Dimensionality

We see things in either 2 or 3 dimensions. But what about a 4th dimension? Physics debates whether we exist within 3 or 4 dimension. Sacred geometry takes all 4 dimensions into consideration.

The progression from point (0-dimensional) to line (1-dimensional) to plane (2-dimensional) to space (3-dimensional) and beyond leads us to the question — if mapping from higher order dimensions to lower ones loses vital information (as we can readily observe with optical illusions resulting from third to second dimensional mapping), does our “fixation” with a 3-dimensional space introduce crucial distortions in our view of reality that a higher-dimensional perspective would not lead us to?

Fractals and Recursive Geometries

Fractals are a relatively new form of mathematics, beginning only in the 17th century. A good example of a fractal form is a fern. Each leaf on a fern is made up of smaller leaves that have the same shape of the larger whole. In recursive geometry the formula making up a form can be used repeatedly.

Most physical systems of nature and many human artifacts are not regular geometric shapes of the standard geometry derived from Euclid. Fractal geometry offers almost unlimited ways of describing, measuring and predicting these natural phenomena. But is it possible to define the whole world using mathematical equations?

This article describes how the four most famous fractals were created and explains the most important fractal properties, which make fractals useful for different domain of science.

Fractals’ properties

Two of the most important properties of fractals are self-similarity and non-integer dimension. What does self-similarity mean? If you look carefully at a fern leaf, you will notice that every little leaf – part of the bigger one – has the same shape as the whole leaf. You can say that the fern leaf is self-similar. The same is with fractals: you can magnify them many times and after every step you will see the same shape, which is characteristic of that particular fractal.

The non-integer dimension is more difficult to explain. Classical geometry deals with objects of integer dimensions: zero dimensional points, one dimensional lines and curves, two dimensional plane figures such as squares and circles, and three dimensional solids such as cubes and spheres. However, many natural phenomena are better described using a dimension between two whole numbers. So while a straight line has a dimension of one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it twists and curves. The more the flat fractal fills a plane, the closer it approaches two dimensions. Likewise, a “hilly fractal scene” will reach a dimension somewhere between two and three. So a fractal landscape made up of a large hill covered with tiny mounds would be close to the second dimension, while a rough surface composed of many medium-sized hills would be close to the third dimension.

There are a lot of different types of fractals. In this paper I will present two of the
most popular types: complex number fractals and Iterated Function System (IFS)
fractals.

Complex number fractals

Before describing this type of fractal, I decided to explain briefly the theory of complex numbers. A complex number consists of a real number added to an imaginary number. It is common to refer to a complex number as a “point” on the complex plane. If the complex number is, the coordinates of the point are an imaginary axis.

The unit of imaginary numbers: $i - \sqrt{-1}$

Two leading researchers in the field of complex number fractals are Gaston Maurice Julia and Benoit Mandelbrot. Gaston Maurice Julia was born at the end of 19th century in Algeria. He spent his life studying the iteration of polynomials and rational functions. Around the 1920s, after publishing his paper on the iteration of a rational function, Julia became famous. However, after his death, he was forgotten. In the 1970s, the work of Gaston Maurice Julia was revived and popularized by the Polish-born Benoit Mandelbrot. Inspired by Julia’s work, and with the aid of computer graphics, IBM employee Mandelbrot was able to show the first pictures of the most beautiful fractals known
today.

Mandelbrot set

The Mandelbrot set is the set of points on a complex plain. To build the Mandelbrot set, we have to use an algorithm based on the recursive formula: $$ Z_{n} = Z_{n-1}^{2} + C $$
separating the points of the complex plane into two categories:

• points inside the Mandelbrot set,
• points outside the Mandelbrot set.

The image below shows a portion of the complex plane. The points of the Mandelbrot set have been colored black.

![[img/sg_img-132.png]]

It is also possible to assign a color to the points outside the Mandelbrot set. Their
colors depend on how many iterations have been required to determine that they are
outside the Mandelbrot set.

![[img/sg_img-133.png]]

How is the Mandelbrot set created?

To create the Mandelbrot set we have to pick a point $C$ on the complex plane. The complex number corresponding with this point has the form: $C= a + b \cdot i$.

