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:
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
fn = fn-2 + fn-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... .
The Golden Rectangle, is a rectangle whose ratio of length to width is Φ. It is relatively easy to construct.
Construct square ABCD.
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.
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.