Hidden Dimensions of Nature
Symmetry in the world around us provides us with the most aesthetically pleasing phenomenon; from its application in the human body and other animals, simple and complex, to plants and a massive number of other phenomena, the symmetry is what appeals to most of us. However, it is not just the aesthetics provided by symmetry that makes it universal, it is the efficiency achieved through it that makes it universally applicable.
Normally, when we speak of symmetry we talk about symmetry under three transformations of reflection, rotation and translation. However, there is a fourth aspect of ‘self-similarity’ that is left out of the conversation. The applications of ‘self-similarity’ are just as widespread as that of the traditional aspects of symmetry. ‘Self-similar’ figures appear to be the same, exactly or approximately, under several levels of magnification i.e. they are made up of smaller copies of themselves and this property is known as ‘scaling symmetry’ and such figures as ‘fractals’. Thus, a fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a reduced copy of the whole. However, it is important to note that not all self-similarity is fractal in nature; certain objects, like spirals, are self-similar about a single point and thus, are not fractals.
The mathematics behind fractals began taking shape in the 17th century when Gottfried Leibniz, a mathematician and philosopher, considered recursive self-similarity, however, it was in the late 19th century that the first mathematical function, that would be considered a fractal by today’s standard, appeared. Gradually, a number of developments took place that gave us more functions such as the Koch Curve.
Beyond the aesthetics
Fractal figures are mesmerizing to the layman as a fractal can be said to be a figure that shows self-symmetry and thus, in case of a perfect fractal created using supercomputers, it is possible to find infinite levels of symmetry, or in simple terms, one can zoom infinitely into the figure. Thus, some say fractals are finite figures that store the infinite in them and this aspect of fractals is what has led to intensive research in mathematics and has been a major argument for the philosophers- ‘Can infinite, indeed, be stored in the finite?’
While watching movies that employ heavy use of animations, have you ever pondered over how the animators create such astonishingly real-looking landscapes in movies like Star Trek or Independence Day? The basics of animation come down to geometrical figures, such as triangles, being repeatedly used in a manner that facilitates the production of such magnificent patterns and figures on the screen. Fractals also help in understanding morphology of the amorphous such as the shape of complex landscapes and coastlines, the formation of clouds, ferns, snowflakes, crystals, mountain ranges, lightning, river networks, bacterial colonies, the structure of cauliflowers and broccoli, the generations of patterns of camouflage and so on.
Thus, fractals are those figures that help define the creation and/ or substantiate the shape and structures of the seemingly random and complex phenomenon around us.