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.

The Hardest Mandelbrot Zoom 2017


Opened YouTube tonight and found “The Hardest Mandelbrot Zoom 2017” under recommended viewing. Fractals leapt off the screen (thanks to https://geneticfractals.wordpress.com/) – admittedly, recognition and comprehension play in different leagues. Mesmerized by the elegance of fractal expansion, I vowed to grasp a foundation of understanding.

Mandelbrot zoom had to wait, instinct demanded clear definition of fractal. Nailing down fractal began with “In mathematics a fractal is an abstract object used to describe and simulate naturally occurring objects”. The key word being “abstract”, as in wrapping your head around a infinite curved line winding through space that appears one dimensional, but in truth harbors a surface defined in fractal dimensions. Further research insisted  “A regular line is conventionally understood to be 1-dimensional; if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake. It is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. The fractal curve divided into parts 1/3 the length of the original line becomes 4 pieces rearranged to repeat the original detail, and this unusual relationship is the basis of its fractal dimension”.

Defining fractal called for a simple analogy, I found this – “imagine that you were to measure the trunk of a tree. We humans look at it at the largest scale let’s say that you wrapped a measuring tape around it. That tape would give you a decent estimate, let’s say 1 meter, but it wouldn’t be getting all the little bumps and crevices. If you were to have, say, a beetle crawling along the surface of the trunk, it would be going up and down those bumps, so it would measure a longer distance, maybe 3 meters. But even that beetle would just be walking over tiny little slits in the bark that are too small for it. If you had something even smaller (an extraordinarily tiny insect, for example), it would have to go up and down even more, and it could measure the distance around the tree at huge numbers, such as 100 meters. And this can continue for a long time, getting smaller and smaller, but seeing basically the same thing at each level, just a lot more of it. Of course, this breaks down when you get too small, since everything is composed of atoms, but for the scales where life occurs, tree trunks are actually fractal in nature.”

Back to YouTube’s recommended viewing of the Mandelbrot Zoom – according to Wikipedia the Mandelbrot set is a nod to mathematician Benoit Mandelbrot who coined the term fractal when he theorized mathematical definition of roughness and self-similarity in nature. Confused? Shake it off and immerse yourself in this clip. It doesn’t matter if it makes sense, all that matters is willingness to accept it exists. For those who need hard facts over abstract wonder, some links –

http://everything.explained.today/Fractal/

http://everything.explained.today/Mandelbrot_set/

Enough talk, sit back and watch the hardest Mandelbrot zoom …