User:Gfdas29/Golden Fractals

This page is dedicated to using fractal properties as a way of studying Golden Rectangles. Fractals are identified by their self-similarity at varying levels of magnification. Since Golden Rectangles are defined by the ability to create other rectangles proportionally similar to itself, they are prime candidates for discussion as fractals. However, relatively little has been done in examining the intersection of Golden Rectangles and fractals. This brings together some of what has been done.

Golden Rectangles
The Golden Ratio is defined as the ratio between 1 and φ, where φ is the number such that


 * $$\frac{\varphi}{1} = \frac{1}{\varphi-1}$$

This ratio comes from a special rectangle called the Golden Rectangle. This is a rectangle with sides 1 by φ. An example can be seen below:

This is how we get the relationship $$\frac{\varphi}{1} = \frac{1}{\varphi-1}$$. Using this relationship we can determine the value of φ.

$$\frac{\varphi}{1} = \frac{1}{\varphi-1}$$

(φ-1) φ = 1			  	        (By multiplying both sides by (φ-1).)

(φ-1) φ - 1 = 0				(By subtracting 1 from both sides.)

φ2 - φ - 1 = 0				(By the distributive property.)

From here we can use the quadratic equation (which we’ve already proven to be valid in the last problem set) and solve for φ.

φ = $$\frac{1 \pm \sqrt{(-1)^2 -4(1)(-1)}} {2(1)}$$

φ = $$\frac{1 + \sqrt{5}} {2}$$ (Reduced and + rather than ± because φ is a distance in this case.)

φ  1.6180339887…

The Eye of God
If we continue to form golden rectangles from our original golden rectangle we will come up with a shape that is divided indefinitely and looks similar to:

n = $$ \frac{\text{Area of Whole}}{\text{Area of Part}} $$

$$ \frac{\varphi * 1}{1 * (\varphi -1)} = 2.618$$

f = φ

We then plug these values into the formula to find the dimension.

fd = n

φ d = $$ \frac{\varphi * 1}{1 * (\varphi -1)}$$

d*ln(φ) = $$ ln( \frac{\varphi * 1}{1 * (\varphi -1)})$$

d = $$ \frac {ln( \frac{\varphi * 1}{1 * (\varphi -1)})} {ln( \varphi)}$$

d = 2           Just as we said earlier.

Golden Fractal
Using the four properties listed above for a standard of an interesting fractal, one can be created using the golden rectangle as our starting point. Our fractal is essentially going to be the Eye of God, except instead of dividing one golden rectangle into a square and another golden rectangle, we dived it into 2 golden rectangles and an empty space in the middle of them. This creates an infinite number of spirals.

Begin with a Golden Rectangle with sides φ by 1.

Since the area of the larger rectangle is made up of 2 smaller golden rectangles (remember the middle area no longer exists) we can say that the number of parts in the whole is 2. n = 2

The magnification factor is still φ. f = φ

$$\varphi^d = 2$$

$$d * ln(\varphi) = ln(2)$$

$$ d = \frac{ln(2)}{ln(\varphi)} $$

d ≈ 1.4404

Therefore, the dimension of this geometric shape is a non-Euclidian fractal dimension.