After calculating the value of previous expression:$$ Z_{1} = Z_{0}^{2} + C $$using zero as the value of $Z_o$ we obtain $C$ as the result. The next step consists of assigning the result to $Z_1$ and repeating the calculation: now the result is the complex number $C_2 + C$. Then we have to assign the value to $Z_2$ and repeat the process again and again.

This process can be represented as the “migration” of the initial point $C$ across the

plane. What happens to the point when we repeatedly iterate the function? Will it
remain near to the origin or will it go away from it, increasing its distance from the
origin without limit? In the first case, we say that $C$ belongs to the Mandelbrot set (it
is one of the black points in the image); otherwise, we say that it goes to infinity and
we assign a colour to C depending on the speed at which the point “escapes” from
the origin.

We can take a look at the algorithm from a different point of view. Let us imagine that all the points on the plane are attracted by both: infinity and the Mandelbrot set. That makes it easy to understand why:

• points far from the Mandelbrot set rapidly move towards infinity,
• points close to the Mandelbrot set slowly escape to infinity,
• points inside the Mandelbrot set never escape to infinity.

Julia sets

Julia sets are strictly connected with the Mandelbrot set. The iterative function that is used to produce them is the same as for the Mandelbrot set. The only difference is the way this formula is used. In order to draw a picture of the Mandelbrot set, we iterate the formula for each point $C$ of the complex plane, always starting with $Z_0 = 0$. If we want to make a picture of a Julia set, $C$ must be constant during the whole generation process, while the value of $Z_0$ varies. The value of $C$ determines the shape of the Julia set; in other words, each point of the complex plane is associated with a particular Julia set.

How is a Julia set created?

We have to pick a point $C$ on the complex plane. The following algorithm determines
whether or not a point on complex plane $Z$ belongs to the Julia set associated with $C$, and determines the color that should be assigned to it. To see if $Z$ belongs to the set, we have to iterate the function $Z_1 = Z_0^2 +C$ using $Z_0 = Z$. What happens to the initial point $Z$ when the formula is iterated? Will it remain near to the origin or will it go away from it, increasing its distance from the origin without limit? In the first case, it belongs to the Julia set; otherwise it goes to infinity and we assign a color to Z depending on the speed the point “escapes” from the origin. To produce an image of the whole Julia set associated with C, we must repeat this process for all the points Z whose coordinates are included in this range: $$2 < 2; -1,5 < y < 1,5 $$The most important relationship between Julia sets and Mandelbrot set is that while the Mandelbrot set is connected (it is a single piece), a Julia set is connected only if it is associated with a point inside the Mandelbrot set. For example: the Julia set
associated with $C_1$ is connected; the Julia set associated with $C_1$ is not connected (see picture below).

![[img/sg_img-148.png]]

Iterated Function System Fractals

Iterated Function System (IFS) fractals are created on the basis of simple plane
transformations: scaling, dislocation and the plane axes rotation. Creating an IFS fractal consists of following steps:

1. defining a set of plane transformations,
2. drawing an initial pattern on the plane (any pattern),
3. transforming the initial pattern using the transformations defined in first step,
4. transforming the new picture (combination of initial and transformed patterns) using the same set of transformations,
5. repeating the fourth step as many times as possible (in theory, this procedure can be repeated an infinite number of times).

The most famous ISF fractals are the Sierpinski Triangle and the Koch Snowflake.

Sierpinski Triangle

This is the fractal we can get by taking the midpoints of each side of an equilateral triangle and connecting them. The iterations should be repeated an infinite number of times. The pictures below present four initial steps of the construction of the Sierpinski Triangle:

![[img/sg_img-149b.png]]

Using this fractal as an example, we can prove that the fractal dimension is not an integer. First of all we have to find out how the “size” of an object behaves when its linear dimension increases. In one dimension we can consider a line segment. If the linear dimension of the line segment is doubled, then the length (characteristic size) of the line has doubled also. In two dimensions, if the linear dimensions of a square for example is doubled then the characteristic size, the area, increases by a factor of 4. In three dimensions, if the linear dimension of a box is doubled then the volume increases by a factor of 8.

