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Koch’s Snowflake

Niels von Koch was a Swedish Mathematician.  He is well known for being of the first to describe a fractal curve.  His postulation demonstrated that an area bounded by a progression remains constant, yet the linear length of the boundary tends to infinity.

In the picture on the left, Koch's Snowflake has been cut/carved from a wheat field in Wiltshire.  Fractals are useful mathematical tools that describe the scaling effect of large and complicated systems.  A good example is the tree.  Take a large branch from the same tree and on upright examination, the branch resembles the tree.  By taking a smaller branch from the upright branch, the tree is resembled again.  In a similar fashion, the linear and volumetric geometry of a mouse is scaled all the way up to the blue-whale.  What is interesting though in the mammal analogy is that the metabolic rate of cellular activity does not increase in a linear fashion.  There is a distinct energetic saving in the larger species.  This law allows larger mammals to live longer than smaller mammals.

Research at the Sante Fe institute, has shown that service delivery, as a measurable unit per inhabitant, is higher in cities than in small communities.  It is tempting to take this observation into the realm of SOG sewage treatment.  The unintended corollary is that a larger system will perform better, requiring less energy.

Koch with his snowflake. By continuously adding equilateral triangles to the centre of equilateral triangles, an infinite distance bounds a finite area.

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