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:

• 
  Golden Rectangle Stage 1

This rectangle can then be divided into a 1x1 square and a rectangle where the Golden Ratio still holds, or a square and another golden rectangle. See Below:

• 
  Golden Rectangle Stage 2

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:

• 
  Eye of God

This spiral towards infinity is called the Eye of God or The Golden Spiral.

From this we can see that there may be some fractal properties in this object. First we need to define what a fractal is.

While it seems that there several competing definitions of what a fractal is, in a very broad sense A fractal is a shape made of parts similar to the whole in some way. We can see that this definition immediately applies to the Eye of God. Each smaller rectangle of the whole is proportional to the whole by some factor of φ. For example:

• 
  Golden Rectangle Stage 2

If you multiply the smaller rectangle by φ then you get the larger rectangle. This is a necessary property of a Golden rectangle. While the Eye of God has the property of self-similarity, it fails to have some other common fractal properties.

Fractals often will have an infinite length or preimeter, but the length of the Eye of God seems to converge towards a specific value. See Table Below:

Side Length	Number of Sides	Total Perimeter
1.618033989	2	3.236067977
1	2	5.236067977
0.618033989	1	5.854101966
0.381966011	1	6.236067977
0.236067977	1	6.472135955
0.145898034	1	6.618033989
0.090169944	1	6.708203932
0.05572809	1	6.763932023
0.034441854	1	6.798373876
0.021286236	1	6.819660113
0.013155617	1	6.83281573
0.008130619	1	6.840946349
0.005024999	1	6.845971347
0.00310562	1	6.849076968
0.001919379	1	6.850996346
0.001186241	1	6.852182588
0.000733137	1	6.852915725
0.000453104	1	6.853368829
0.000280034	1	6.853648862
0.00017307	1	6.853821933
0.000106963	1	6.853928896
6.6107E-05	1	6.853995003
4.08563E-05	1	6.854035859

Additionally, the areas of fractal shapes can often be unique, but the area of the Eye of God does not change with each new rectangle added to it. It remains φ square units and is 2-Dimensional.

While the Eye of God may technically be a fractal, it lacks many of the properties that make fractals interesting.

Hausdorff-Besicovitch Dimension

One of the most interesting properties of fractals are their strange dimensions. While Euclidian shapes have integer dimensions (a line: D=1; a square: D=2; a cube D=3) fractals can have non-integer dimensions. These can be determined by the use of the equation

Where, n is the number of self-similar parts of a shape in the whole f is the factor that the self-similar part must be multiplied by to get the original shape. d is the dimension of the shape.

This is used to find the Hausdorff-Besicovitch dimension of any geometric shape. For example, let's examine the dimension of the Eye of God. We'll use just the first two golden rectangles as our part and whole.

• 
  Golden Rectangle Stage 2

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.

f^d = 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.

• 
  Golden Fractal Stage 1 (R1)

Next divide the square into two golden rectangles with sides 1 by (φ-1).

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

We can do this by the definition of the golden rectangle and φ. Remove or color in the section in between the two rectangles. This area is (1-(φ-1)) by 1. (Note: (1-(φ-1)) + (φ-1) + (φ-1) = φ which is the side of our original rectangle)

• 
  Golden Fractal Stage 2 (R2)

Now that we have removed the middle we perform the same operation on the two golden rectangles available to us.

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

We have now created 4 new rectangles with sides (φ-1) by (1-(φ-1)). Color in the non-golden rectangle area.

• 
  Golden Fractal Stage 3 (R3)

Continue this pattern indefinitely.

• 
  Golden Fractal Stage 4 (R4)

Not all other iterations at each stage are shown. With a little bit of imagination one can fill in the empty spaces with ever-smaller golden rectangles, though.

Self-Similarity

This fractal has some particularly interesting properties. First, we can tell that it is self-similar because all the divisions are into golden rectangles and by definition all golden rectangles are similar to each other. Additionally, the number of rectangles added grows by a factor of 2 with each iteration. So, if x is the number of iterations, the number of new rectangles is 2^x.

