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GOLDEN RATIO

Definitions of Golden Ratio
1) Numeric definition

Here is a 'Fibonacci series'.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..

If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers.

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...

The ratio seems to be settling down to a particular value, which we call the golden ratio(Phi=1.618..).



2) Geometric definition

We can notice if we have a 1 by 1 square and add a square with side lengths equal to the length longer rectangle side, then what remains is another golden rectangle. This could go on forever. We can get bigger and bigger golden rectangles, adding off these big squares.

Step 1 Start with a square 1 by 1
Step 2 Find the longer side
Step 3 Add another square of that side to whole thing



Here is the list we can get adding the square;
1 x 1, 2 x 1, 3 x 2, 5 x 3, 8 x 5, 13 x 8, 21 x 13, 34 x 21.
with each addition coming ever closer to multiplying by Phi.

start 1 by 1, add 1 by 1 => Now, it is 2 by 1, add 2 by 2



Now, it is 3 by 2, add 3 by 3 => Now, it is 5 by 3, add 5 by 5










Now, it is 8 by 5.










3) Algebraic and Geometric definition

We can realize that Phi + 1 = Phi * Phi.


















Start with a golden rectangle with a short side one unit long.
Since the long side of a golden rectangle equals the short side multiplied by Phi, the long side of the new rectangle is 1*Phi = Phi.

If we swing the long side to make a new golden rectangle, the short side of the new rectangle is Phi and the long side is Phi * Phi.

We also know from simple geometry that the new long side equals the sum of the two sides of the original rectangle, or Phi + 1. (figure in page4)

Since these two expressions describe the same thing, they are equivalent, and so
Phi + 1 = Phi * Phi.



II. Some Golden Geometry

1) The Golden Rectangle








A Golden Rectangle is a rectangle with proportions that are two consecutive numbers from the Fibonacci sequence.

The Golden Rectangle has been said to be one of the most visually satisfying of all
geometric forms. We can find many examples in art masterpieces such as in edifices of ancient Greece.





2) The Golden Triangle



If we rotate the shorter side through the base angle until it touches one of the legs, and then, from the endpoint, we draw a segment down to the opposite base vertex, the original isosceles triangle is split into two golden triangles. Aslo, we can find that the ratio of the area of the taller triangle to that of the smaller triangle is also 1.618. (=Phi)








If the golden rectangle is split into two triangles, they are called golden triangles suing the Pythagorean theorem, we can find the hypotenuse of the triangle.

3) The Golden Spiral









The Golden Spiral above is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with the golden rectangle.
If you take one point, and then a second point one-quarter of a turn away from it, the second point is Phi times farther from the center than the first point. The spiral increases by a factor of Phi.





This shape is found in many shells, particularly the nautilus.








4) Penrose Tilings





The British physicist and mathematician, Roger Penrose, has developed an aperiodic tiling which incorporates the golden section. The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B).



When a plane is tiled according to Penrose's directions, the ratio of tile A to tile B is the Golden Ratio.








In addition to the unusual symmetry, Penrose tilings reveal a pattern of overlapping decagons. Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean.



5) Pentagon and Pentagram





We can see there are lots of lines divided in the golden ratio. Such lines appear in the pentagon and the relationship between its sides and the diagonals.



We can get an approximate pentagon and pentagram using the Fibonacci numbers as lengths of lines. In above figure, there are the Fibonacci numbers; 2, 3, 5, 8. The ratio of these three pairs of consecutive Fibonacci numbers is roughly equal to the golden ratio.