This relationship between dimension $D$, linear scaling $L$ and the result of size increasing $S$ can be generalized and written as: $$S = L \cdot{D}$$Rearranging of this formula gives an expression for dimension depending on how the size changes as a function of linear scaling: $$ D = \frac{\log(S)}{\log(L)} $$In the examples above the value of $D$ is an integer $-1$, $2$, or $3$ depending on the dimension of the geometry. This relationship holds for all Euclidean shapes. How about fractals?

Looking at the picture of the first step in building the Sierpinski Triangle, we can notice that if the linear dimension of the basis triangle $L$ is doubled, then the area of whole fractal (blue triangles) increases by a factor of three $S$.

Using the pattern given above, we can calculate a dimension for the Sierpinski Triangle: $$ D = \frac{\log(3)}{\log(2)} = 1,585 $$The result of this calculation proves the non-integer fractal dimension.

Koch Snowflake

To construct the Koch Snowflake, we have to begin with an equilateral triangle with sides of length, for example, $1$. In the middle of each side, we will add a new triangle one-third the size; and repeat this process for an infinite number of iterations. The length of the boundary is $3 \cdot \frac{4}{3} \cdot \frac{4}{3} \cdot \frac{4}{3} \cdot \dots$ -infinity. However, the area remains less than the area of a circle drawn around the original triangle. That means that an infinitely long line surrounds a finite area. The end construction of a Koch Snowflake resembles the coastline of a shore.

Four steps of Koch Snowflake construction:

![[img/sg_img-157.png]]

Another IFS fractals:

![[img/sg_img-158a.png]]

Fractal applications

Fractal geometry has permeated many area of science, such as astrophysics, biological sciences, and has become one of the most important techniques in computer graphics.

Fractals in astrophysics

Nobody really knows how many stars actually glitter in our skies, but have you ever wondered how they were formed and ultimately found their home in the Universe? Astrophysicists believe that the key to this problem is the fractal nature of interstellar gas. Fractal distributions are hierarchical, like smoke trails or billowy clouds in the sky. Turbulence shapes both the clouds in the sky and the clouds in space, giving them an irregular but repetitive pattern that would be impossible to describe without the help of fractal geometry.

Fractals in the Biological Sciences

Biologists have traditionally modeLled nature using Euclidean representations of natural objects or series. They represented heartbeats as sine waves, conifer trees as cones, animal habitats as simple areas, and cell membranes as curves or simple surfaces. However, scientists have come to recognize that many natural constructs are better characterized using fractal geometry. Biological systems and processes are typically characterized by many levels of substructure, with the same general pattern repeated in an ever-decreasing cascade.

Scientists discovered that the basic architecture of a chromosome is tree-like; every
chromosome consists of many ‘mini-chromosomes’, and therefore can be treated as fractal. For a human chromosome, for example, a fractal dimension D equals 2,34 (between the plane and the space dimension).

Self-similarity has been found also in DNA sequences. In the opinion of some biologists fractal properties of DNA can be used to resolve evolutionary relationships in animals. Perhaps in the future biologists will use the fractal geometry to create comprehensive models of the patterns and processes observed in nature.

Fractals in computer graphics

The biggest use of fractals in everyday live is in computer science. Many image compression schemes use fractal algorithms to compress computer graphics files to less than a quarter of their original size. Computer graphic artists use many fractal forms to create textured landscapes and other intricate models.

It is possible to create all sorts of realistic “fractal forgeries” images of natural scenes, such a lunar landscapes, mountain ranges and coastlines. We can see them in many special effects in Hollywood movies and also in television advertisements. The “Genesis effect” in the film “Star Trek II - The Wrath of Khan” was created using fractal landscape algorithms, and in “Return of the Jedi” fractals were used to create the geography of a moon, and to draw the outline of the dreaded “Death Star”. But fractal signals can also be used to model natural objects, allowing us to define mathematically our environment with a higher accuracy than ever before.

The Geometry of Fractal Shapes

• To explain the process by which fractals such as the Koch snowflake and the Sierpinski Gasket are constructed.
• To recognize self-similarity (or symmetry of scale) and its relevance.
• To describe how random processes can create fractals such as the Sierpinski Gasket.
• To explain the process by which the Mandelbrot set is constructed.

