Fractals: No Matter How Close You Look, It’S The Same

Have you ever looked at a snowflake and zoomed into it, and you saw that it was had the same pattern as before you zoomed in, and you wondered why that happened? This is due to the mathematical term called fractal patterns. Fractals come in all shapes and sizes, from something as big as a mountain range to a human cell. They can be two-dimensional or three-dimensional, physical or digital, however, just looking at something isn’t enough to learn about fractals. You need to look into the background of fractals, and the mathematical concepts to further your comprehension. Only then, can you fully learn what a fractal is and how to be able to apply it in the real world. The information in the following pages are from research and experiments done by mathematicians, biologists, astronomers, and computer scientists in their respective fields.

Introduction

A fractal pattern is a shape that stays the same at any magnification no matter how far you zoom in. It has been recently proven in the 20th century that fractals have been in almost everything in nature, “the word ‘fractal’ was coined by Benoit Mandelbrot, sometimes referred to as the father of fractal geometry”. Fractals, however, are not only shapes. Rather, they are found all around us and are used in many fields of study. Fractal patterns are not what is called traditional geometry and that it is in its own classification of geometry. Fractal patterns are also called chaotic because “fractal geometry links…chaotic dynamics”. In this paper, we will discuss the uniqueness of fractals along with some mathematical examples as well as how they can be used in the real world as well as in other disciplines, besides, mathematics.

Self-Similarity – It’s the Same!

Fractals are unique in that they are self-similar. At first glance, a fractal pattern is just a normal object, i. e. , a leaf, a mountain, a snowflake, however, as you zoom in on the object, you realize that the object is of the same pattern/shape. An object is self-similar when it “possesses symmetry across scale, with each small part of the object replicating the structure of the whole”. They are duplicates of the original entity. There are two types of self-similarity: statistical and exact. Statistical self-similarity pertains to “(the same degree of ruggedness) as we zoom in”. At every magnification, the fractal pattern is repeated, however, the pattern is not completely the same to the original. The copied object is not like the original. Examples of statistical self-similarity are "the boundary of clouds, wall cracks, a hillside silhouette and a fern”.

Exact self-similarity relates to a fractal that is self-similar and is an exact copy of the original pattern in every aspect of the fractal. No matter where you zoom in, the pattern of the original is seen in the duplicate. An exact self-similar object is also known as a normal fractal. An example of exact self-similarity is a fern because “each frond of the fern is a mini-copy of the whole fern, and each frond branch is similar to the whole frond”.

Examples of Fractals

Fractals are everywhere. They can come in many forms. In math and in nature, fractals can be found, and most can utilize math to calculate their dimensions, and the number of patterns. Some fractals that are well known are the Cantor set, Koch Snowflake, Sierpinski’s carpet and gasket, and the Menger sponge.

Cantor Set

The Cantor set is a fractal that is “one of the most frequently quoted objects”. The Cantor set is an easy fractal to learn is because there aren’t any difficult or complicated shapes. They only utilize an infinite number of lines, and those lines disappear. The most straightforward Cantor set is the triadic Cantor set. To create a Cantor set, the first thing is to do is “remove the middle third of the unit line…From the two remaining line segments…the middle thirds are again removed…and so on to infinity”. The limitation of the Cantor set is its ability to be seen after a certain point.

Koch’s Snowflake/Curve

Another example that is similar to the Cantor set is the Koch curve also known as the triadic Koch curve. It is similar in that it is “simply constructed using an iterative procedure beginning with the initiator of the set as the unit line segment (step k = 0 in the figure)”. The same pattern will repeat infinitely many times. This illustrates the object’s self-similarity. The dimension that is used to describe the Koch Snowflake or the Koch curve is called the similarity dimension. To calculate the similarity dimension of the Koch curve, we use.

In addition to the triadic curve there is also the quadratic curve. Similar to the triadic Koch curve, equation (1. 0) can be used to compute the similarity dimension of the quadratic curve.

Sierpinski’s Gasket and Carpet

Another mathematical object that has the property of self-similarity is the Sierpinski’s carpet. “The initiator in this case is a filled triangle in the plane. Then the middle triangular sections are removed from the remaining triangular elements and so on. After infinite iterations the Sierpinski gasket is formed”. Similar to Sierpinski’s gasket is Sierpinski’s carpet, however, “this time the initiator is a square and the generator removes the middle square, side length one-third, of the original square”. The similarity dimensions of the Sierpinski carpet and Sierpinski gasket is calculated using equation (1. 0) similar to the Koch triadic curve similarity dimension and the Koch quadratic curve similarity dimensions are.

