Fractals are both fascinating and essential to life in their various mathematical, scientific and medical applications.
Fractals have an undeniable role in the advanced technological era we are living in and they are also an essential element in the formulation of breath taking artistic and architectural accomplishments. Human-kind has been exposed to the complex beauty of patterns and shapes, throughout history, both which have inspired extensive mathematical research as well as ignited fierce iconic arguments. To better understand fractals and their increasingly important uses in the modern era we live, it is essential to know where they originated and why.The PythagoreansPopularized in the year 1975 by Mathematician Benoit Mandelbrot, the term “fractals” is used to “characterize irregular shapes in mathematics and nature, which display self-similarity” (Matson 2010).
Although the term fractals was born in the twentieth century, their mathematical evolution and practicality throughout history can be traced back to famous Greek Mathematician Pythagoras of Pythagorean fame. The Pythagoreans believed that all things in the world can be expressed by using indefinite sets of whole numbers, and the Pythagorean Theorem established the basis that “the square of the longest side of the right-angled triangles is equal to the sum of the square of the two opposite sides” (Mastin 2010). However, the Pythagoreans did not realize that the theorem works with all shapes, regular or irregular, which established that the diagonal, and the side of the square, are unable to be expressed as whole, or rational number multiples. Evidence of non-integer solutions enticed Pythagorean student Hippasus to explore the value of the square root of 2, where he determined it was not possible to express as a fraction, thus revealing the concept of irrational numbers and the possibility of decimal expansion, which does not end. The notion of divine numbers disproven through the work of Hippasus was essential to the birth of Greek Geometry and the continued development of mathematics for centuries to come.
Until the 19th century, mathematicians believed that all continuous functions must be differentiable, or smooth, at one point. It was during the last half of the century when Mathematician Karl Weierstrass created a function which was not smooth, yet still continuous. This new function shook the mathematical community as it was considered unconventional, and at the time, “there was no conceived use for it” (Turner, 1998). Earlier in his life, Karl Weierstrass seemingly lacked initiative and direction, and spent many years living in relative obscurity from that of his contemporaries. Although Weierstrass was apart from his peers, his time away was well invested in his interest in the theory of functions, which would later include the convergence of series and converging infinite products, thus implementing the idea of power-sets and directly contributing to the development of the study of fractals. Karl Weierstrass and Power SeriesMathematician, Karl Weierstras is commonly regarded as the “Father of Modern Analysis” (Britannica.
com 2018). Though he had a reputation for contributing minimal products of his works to be published, his various lectures left an undeniable impression on the mathematics community which are still felt today. Karl Weierstrass died in 1897 in Germany having influenced numerous students and mathematicians around the world. The recipient of an Honorary Doctorate Degree from The University of Konisberg, Karl Weierstrass guided other future mathematicians such as Georg Cantor, who discovered “set-theory” and the concept of indefinitely large collections of objects, commonly known as power-sets.
Power sets thus proved the set of real numbers has more elements than the set of natural numbers. His progressive notion of regarding infinite sets the same as objects, which may not be of mathematical nature, set the foundation for incoming pioneer Mathematician, Benoit Mandelbrot and his concept of fractals. The Father of FractalsNotwithstanding a formal education, Mathematician Benoit Mandelbrot, was also partially self-taught, which served as the inspiration for his interests in more non-traditional mathematics. Mandelbrot’s interests were also piqued by his uncle, who was responsible for introducing him to Mathematician Gaston Julia’s, “Julia Sets.” Dismayed at the conclusions drawn from theories of Julia’s Sets, Mandelbrot continued to work within many diverse fields in which his mathematic prowess routinely garnered recognition. However, it was not until 1975 that Benoit Mandelbrot was able to develop and utilize computer aided graphics to visualize the theory of Julia Sets. With the ability to see previously theorized sets as uneven; Mandelbrot also demonstrated and initiated the idea that non-uniform and uneven objects could be utilized for a host of affairs in fields such as Medicine, Agriculture and Economics, when applied mathematically. Mandelbrojt (2010) suggests, when Benoit Mandelbrot saw the “complex shapes appear on his computer screen as the result of an equation”, fractals were essentially born and re-born on the same day (Mandelbrojt 2010).
Throughout history, the evolution of fractals has sparked heated discussions and has been studied and theorized by many mathematicians using items such as the “von Koch snowflake curve” and the “Sierpinski gasket”, as well as the “Julia Sets.” Aided by their formulation and theoretical conclusions, and as a result of his dedication to mathematics the term “fractals” became relevant throughout many areas of study which include chemistry, science and agriculture and earned Benoit Mandelbrot the title “Father of Fractals.”