The Koch Snowflake (Recursive Construction)

• Start. Start with a solid equilateral triangle (a). The size of the triangle is irrelevant, so for simplicity we will say that the sides of the triangle are of length 1.
• Step 1. To the middle third of each of the sides of the original triangle add an equilateral triangle with sides of length 1/3, as shown in (b). The result is the 12-sided ―Star of David‖ shown in ©.
• Step 2. To the middle third of each of the 12 sides of the star in Step 1 add an equilateral triangle with sides of length one-third the length of that side.
• Step 2 (cont). The result is a ―snowflake with 12 x 4 = 48 sides, each of length (1/3)² = 1/9, as shown in (a). (Each of the sides ―crinkles into four new sides; each new side has length 1/3 the previous side.)
• Step 3. Apply Procedure KS to the “snowflake” in Step 2. This gives the more elaborate ―”snowflake” shown in (b). Without counting we can figure out that this snowflake has 48 x 4 = 192 sides, each of length (1/3)³ = 1/27.
• Step 4. Apply Procedure KS to the “snowflake” in Step 3. This gives the “snowflake” shown in ©. (You definitely don’t want to do this by hand – there are 192 tiny little equilateral triangles that are being added!)
• Step 5, 6 etc. Apply Procedure KS to the “snowflake” obtained in the previous step.

At each step of this process we create a new “snowflake”, but after a while it’s hard to tell that there is any change. For all practical purposes we are seeing the ultimate destination of this trip: the Koch snowflake itself as shown by the figure on the right.

The Koch snowflake is a fairly complicated shape, but we can define it in two lines using a form of shorthand we will call a replacement rule – a rule that specifies how to substitute one piece for another.

• Start: Start with a solid equilateral triangle $\triangle$ .
• Replacement Rule: Whenever you see a boundary line segment, apply Procedure KS to it.

If we only consider the boundary of the Koch snowflake and forget about the interior, we get an infinitely jagged curve known as the Koch curve (or sometimes called the snowflake curve) shown in (a).

Clearly (a) is just a rough rendering of the Koch curve, so our natural curiosity pushes us to take a closer look. We’ll just randomly pick a small section of the Koch curve and magnify it (b). The surprise (or not!) is that we see nothing new– the small detail looks just like the rough detail. Figure © shows a detail of the Koch curve after magnifying it by a factor of almost 100.

To compute the boundary of the Koch snowflake, let’s look at the boundary of the figures obtained in steps 1 and 2 of the construction in the above figure. At each step we replace a side by four sides that are 1/3 as long.

Thus, at any given step the perimeter is 4/3 times the perimeter at the preceding step. This implies that the perimeters keep growing with each step, and growing very fast indeed. Therefore,

• The Koch snowflake has infinite perimeter.

To compute the exact area of the Koch snowflake is considerably more difficult, but,
as we see from the above figure, the Koch snowflake fits inside the circle that
circumscribes the original equilateral triangle. Therefore,

• The area of the Koch snowflake is 1.6 times the area of the starting equilateral triangle.

The Sierpinski Gasket

• Plurality method

Election of 1st place votes

• Plurality candidate

The Candidate with the most 1st place votes.

The Sierpinski Gasket (Recursive Construction)

• Start. Start with any solid triangle ABC (a). (Often an equilateral triangle or a right triangle is used, but here we chose a random triangle to underscore the fact that it can be a triangle of arbitrary shape.)
• Step 1. Remove the triangle connecting the midpoints of the sides of the solid triangle. This give the shape shown in (b)– consisting of three solid triangles each a half-scale version of the original and a hole where the middle triangle used to be.
• Step 2. To each of the three triangles in (b) apply Procedure SG. The result is the “gasket” shown in © consisting of 32 = 9 triangle each at one-fourth the scale of the original triangle, plus three small holes of the same size and one larger hole in the middle.
• Step 3. To each of the three nine triangles in © apply Procedure SG. The result is the gasket shown in (d) consisting of 33 = 27 triangle each at one-eighth the scale of the original triangle, nine small holes of the same size, three medium-size holes and one large hole in the middle.
• Step 4, 5, etc. Apply Procedure SG to each triangle in the “gasket” obtained in the previous step.

You can think of the figure on the right as a picture of the Sierpinski gasket (in reality it is the gasket obtained at step 7 of the construction process.

