Fractals 

Portion of a Mandelbrot Set

A fractal is an entity that exhibits a repeating pattern.  Many patterns in nature exhibit fractal phenomenon and computer simulations are used to generate fractal patterns artificially.  Natural phenomena such as coastlines  exhibit a fractal pattern, as the pattern displayed may be exact (self-similar) or perhaps just similar at various levels of detail or magnification.  Snowflakes and trees property of continuing detail at higher levels of magnification.  Because of this, fractals are considered "nowhere differentiable" because of their inability to be measured traditionally.

Fractals are used in many fields, including physics, biology, medicine and physiology, imaging and financial fields. Fractals may apply in economic contexts such as the stock market Standard and Poors 500 Index, when examining longer term patterns (years) vs shorter terms (months, days, intra-day trading).  Fractals may be used in cinema, advertising, graphic design and climate science .  Fractals are a beautiful representation of art in their own way, in the visual arts, including the Droste effect, which is a picture within a picture. 

Fibonacci numbers, the basis of the Fibonacci Sequence appears in fractal geometry in a wide variety of ways.  Fractal dimension is a measure used to quantify complexity.  It measures  how detail changes with scale and the capacity of the fractal to fill space.  Various definitions of fractals and mathematical indicators exist, including a definition by mathematician Benoit Mandelbrot  who characterized a fractal as an object whose Hausdorff-Besicovitch dimension  is greater than its topological dimension. 

Wikipedia lists the Hausdorff-Besicovitch dimensions of a number of common fractals, including the Koch snowflake, Sierpinski Triangle, Quadric Cross, Julia Set and the Boundary of the Mandelbrot Set. Values for natural processes such as  the Coastline of Ireland , Great Britain and Norway are listed, as are values for various Brownian motion and random walk processes.  Dimensions are shown for biological models such as Cauliflower, Broccoli, the surface of the Human Brain , and the Human Lung.  Higher numbers indicate increasing complexity.

Fractals may be use in diagnostic medicine and physiology.  For example, blood vessels may exhibit fractal characteristics, as may the lung and surface of the human brain.  Tortuosity, another metric, relates the ratio of the actual length of a curve or segments of a curve to the distance between the two ends. Tortuosity may also reflect the degree to which a curve crosses over itself.  
  
Tortuosity was used for characterizing animal trails of mites  with regards to Brownian motion pathways.  Fractal dimension and tortuosity may both be used in measurement of blood vessels, as is shown in this article from the medical journal PubMed in a study of pulmonary hypertension.  In that study, distance metric, a measurement of tortuosity, was statistically more significant than the fractal dimension in correlating clinical patient parameters with the particular metric.   This goes to show that the use of different metrics may produce differing correlations, perhaps a clue in itself to underlying characteristic studied.

Fractals  form the basis of many aspects of life and the world around us, igniting our curiosity, aiding our research, informing us, and conveying a sense of beauty, form and function.