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.

The Ginko Biloba Tree

Ginkgo Leaf Fossil, Burke Museum, University of Washington, Seattle, Washington

The above photo of the Ginkgo Leaf Fossil was taken at the Burke Museum, University of Washington, Seattle, Washington and  represents a fossil of a Ginkgo taken in Smithers, British Columbia, Canada.  Driftwood Canyon Provincial Park, in Smithers, is one of the world's most significant fossil beds.   

The Ginkgo Biloba tree appears naturally, in the wild, only in China.  It is grown in many place worldwide, as a cultivated tree, and adapts well in well watered and well drained habitats. Ginkgo Biloba is a living fossil , dating back 270 millions to the Permian Period  .  Ginkgo has managed to evolve along a very long evolutionary time period without much speciation.

It is an interesting example of a plant that has had extreme longevity, slow growth, late reproductive maturity and has survived through many diverse and disturbed environments, including the Ice Age .  It has  managed, despite  a narrow base of speciation to support it.  It has two sexes, male and female, and has its ability to exhibit clonal reproduction, a process which produces a population of identical units which reproduce from the same ancestral line. This has helped it survive evolutionary challenges.  The Ginkgo is also highly resistant to air pollution and grows in areas where air pollution has damaged other species.

A HHMI BioInteractive presentation discusses the issue of clonal reproduction in a video. "Are Males Really Necessary?"  using fruits and vegetables as props.   The video is thought provoking and uses creative use tools to discuss an educational subject.  The 23rd or "sex chromosome"  is a specialized area of interest not explicitly discussed.  Protocols for tossing out wilted lettuce or black bananas present an analytic challenge.

The black banana we consider tossing out might be more useful than the newest banana, especially if we are considering  baking banana bread to bring to the next potluck.  It takes awhile to produce the necessary senescence in a banana to get the right flavor and texture for banana bread.  Thus,  Interoperability, whether it be in bread making, computer engineering or other systems, is an important evolutionary issue, involving mutations along with a stochastic process.

One has to wonder at the combination of processes that has sustained the Ginkgo for million years as well as the diverse environments that the Ginkgo tree  has experienced.   Ginkgo is an herb.  Its leaves and sometimes its seeds are used to make extracts for medicinal purposes, including memory disorders such as Alzheimers and dementia.  Even as we consider the marvel that is the Ginkgo Biloba, we find fascination and beauty in its foliage, and comfort in the shade that it produces.  It is a tree and much more.