Fractal geometry was essentially discovered by Benoit Mandelbrot in the nineteen sixties and seventies, based partially on the work of Gottfried Leibniz, Georg Cantor, Henri Poincare, and Helge von Koch. Mandelbrot coined the term fractal in 1975 in his foundational book, revised and published in English first in 1977 as Fractals: Form, Chance, and Dimension, and then further revised and expanded in 1982 as The Fractal Geometry of Nature. In his book, Mandelbrot elucidates the mathematical sets that produce fractals along with a conceptual framework for their understanding, which demonstrate that many objects and processes found in both nature and human culture develop self-similarly, or through the closely related concept of self-affinity, which he describes as “the resemblance between the parts and the whole.” That is, coastlines, fern leaves, stock markets, turbulence, and many other complex processes and entities all recursively repeat similar structural patterns across different scales of magnitude, so that one part of the leaf magnified looks very much like the larger leaf, or the distribution of stars in one galaxy looks very much like the distribution of many galaxies in a galactic cluster.
Furthermore, Mandelbrot shows that fractals possess fractional dimensionality, which, along with the fractured topology of the objects fractality describes, is why he named them as such. For instance, a Koch snowflake, a triangle with successively smaller triangles symmetrically added to the object’s outer surface, at first producing a Star of David, and then increasingly resembling a snowflake with each successive iteration, exhibits a dimensionality of approximately 1.26. This mathematical object can be extended into a third-dimensional space by substituting pyramids for triangles, which exhibits a dimensionality of approximately 2.58. The degree of the object’s or process’ fractional dimension is significant, so that a 1.1 dimensional object traces a curved line, while a 1.9 dimensional object bends back upon itself in complex convolutions so that it almost fills a two-dimensional plane.
[This post is an excerpt from The Dynamics of Transformation: Tracing an Emerging World View.]