The Menger Sponge

A third mathematical fractal is the Menger sponge. The Menger sponge is an “object constructed in 3D space…The initiator is a cube”. The creation of a Menger sponge “is closely related to the Sierpinski carpet”. Applying the same method done to Sierpinski’s carpet and gasket, a cube is taken out of the original cube and cubes of increasingly smaller sizes are taken out around where the first cube was taken out. Like the Sierpinski’s gasket and carpet, as well as Koch’s triadic and quadratic curves, the Menger sponge’s similarity dimension utilizes the same equation, equation (1. 0).

Natural Fractals

Fractals are also found in nature. ‘The fractals of figure 1. 0 require a two-dimensional (2D) plane to ‘live in’, that is all the points on them can be specified using only two co-ordinates”. This says that many fractals are 2D but there are some that are 3D. “However, many natural fractals need a 3D world in which to exist”. An example of a three-dimensional fractal is “a tree whose branches weave through three dimensions; see the tree branching in 3D in figure 1. 5”. Natural fractals need to be in the third dimension because they have length, width, and height. They are objects that can be held, however, fractals can be taken from the second and third dimensions to be utilized not only in mathematics but also in other fields of study.

Fractal Applications

Fractals is a mathematical term, and the proofs to prove them reside in the mathematics field, however, their applications are not just in mathematics. Aside from mathematics, fractals can be applied in many other different fields of study. In the fields of biology, astronomy, computer science, and many, many more fields have used fractal patterns to further not only their individual field but also the world.

Fractals in biology

The world is filled with fractals created by nature. They are all around us and some are more easy to see than others. A couple of nature made fractals are turbulence and wildfires These fractals are chaotic and self-similar patterns can be found. In cardiology, “a cardiovascular surgeon can detect with the regular (and more so, irregular) beating of the human heart”. The beat of the heart is seen to have a pattern that either normal or not is a fractal pattern. This is because the beat of a heart is the same, in the sense of beating, therefore, there is a pattern that repeats until the heart stops beating.

Fractals in Astronomy

Fractals are not only on earth; they are also in space. The stars in the night sky form self-similar patterns. Due to gravity, every star rearranges themselves in the night sky so that they are far enough away from each other, so they do not collide. Collisions occur because every star has their own gravitational field and thus, create patterns or constellations that can be viewed on Earth at night. “The behavior of correlation function over a range of length-scale is a power law in nature. This observation led Pietronero (1987) 15 to propose that the distribution of galaxies follows closely the fractal distribution which themselves are self-similar in nature”. Almost everything no matter where, can be a fractal as observed by Pietronero and later confirmed by “a study by Sylos Labini et al (1998) 17 ruled the galaxy distribution to be of fractal nature with no transition to homogeneity on any length-scale”. These two observations and research done by Pietronero and Labini proves that fractals exist outside of this world. The arrangement of the stars is something of a natural work of art because constellations form pictures and shapes in the night sky.

Fractals in Computer Science

Fractals are also in the digital world as well. They are the backbone of many programs we use today and the pictures and movies we see in entertainment. In computer science, fractals have been utilized in a “space filling algorithm described here differs from the usual packing algorithms”. The use of fractals in the digital world is to be able to make a digital object and be able to fill it without “the shapes having any contact points”. Fractals play a key part because when there are repeated patterns, the patterns never touch, unless the original pattern had a part that intersected.

Another use of fractals in computer science deals with digital media on the internet. This usage is “fractal image coding which is based on the iterated function system”. Image coding have been used “To protect against this illegal piracy and malicious manipulation of digital images, digital watermarking techniques that embed a digital signature into an image have been proposed”. The use of fractals in this manner is both for computer science and for law enforcement because digital images are being protected by water marks and are preventing “illegal distribution and modification of digital content illegal distribution and modification of digital content”.

Conclusion

Fractals, in the end, are a very interesting part of mathematics. They apply to almost every field of study. The mathematical examples are the best and easiest way to begin your study into the topic of fractals, however, you need to start off with definition of self-similarity so that you can understand what a fractal really is.