Perimeter

Second, let’s look at the method of growth for the perimeter of the rectangles. We begin with a perimeter of 2(φ) + 2(1) which is the perimeter of a standard Golden Rectangle (R1). Next we add 2 more lines of length 1 each to divide the rectangle into two more rectangles (R2), so we get (2(φ) + 2(1) + 2(1)) as our total perimeter. This next time we add 4 lines to divide 2 rectangles into 4 (R3) and the lines are the length of the height of R2 or 4(φ-1) so our new total perimeter is: 2(φ) + 2(1) + 2(1) + 4(φ-1) Next we divide R3s and get 8 new rectangles with new lengths of (1-(φ-1)). Our new Perimeter is: 2(φ) + 2(1) + 2(1) + 4(φ-1) + 8(1-(φ-1))

Continuing this a few more times we get a perimeter of: 2(φ) + 2(1) + 2(1) + 4(φ-1) + 8(1-(φ-1)) + 16((φ-1) -(1-(φ-1))) + 32((1-(φ-1))-((φ-1) -(1-(φ-1))))

If you notice there is a pattern to both the coefficient and the expression with the φ’s of each term after the initial perimeter. The coefficients are the number of rectangles and we’ve already determined that this is an exponential function 2^x where x is the number of iterations. The second part of the term is the n-1 term subtracted from the n-2 term. This makes sense given the method of creating new golden rectangles involves subtracting the length from the height of the original golden rectangle. (See "Golden Rectangles" Above)

However, this process gets even more interesting once we simplify all the φ’s in each term. 2(φ) + 2(1) + 2(1) + 4(φ-1) + 8(2-φ) + 16(2φ-3) + 32(5-3φ) + 64(5φ-8) + 128(13-8φ) +…

From here we see that there seems to be a Fibonacci sequence in play between the difference terms, where the integer in each term is the sum of the previous term’s integer and the previous term’s φ coefficient. The coefficient of φ in the new term is simply the integer in the previous term. The series then finds the positive difference between them and multiplies that by the number of rectangles made in that iteration.

The significant question here is “What does this series sum to as the fractal iterations go to infinity?” We know that the 2^x is increasing infinitely as x increases (once again x is the number of iterations), but we have to see what the (I ± Cφ) portion is going to do before making a determination of where the summation is going.

We know that the difference is going to be less than or equal to 1 and will continue to get smaller as time goes on because the first term that isn’t our initial perimeter is 1 and φ-1 ≈ 0.6180 and as the Fibonacci sequence increases, the actual difference between the numbers in the sequence increases but the proportional difference does not. For example, 5 is 2 more than 3 and 13 is 5 more than 8, but 13 is only (8/13) = 1.625 time more than 8 while 5 is 1.666... times more than 3. We also know, based on our study of (φ) and Fibonacci sequences, that the ratio between 2 sequential Fibonacci numbers approaches φ as the sequence goes to infinity. Therefore as iterations continue the difference (I ± Cφ) will go to 0. This means we have to determine which between (I ± Cφ) and 2^x is growing or shrinking faster, because that will determine where our perimeter sequence goes to.

Because the Fibonacci sequence increases based on the addition of previous numbers, and the difference of those numbers in (I ± Cφ) approach a decreasing rate of 1/φ we then know that the (I ± Cφ) part of the term decreases at a slower rate than 2^x increases because it would have to be decreasing at least at a rate of .5 to match it.

Therefore, the series’ terms increase in size as the iterations increase and the perimeter of the fractal goes to infinity. This is demonstrated in the approximation of the series.

2(φ) + 2(1) + 2(1) + 4(φ-1) + 8(2-φ) + 16(2φ-3) + 32(5-3φ) + 64(5φ-8) + 128(13-8φ) +…

3.236+2+2+2.472+3.056+3.777+…

Area

Additionally, we can say that the area goes to 0 since as the fractal increases, all the area available gets divided up and goes to a single point where it isn’t just removed.

This is demonstrated in the following table:

Iteration	Area Removed	Total Area
0	0	1.618033989
1	0.381966011	1.236067977
2	0.291796068	0.94427191
3	0.22291236	0.72135955
4	0.17028989	0.55106966
5	0.1300899	0.42097976
6	0.09937984	0.321599919

Dimension

Finally, we can calculate the dimension of the fractal in the same way that we calculated the dimension of the Eye of God, f^d = n. Using the first iteration:

• 
  Golden Fractal Stage 2

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.