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?


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.

26 thoughts on “Koch Snowflake

      • It’s worth every penny! I’ve had to do it for family occasions (a few weddings and birthday/anniversary celebrations) — so much so that I’ve told the remaining unmarried daughter that she’d better have a ‘destination’ wedding. …her mother has done it enough! 🙂

      • Good for you! My best friend’s only daughter did that and she was thoroughly pissed. .. I told her Jessica did her a great favour! 🙂

        We have a wedding in mid-August – two sixty plus people. ..like, what are they SMOKING??? *headshake*

  1. Since you’re still up, notes – and it’s time I started my day – I’ll tell you what I’m doing. Hubby dug a hole for the neighbour’s dead dog first thing this morning with his backhoe. (We’re do-it-yourselfers around here; dispenses with vet fees) He then drove the machine up to our woodlot where it ‘lives’. I have to take a vehicle up there, drop it off, and walk back down the road (5 k). Then I must head to the city to retrieve my 79-year-old uncle who had a spinal fusion (his 4th) yesterday. Yes, you read that right. They fix ‘em up and chuck ‘em out early in this neck of the woods.

  2. Ahhhh … you and I, my friend, are polar opposites! I have to have everything matched and symmetrical, towels must be folded just so, and stacked just such. A candlestick on either side of the picture, perfectly equidistant, to the nth of a millimeter. I would drive you crazy! 🤣

    • Fascinating and oh so true! Symmetry hurts my head! At work staff stands back to watch me design food presentations, hang on they say, it will make sense in a minute. Then – wow that looks incredible! Threes and fives I repeat, unexpected angles, height and placement, we’re creating a masterpiece, not a postcard. Sigh. Go figure. 🙂

      • Interesting to ponder … methinks perhaps you are far more artistic than I am. I am very much a pragmatist, I take everything literally and for me, seeing is believing. When I was a child, many moons ago, art teachers in school simply shook their heads at my efforts and gave me some toys to play with instead! 😉

      • Unimaginative art teachers are responsible for scores of symmetrical thinkers. Sigh. My grandmother was a fashion illustrator for Vogue magazine in 1920s San Francisco, my mother a published children’s story author and illustrator. Suffice to say I grew up in a asymmetrical world. 🙂

      • Symmetry – asymmetry: I think there’s always room for both. A Holstein cow is essentially a symmetrical unit until you focus on the pain job.

      • Ahhhh … so you are from a long line of artistic people! Indeed you did grow up in an asymmetrical world! I grew up in a world of chaos, and I suspect that is why I am now a stickler for order. 😉

  3. The “beauty” of fractals gets boring pretty quickly. On the other hand, humans aren’t symmetrical. One side of the face is not a mirror image of the other. Feet on the same body aren’t necessarily the same size, nor are breasts, nor are ears. When we humans make a pin, the end is rounded. We don’t know how to make a truly pointed-to-the-end piece of anything, ours are always round at the end. No one plants a garden with even numbers of plants, odd is always more appealing. There is balance in randomness. The balance of being unequal, random, slightly haphazard.

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