Koch Snowflake


Yesterday work found me on a luxury yacht, a 60th wedding anniversary celebration with finicky moving parts. As chefs began plating passed canapes I voiced dissatisfaction with presentation – no symmetry please! Later that night one of the chefs, a close friend and co-worker of nine years messaged – in all our years working together why haven’t you corrected my symmetrical arrangements? Adding, “Google informs me  “Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance”. He asked “what would you call your preference? Randomness, disorder or perhaps asymmetry “. I replied, “Ask any staff member what makes me crazy, I guarantee one of two answers – symmetry or bartenders who put caps on empty wine bottles.” My preference? Asymmetry of course!

Why asymmetry? What compels me to hammer notions of symmetry out of new staff? Why do long-time staff members laugh out load when they hear me train new staff, “pay attention” they chime, “she hates symmetry, no bookends, twos or fours, only threes and fives”. Cheekier staff punctuate with “relax, as long as it’s random she’ll be happy”.

Random? Asymmetry isn’t random, it’s pleasing and calculated to my eye! Without warning a fractal bomb went off – wait a minute, fractal symmetry is absolute perfection!

Ponders scurried from Mandelbrot Sets to Koch Snowflakes.  From https://fractalfoundation.org/resources/what-are-fractals/ “A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.”

In 1904 Swedish mathematician Helge von Koch published a paper titled “On a Continuous Curve Without Tangents, Constructible from Elementary Geometry” – translation, one of the first published fractal theories. Koch Snowflake is an elaboration of the Koch Curve. Be it curve or snowflake, fractal mathematics are the same – whenever you see a straight line divide it in thirds, build a equilateral triangle on the middle third, erase the base of the triangle so it looks like the shape to the right.

Animation of the first seven Koch Snowflake iterations –

Koch Snowflake

Shortly after his first query, my friend reminded me of mutual affinity for Mandelbrot sets (example below). So why asymmetry, he pressed. Why, indeed?

https://en.wikipedia.org/wiki/Mandelbrot_set

Oh man, I replied! It’s too late for this ponder! Obviously fractal symmetry warms my heart, but until the day chefs definitively represent fractal perfection with smoked beet tartare on a passing platter – asymmetry remains an art form, symmetry makes me cringe. Go figure.