The Golden Mean
If the sides of a rectangle are in this proportion then adding or subtracting a square leaves a rectangle in the same proportion.
This diagram might show subtracting the square from the big rectangle to leave the smaller one: or it might show adding the square to the smaller rectangle to get the bigger one. The sides of both rectangles are in the golden ratio.
The golden mean is also closely related to the pentagon. If we form a 5-pointed star by joining alternate vertices of a regular pentagon then the side of the pentagon to the side of the star is in the golden mean.
Imagine unfolding the star across AB so that C is reflected to point down. Then triangles ABC and DECwould be similar and show the sides to be in the golden ratio.
Another interesting geometrical feature of the golden mean is demonstrated by the spiral in figure 3.
The Fibonacci Sequence
In the Fibonacci sequence successive terms are the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 etc. As the number of terms increases, the ratio of the last two terms approaches the golden ratio (or mean). Since we are concerned with the ratio of two numbers, we might express it as a 2-dimensional vector.
Fibonacci Sequence
|
| 0 | |
1 | |
1 | |
2 | |
3 | |
5 | |
8 | |
13 | |
21 | |
34 | |
55 | |
89 | |
144 | |
233 | |
377 | |
610 | |
987 | |
1597 | |
2584 | |
4181 | |
6765 | |
10946 |
| 1 | |
1 | |
2 | |
3 | |
5 | |
8 | |
13 | |
21 | |
34 | |
55 | |
89 | |
144 | |
233 | |
377 | |
610 | |
987 | |
1597 | |
2584 | |
4181 | |
6765 | |
10946 | |
17711 |
The ratio of consecutive terms approaches the golden mean.
The Matrix Aspect
The sequence can also be seen as consecutive transformations of a base vector by a triangular matrix:
The same result comes from combining the matrix operations and applying the resultant to the vector. This gives us a sequence of matrices with certain properties:
- they are symmetric
- they commute
- they maintain the ratio of the vector limits (the golden mean)
| … | | 2 | -1 | | -1 | 1 | | 1 | 0 | | 0 | 1 | | 1 | 1 | | … |
| ← | | -1 | 1 | | 1 | 0 | | 0 | 1 | | 1 | 1 | | 1 | 2 | | → |
In the sequence above the centre matrix is the identity matrix and subsequent matrices in either direction are mutually inverse. The sequence could be extended in both directions.
The 2-D Equations
The inverse of the golden mean is formed by adding 1 for s/l or subtracting 1 for l/s:
r-1 = 1 + r
1 = r + r2 (multiplying by r)
r2 + r + 1/4 = 5/4 (adding 1/4 to each side)
(r+ 1/2)2 = 5/4
r + 1/2 = ± √5/2
r = (± √5 - 1)/2
r-1 = r - 1
1 = r2 - r (multiplying by r)
r2 - r + 1/4 = 5/4 (adding 1/4 to each side)
(r - 1/2)2 = 5/4
r - 1/2 = ± √5/2
r = (1 ± √5)/2
The golden mean is an irrational number, thus bewildering the ancient Greeks, who much admired it. We can associate with the Golden Mean some interesting geometry, the Fibonacci Sequence, a sequence of matrices and two equations.
Graphical Interpretation
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