The Golden Section, The Fibonacci Sequence, and .
- by Paul Hildebrandt


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 the ratio 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,
or (Tau).


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 preceeding 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:

   number  ratio         number    ratio
    1                        89     1.617978
    1      2.000000         144     1.618056
    2      1.500000         233     1.618026
    3      1.666667         377     1.618037
    5      1.600000         610     1.618033
    8      1.625000         987     1.618034
   13      1.615385        1597     1.618034
   21      1.619048        2584     1.618034
   34      1.617647        4181     1.618034 
   55      1.618182        6765     1.618034

Let's look at those ratios in the form of a graph...

And 1.618034... =  = the Golden Mean.