 ≈ 1.61803398875

Φ² = Φ + 1

Φ^-1 = Φ - 1 = φ

Even though this number is irrational, it can be constructed using compass and straightedge! It can also be found by repeated square rooting. Simple take ANY positive number, find the square root, add one to it, then take the squareroot again, repeating it over and over. It will always converge to Φ!

Enter any positive number:

Golden Section

To construct the Golden Ratio, all you need to do is construct the √5, add 1 to it, and cut it in half.

Draw AB (yellow).
Draw BC (magenta), so that BC _|_ AB and BC is half of AB.
Draw a line through points CA (red).
Draw a circle at C, crossing B. Circle C crosses CA at D.
Draw a circle at A, crossing D. Circle A crosses AB at E.
The ratio of AE to EB is equal to Φ.

Another way of calculating the Golden Ratio, is the use of Fibonacci numbers. A Fibonacci sequence is of this form:

f₁ = 1
f₂ = 1
f_n = f_n-2 + f_n-1, for n > 2.

It has a sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55..., where the next term is the sum of the two previous terms. Notice, that if you divide consecutive terms, they approach Φ. It approaches quite fast actually, where 55/34 = 1.61764... .

Golden Rectangle
The Golden Rectangle, is a rectangle whose ratio of length to width is Φ. It is relatively easy to construct.

Construct square ABCD.
Extend DC.
Find the midpoint M of DC.
Draw a circle at M, crossing B. It intersects DC at E.
DE is the length of the new rectangle. The ratio of DE to AD is Φ.

Golden Spiral
The Golden Spiral is created by constructing golden rectangles embedded in each other. A quarter arc is then drawn across each square. A golden spiral can also be drawn using a 36° - 72° - 72° isoceles triangle. A Golden Triangle is a triangle found inside a regular pentagon.