The Sierpinski gasket is clearly a fairly complicated geometric shape, and yet it can be defined in two lines using the following recursive replacement rule.

The Sierpinski Gasket

• Start: Start with an arbitrary solid triangle $\triangle$.
• Replacement rule: Whenever you see a $\triangle$ apply Procedure SG to it.

As a geometric object existing in the plane, the Sierpinski gasket should have an area, but it turns out that its area is infinitely small, smaller than any positive quantity. Therefore,

• The Sierpinski gasket has zero area but infinitely long boundary.

The Chaos Game

This example involve the laws of chance. We start with an arbitrary triangle with vertices A, B, and C and an honest die (a). Before we start we assign two of the six possible outcomes of rolling the die to each of the vertices of the triangle.

• Start. Roll the die. Start at the “winning” vertex. Say we roll a 5. We then start at vertex C at figure (b).
• Step 1. Roll the die again. Say we roll a 2, so the winner is vertex A. We now move to the point M¹ halfway between the previous position C and the winning vertex A. Mark a point at the new position M¹ (see figure c).
• Step 2. Roll the die again, and move to the point M¹ and the winning vertex. Say we roll a 3– the move then is to M2 halfway between M1 and B as shown in(d). Mark a point at the new position M².
• Step 3, 4, etc. Continue rolling the die, each time moving halfway between the last position and the winning vertex and marking that point.

The Twisted Sierpinski Gasket***

Our next example is a simple variation of the original Sierpinski gasket. We will call it the twisted Sierpinski gasket. For convenience, we will use the term Procedure TSG to describe the combination of the two moves (“cut” and then “twist”).

• Cut. Cut the middle out of a triangle (b).
• Twist. Translate each of the midpoints of the sides by a small random amount and in a random direction ©.

When we repeat Procedure TSG in an infinite recursive process, we get the twisted Sierpinski gasket.

The Twisted Sierpinski Gasket*** (Recursive Construction)

• Start. Start with an arbitrary solid triangle such as shown in figure (a).
• Step 1. Apply Procedure TSG to the starting triangle. This gives the “twisted gasket” shown in (b), with three twisted triangles and a (twisted) hole in the middle.
• Step 2. To each of the three triangles in (b), apply Procedure TSG. The result is the “twisted gasket” shown in ©, consisting of nine twisted triangles and four holes of various sizes.
• Step 3, 4, etc. Apply Procedure TSG to each triangle in the “twisted gasket” obtained in the previous step.

The figure above shows an example of a twisted Sierpinski gasket at step 7 of the construction. Even without touch-up, we can see that this image has the unmistakable look of a mountain.

The construction of the twisted Sierpinski gasket can be also described by a two-line recursive replacement rule.

Twisted Sierpinski Gasket

• Start: Start with an arbitrary solid triangle.
• Replacement rule: Wherever you see a solid triangle, apply Procedure TSG to it.

The Mandelbrot Set

Complex Numbers and Mandelbrot Sequences

The Mandelbrot set can be described mathematically as a recursive process involving simple computations with complex numbers. The complex number (a + bi) can be identified with the point (a,b) in a Cartesian coordinate system as shown to the right.

Mandelbrot Sequence

The key concept in the construction of the Mandelbrot set is that of a Mandelbrot sequence.** A Mandelbrot sequence** (with seed s) is an infinite sequence of complex numbers that starts with an arbitrary complex number s and then each successive term in the sequence is obtained recursively by adding the seed s to the previous term squared.

Much like the Koch snowflake and the Sierpinski gasket, a Mandelbrot sequence can be defined by means of a recursive replacement rule:

Mandelbrot Sequence

• Start: Choose an arbitrary complex number s, called the seed of the Mandelbrot sequence. Set the seed s to be the initial term of the sequence (s⁰ = s).
• Procedure M: To find the next term in the sequence, square the preceding term and add the seed (s ^N+1 = s^2N + s).

The Mandelbrot Set

If the Madelbrot sequence is periodic or attracted, the seed is a point of the Mandelbrot set and assigned the color black; if the Mandelbrot sequence is escaping, the seed is a point outside the Mandelbrot and assigned color that depends on the speed at which the sequence is escaping (hot colors for slowly escaping sequences, cool colors for fast escaping sequences).