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In mathematics, fractals can be defined as any class of complex geometric shapes that are commonly known to have a “fractional dimension”. The concept of fractals was first introduced by the mathematician Felix Hausdorff in the year 1918.

Fractals are distinct from the simple figures of classical / Euclidean, geometry—that is various figures like the square, the circle, the sphere, etc.

These fractals are basically capable of describing spatially nonuniform phenomena or many irregularly shaped objects in nature such as coastlines as well as mountain ranges.

The term fractal is derived from the Latin word fractus (which means “fragmented,” or “broken”), was coined by the Polish-born mathematician.

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Fractals have a fine structure at arbitrarily minute scales.

It is self-similar (at least approximately or stochastically).

Fractal has a Hausdorff dimension which is higher than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).

It has a simple and recursive definition.

We can create Images of fractals using fractal-generating software. Images that are produced by such software can be normally referred to as fractals even if they do not have the characteristics listed above. Also, these may include display artifacts or these may include calculations that are not characteristics of true fractals.

Escape-Time

Iterated Function Systems

Random

The Major Different Types of Categories of Fractals are Given Below -

Fractals in Nature

Fractals in Computers

Fractal Shapes

Fractals in Math

Fractals in 3D modelling

Fractal in Information and Data Management

Fractals in Computer System Architecture

Fractals in other areas of Technology

Fractals in Physical Structures

Fractals and Human Psychology

Fractals in Time

Fractals in Sound

Fractals in Art

Fractals in Law

We can see fractals in the circulatory and respiratory systems of animals. If you take the example of a human respiratory system, then you will be able to see a Fractal that begins with a single trunk (which is similar to the tree) that branches off and it expands into a fine-grained network of cavities.

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Fractals can be seen in the branches of trees from the way a tree grows limbs or branches. In a tree, the origin point for the Fractal is the main trunk of the tree and each set of branches that grows off the main trunk of the tree. Eventually, the branches become small enough and they become twigs, and these twigs will eventually grow into bigger branches and these will have twigs of their own. This cycle generally creates an “infinite” pattern of tree branches where each and every branch of the tree resembles a smaller scale version of the whole shape.

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You might have heard that each and every snowflake is unique. This uniqueness of snowflakes is said to be one of the contributing factors that they form in fractal patterns which can allow for incredible amounts of detail as well as variation. In the case of the formation of ice crystals, the center is said to be the starting point of the Fractal and the shape expands outward in all directions.

As and when the crystal expands, these Fractal structures are formed in each and every direction. Just like the other different examples of Fractals, each iteration of the shape gets smaller as well as with each iteration it gets more detailed, which also contributes to the overall complexity of the shape.

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Clouds also display characteristics of Fractals. The turbulence that is found within the atmosphere has an interesting impact on the way water particles interact with each other. Turbulence is known to be Fractal in nature and therefore it is known to have a direct impact on the formation and visual look of clouds.

The amount of condensation, the number of ice crystals, as well as the amount of precipitation expelled from the clouds, impact the state of the cloud as well as the system’s structure and therefore the turbulence.

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Hundreds and thousands of books have been written exploring the mathematical intricacies of Fractals, let’s go through some of them. We will explore Fractals as they are represented by math formulas, the concept of dimensionality, as well as how Fractals exhibit Fractional Dimensions, and how some of the most iconic Fractal shapes were created using math.

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Just like Fractals in nature, the Fractal shapes shown in the given image above are self-similar and they are identical regardless of what scale you look at.

The solid Triangle in the picture given above is known as a Sierpinski Gasket. To create a Sierpinski Gasket you simply need to begin with a single triangle, and with each iteration, you will start to remove the center of the triangle. With each and every repetition, you will see that more and more of the triangles become empty space.

FAQ (Frequently Asked Questions)

Question 1) Are Humans Fractals? What are Examples of Fractals?

Answer) We are fractal. Our lungs, our circulatory system as well as our brains are like trees. They are fractal structures. Most natural objects - which include us human beings as well as nature - are composed of many different types of fractals woven into each other, each with parts that have different fractal dimensions.

**A Few Examples of Fractals in Nature are Listed below -**

Fractals in branches of trees, fractals in clouds, and crystals, animal circulatory systems, snowflakes, fractals in lightning and electricity, fractals in plants as well as leaves, geographic terrain, as well as river systems.

Question 2) What are the Characteristics of a Fractal?

Answer) A fractal often has the following features:

A fractal has a fine structure at arbitrarily minute scales.

A fractal is self-similar (can be said at least approximately or stochastically).