Historical Development and Background
Fractals have been around for as long as the universe has, but it wasn't until the 1970s that fractals were defined as we know them today. The journey to our modern knowledge about fractals has been built over the centuries by mathematicians from around the world. Many of the following early encounters with fractals paved a way for the formalization of fractals in 1975. To begin this historical journey, let's go back to 1200 AD. It was around this time that Leonardo de Pisa (more commonly known as Fibonacci) wrote a book that included thoughts about the Fibonacci sequence. Fibonacci's writings later brought about the Fibonacci Spiral. Interestingly enough, many of Fibonacci's thoughts were based on ideas he heard of when traveling in North Africa. The Fibonacci Spiral will be discussed later in further detail, but it is important to note that these early thoughts about recursion and iteration helped in the development of fractals.
Fibonacci Spiral -- Image Source: Wikimedia Commons
Fast forward to the late 1800s. In 1883, a German mathematician named Georg Cantor introduced a set of numbers called the Cantor set. Again, this set of numbers can be viewed as a fractal, but it wasn't formalized. In explanation of mathematics, the mechanics of the Cantor set will be discussed further, but for now, remember that this came about in the 1800s (Wolfram Math World).
Cantor Set -- Image Source: Wikimedia Commons
In the early 1900s, several other mathematicians including Helge von Koch from Sweden, Waclaw Sierpinski from Poland, and Gaston Julia from France developed fractals or fractal-like sets of numbers including Koch's snowflake, Sierpinski's triangle, and the Julia set. (See more about these fractals here.) The issue with many of these discoveries was that the computing power needed to continue their exploration had not yet been invented. Many of these mathematicians had the right ideas about fractals but because there was a gap in the technology, many of these ideas faded out of the spotlight.
Another notable discovery was made by Lewis Fry Richardson, an English mathematician in the early 1900s. He was studying the length of the coast of England. He recognized that if you measure the coastline with a yardstick, you will get one measurement. If you then measure with a one foot ruler, you will be a larger number. The smaller the measurement tool, the more descriptive information you can get (How Stuff Works). This again provided a fertile ground for Benoit Mandelbrot to learn and discover more about fractals.
Coast of England Paradox -- Image Source: Research Gate
In the early 1960s, a young Polish-French-American man by the name of Benoit Mandelbrot was working at International Business Machines Corporation - IBM. He had studied mathematics while in college in France, but he found the rigorous environment of those mathematics departments to not fit his style. He later immigrated to America and began his work.
The Mandelbrot Family -- Image Source: atomosyd.net
While working for IMB, Mandelbrot met with Hendrik Houthakker, a Harvard economics professor. Houthakker had been analyzing cotton prices. A common economic belief at the time was that over long periods of time, prices would fluctuate according with what was happening in the real world, in the economy, while short term fluctuations would happen more randomly. After a quick visual inspection of these trends, Mandelbrot recognized that the price of cotton fluctuated in a similar way across scales. This turned into a large endeavor, and Mandelbrot came to realize that the changes happening in the economy were similar across scales large and small. This was a springboard for Mandelbrot's work on fractals.
Later in his work at IBM, telephone engineers were recognizing errors in the telephone transmission signals. At first, the errors were seemingly random, but Mandelbrot took it upon himself to study them. On inspection, the errors were self- similar across scales. Interestingly, these error patterns are similar to the Cantor set that had been discovered nearly 80 years earlier. The following video describes this situation in more detail.
PBS Nova - Fractals - Hunting the Hidden Dimension cropped via ytCropper
Building on his experiences at IBM and drawing upon discoveries and knowledge from centuries before, Mandelbrot continued to develop his ideas about this new geometry. He wanted a word to describe what he was working on. In 1975, while looking through his son's Latin dictionary, "he came across the adjective fractus, from the verb frangere, to break. The resonance of the main English cognates - fracture and fraction - seemed appropriate. Mandelbrot created the word fractal" (Gleick, p. 98). That is the term we use now to describe both pure mathematical fractals and simple or limited fractals found in nature. As mentioned in above, people had recognized iterations and fractal like structures for a long time, but Mandelbrot was able to use the computers at IBM to further iterate and calculate things that had initiated before his time. Thus began the new geometry - fractals.
Since that time, Mandelbrot wrote several books about fractals and worked hard to make a name for himself in the mathematical community. As groundbreaking as these ideas may seem, it took a while for a majority of the mathematical community to get on board with this line of thought. So much of classical mathematics is dependent on smoothness, yet fractals are an exploration of roughness. While classical geometry describes many man-made objects and features, fractal geometry can provide a description of the natural world. Mandelbrot was mocked and not considered to be a true mathematician for a while, since his new geometry seemingly could not be used or proven in ways that classical geometry or algebra can.