The Golden Spiral

To get to this point, we have defined the Fibonacci Sequence and then the Golden Ratio. Our next step is to apply the Fibonacci sequence in a geometric representation.

We begin by enscribing a square below, with a perimeter of 1 unit.

Next, in keeping with the logic of Fibonacci, we draw a square with sides equal to the sum of the previous two square sides, in this case 0 and 1, add them together and produce a 2x1 unit rectangle.

The progression continues... 1 + 1 = 2 units and, rotating clockwise, we draw a square of two units, on top of our 2x1 rectangle.

We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the last two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, and thus called Fibonacci Rectangles.

fibspiral2.gif Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangement of seeds on flowering plants too. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the centre are 1.6184 = 6.854 times further out than when the curve last crossed the same radial line.


Below are images of cross-sections of a Nautilus sea shell. They show the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows. The chambers provide buoyancy in the water. Draw a line from the centre out in any direction and find two places where the shell crosses it so that the shell spiral has gone round just once between them. The outer crossing point will be about 1.6 times as far from the centre as the next inner point on the line where the shell crosses it. This shows that the shell has grown by a factor of the golden ratio in one turn.

Certainly the Golden Logic is pervasive in the construnction of the Universe, from the cosmological to the atomic perspective. To understand why this is so and how this mathematical function ties into thermodynamics, we need to first generalize the Fibonacci sequences, in order to better understand the underlying meaning of this mathematical behavior.

 

Thermodynamics, Number Theory and The Goilden Ratio
Creation, Evolution and the Golden Rule
Theory of Order
Why Fibonacci and Gibonacci sequences appear everywhere in nature,
and how simple combinatoric math can describe how a Universe with simple beginnings evolved into a complex form