We are all familiar with Euclidean geometry. This means we understand circles, square, rectangle... shapes that are easily defined using conventional mathematics. Fractal geometry addresses those shapes that are not so conventional. Lets talk about coastlines. These are not regular shapes. In the past the coast line of England (for example) has been measured by outlining the coast and then assigning an amount to that outline. To be truly accurate one needs to measure ALL parts of the coast line. Than means every nook and cranny, every irregular twist and turn. The conventional measurement lists the length of England's coastline at about 1,000 miles. Include all the nooks and crannies and the actual coast is closer to 3.000 miles. Fractal geometry can be much more precise.
The image above shows conventional measurement on the left and fractal measuring on the right.
So Koch and Sierpinski were experimenting with ideas that turned out to be fractal in nature. They did not know this. No one knew this because the concept of fractals and fractal geometry had not been conceived of. These concepts were brought together and refined by Benoit Mandelbrot who put these concepts together to form what we now consider as fractals and fractal geometry. By using these concepts Benoit was able to create the Mandelbrot Set. The Mandelbrot Set uses the ideas of infinity and zero mass. He is the father of fractals and fractal geometry.