In the field of arts and mathematics, fractals are a reasonably new concept. That is probably why most people can only identify them as visually pleasing pictures worth using either computer screen backgrounds or as highly unique patterns on postcards. But what exactly are fractals all about?
A brief introduction:
Contrary to popular belief, fractals extend way beyond the idea of complicated shaped and attractive digitally generated pictures, as is generally attributed to them. Originating in the 1970s by Polish-born French and American mathematician, Benoit Mandelbrot, a fractal is in essence, a never-ceasing pattern. Anything that seems random and somewhat irregular can merit \ as a fractal. In their ubiquitousness, fractals make their way into various aspects of our lives. From the tiny confines of a cell membrane to the grand spectacle of a solar system, fractals make their presence felt in one way or the other. In a way, they are the irregular and unique footprints cast by the various unforeseeable forces that contribute to our world's inherent chaos. The term fractals have always found mention alongside the term chaos.
The universality of fractals:
Described by most authors as "patterns of chaos', fractals exist traditionally as a vastly convenient means of depicting and defining chaotic behaviour. However, if one were to look closely, they would observe that they carry unmissable traces of self-similarity and recurrence, which can be spotted within their arrangement. For many chaologists out there, the discipline of fractals and chaos is no less than a revolution unto itself. Besides being a new scientific field of study, this discipline exists to unify other academic ventures, such as computer science, mathematics, art, and theoretical physics. Thus, many consider it to be akin to discovering a new field of geometry- one that increasingly strives to explain the ceaselessness of the very universe that we thrive in.
A historical perspective:
Although a large chunk of fractal geometry is held synonymous with and is observed to have close associations with, the computer and digital techniques, the history of humanity's fascination with fractals can be traced back to when computers' invention was still a distant possibility. Since long fractals have extended beyond the conventional perception of mathematics as a compilation of monotonous and complicated formulae and postulates to exist as a field of study that combined mathematics with art. In this way, it successfully demonstrated that there was indeed more to mathematical expressions than a just representation of a collection and a sequence of numbers. What added more interest to their study was their ability to act as the most comprehensive mathematical descriptions of almost every natural form out there, such as the anatomy of living organisms, mountains, and coastlines.
The earliest British cartographers encountered countless difficulties in successfully estimating the measure of the British coast's entire stretch. When calculated on a large scale map, the coastline ended up being approximately half the measure of the coastline estimated with the help of a detailed plan. Consequently, they stumbled upon the realization that the closer they examined, the longer and more complicated the coastline appeared to be. Inadvertently, hence, in their study of coastlines, they had arrived at one of the most fundamental characteristics of fractals.
Need for fractal geometry:
Many human artefacts and the physical systems that exist in nature do not fall under the category of regular geometric forms that Euclid defined in his framework of standard geometry. This is where fractal geometry comes into the picture. It exists to provide seemingly endless ways of approximating, predicting, and describing this series of natural phenomena. In theory, it could be argued that anything and everything we behold under us can be classified as fractals; From the veins in our hands and the branching of our tracheal tubes to the leaves of a tree and the stock market machinations.
The geometry that we learned in school taught us to identify and construct an assortment of various shapes. In that sense, the study of fractal geometry is not much different either. The only distinction lies in the fact that while classical geometry concerned itself with the definition of shapes, traditionally considered to be regular and smooth such as a triangle or a circle, fractal geometry mainly exists to explain infinitely complex and rough shapes. However, at its heart, fractal geometry is all about defining, calculating, and approximating shapes, much like its classic counterpart.
The relevance of fractal geometry:
Of the many reasons that make themselves sufficiently consequential, at the outset, there are a set of few pivotal ones which elaborate on why one should concern themselves with the study and research of fractal geometry.
- The methodology used to construct shapes in fractal geometry is vastly different from what traditional methodology to construct forms under classical geometry. Although both processes are incredibly simple in their capacities, they present a stark contrast to each other.
- While classical geometry typically uses an assortment of defined formulas to define the various shapes, fractal geometry employs the concept of iterations to lend a voice to its set of complicated and rough figures.
- As a result, the study of fractal geometry involves taking a break from any existing conventional frameworks of geometry, as compiled by Plato, Pythagoras, and Euclid's collective prodigious minds.
- Fractal geometry has only witnessed 40 years of investigation compared to the lofty 200 years of scrutinization and exploration in classical geometry.This further puts matters into perspective.
- The incredible fact that most shapes in fractal geometry look natural. It is a well-known fact that there are neither perfect circles nor any ideal squares and triangles in the heart. To put it broadly, none of the shapes defined by classical geometry has a flawless occurrence in reality- a matter that highlights the relevance of fractal geometry more than anything else.
- None of the shapes that we see around us, ranging from trees and mountains to roads and buildings, even remotely resemble the large assortment of various structures and forms defined under conventional mathematics. Using fractal geometry, however, along with the aid of a few simple principles that are iterated several times, one can model this series of natural phenomena with an accuracy that is as alarming as it is insightful.
- In essence, fractal geometry gives you the ability to view the entire world around us through the lens of simple mathematics. In the course, it unravels various mysteries of the universe for us and ends up imbuing in us a sound practical understanding of things.
Applications of fractal geometry:
- Astronomy: Cosmologists are increasingly reliant on the study of fractals in general, and fractal geometry in particular, to shape their understanding of the machinations of the universe. The homogeneity of the current framework of statistical methods proving to be an inconsistent matter has pushed scientists to consider the possibility of fractals a means of resolving their problem statements.
- Nature: Ranging from a bundle of leaves to a snowflake, the ubiquitousness of fractals helps us gain valuable insights into the various forces that are in play in nature.
- Computer Science: Fractals are being increasingly used in the field of computer science to drive innovations in the field of fractal image compression.
Now that you have read about the various properties of fractal geometry and its relevance in our daily tidings, you must have understood the overall importance of geometry around us. Consequently, it becomes very important for us to impart the same sense in our children right at the very beginning of their academic journeys. Students often consider mathematics as an overwhelming and intimidating subject, and read it just for the sake of examinations; not to expand their practical outlook. With the help of Cuemath's immersive learning curriculum, students can now understand various geometry formulas and examples with ease and visualize its relevance in a practical context. They have a personalized learning plan for every student and focus on building the concepts so that instead of fearing math, students feel at ease with it. Subsequently, they grow increasingly familiar with the various concepts that the subject entails.
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