The Golden Section and The Fibonacci Sequence
The Golden Section
The Golden Section (also called the Divine Proportion), said by Kepler to be "one of the two treasures of geometry," appears repeatedly in growth patterns in nature and has fascinated mathematicians and artists for centuries.
What is the Golden Section?
It means that a certain length is divided in such a way that the ratio of the longer part to the whole is the same as that of the shorter part to the longer part.
In the illustration at right, Line AB is divided so that the ratio of AC to AB is the same as the ratio of CB to AC. If AC is 1.000, then AB becomes 1.618, which is also known as the Golden Mean, also known as (Tau) or (Phi).
The Golden Rectangle
There is a special rectangle with proportions corresponding to the Golden Section. It's called the golden rectangle.
To construct a Golden Rectangle on paper, you'll need a pencil, ruler, compass, and a right-angle triangle. First draw a square, AEFD, of arbitrary size. Then divide the line AE in half at A'. Then, with the compass and using A' as center, draw an arc from F up to B, which intersects the extension of line AE at B. With your triangle, draw BC perpendicular to AB, meeting the extension of line DF at C. The new ABCD rectangle is a golden rectangle, in which AB is divided by E in exactly the golden section:
AE:AB = EB:AE
That is, the ratio of the longer part to the whole is equal to the ratio of the shorter part to the longer part. Constructing a golden rectangle with Zometool is a bit simpler: Use four blue sticks, two large and two medium (OR two medium and two small), and join them with four balls. Notice anything familiar about the ratio between sticks of the same color? That's right - it's equal to the golden mean.
The Fibonacci Sequence
Leonardo Fibonacci of Pisa was a mathematician in 13th century Italy. By charting the population of rabbits, he discovered a number series from which one can derive the Golden Mean (see above). But first, here's the beginning of the sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
Each number is the sum of the two preceding numbers, as follows: A neat trick, but here's the amazing part: Dividing each number in the series by the one which preceeds it produces a ratio which stabilizes around 1.618034:
