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# 2017 Advent Calendar of Fascinating Websites – Day 11 – Fractals and Snowflakes

Brrrr.. it’s cold outside!

So have you been enjoying the snow? My daughter sledged, created snow angels, did cartwheels and handsprings in the snow – and I watched from the comfort of the living room window!

After that we settled down to watch Frozen, it just seemed the right film for the right day (header video clip: Let It Go from Disney’s FROZEN as performed by Idina Menzel | Official Disney HD by Disney UK).   In what is now one of the most famous songs in existence the ultimate six-sided star-shaped snow crystal builds the ultimate snow palace and (2:20 mins onwards on the video) we hear the line “My soul is spiralling in frozen fractals all around”. Snowflakes don’t develop using the algorithms of fractal repetition, so what is the relationship between fractals and snow?

Fractals are geometric figures with “self-similarity”, irregular geometric shapes that have the same degree of irregularity on all scales, that is, no matter at what scale you look at an image you see the same features. Repetition of patterns such as branching and spirals are common in Nature -just look at leaves, lightning, even pineapples! Fractals can also be created by repeatedly calculating a simple equation over and over thousands or millions of times, creating the most amazing images you can zoom into forever. The maths involved is helping examine things as far apart as the human heartbeat and drawing animation.

If you want to find out about the fascinating world of fractals then see http://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf or visit http://fractalfoundation.org where you can find links to great fractal generators.

Snow crystals begin as either a single ice crystal or an aggregation of ice crystals falling through high-humidity cold air. As the crystal falls it rotates, and the snowflake grows at its perimeter.  Complex shapes emerge as the flake moves through different temperatures and humidity, creating individual snowflakes that are nearly unique in structure at the molecular level. Many snowflakes form structures that lack symmetry and therefore aren’t fractals. However, fractal geometry has long been associated with snowflakes as the patterns fractals create approximate real snowflakes very closely, and it is fractal mathematics which is of most use in the analysis of snowflake structures.

The Swedish mathematician Helge von Koch (1870-1924) formulated one of the first mathematical fractals now known as the Koch snowflake. In it you start with an equilateral triangle, then with each iteration remove the middle third of each side and add two line segments to make new equilateral triangles on each side of the previous triangle, this builds up the snowflake shape. If you zoom into the edge of the snowflake you won’t be able to tell how many iterations have been performed. Infinite repetition of the iteration equals a snowflake with an infinite perimeter, even though the snowflake itself only has a finite area. To find out more about Koch snowflake development and geometry see this lesson on YouTube by Khan Academy. https://www.youtube.com/watch?time_continue=210&v=azBNsPa1WC4

There are 35 types of snowcrystal shapes, such as prisms, dendrites, columns, plates and needles with 121 sub-categories, but ask someone to draw a snowflake and they will draw a six-sided flake, like that used in the Frozen video. Why do we associate this shape so much with snowflakes?

Our image of the perfect snowflake is influenced by the work of Bentley, The Snowflake Man. In 1885 Wilson Bentley became the first person to photograph a single snow crystal. A pioneer in the field of photomicrography, his experiments with a microscope and a bellows camera, and his passion for his subject matter, led him to photograph more than 5000 snowflakes over the space of his lifetime. It was his work that led to the common belief that no two snowflakes are exactly alike. Bentley’s photographs show images of great diversity in shapes, but are dominated by beautifully symmetrical, six-sided star-like crystals and this is what we think of when we think of snowflakes. His legacy can be explored at http://snowflakebentley.com/index.htm When later teams copied his work, they realised that there were more imperfect snowflakes than perfect – Bentley admitted to an early form of “photoshopping” – touching up his negatives and adapting the images, saying in mitigation that untouched pictures misrepresented the crystals that in their truest form were nothing short of perfection!

We are used to beautiful images of two-dimensional single snowflakes, however, the majority of snow falls in clumps. Recently, a camera system called the Multi-Angle Snowflake Camera, capable of capturing images of snowflakes in 3D as they fall from the sky was developed. Three high-speed cameras are triggered by infrared sensors and use extremely fast exposures (up to 1/2,500th of a second) to photograph snowflakes and monitor their path and speed as they fall. 3D imagery is helping to create a better understanding of snowcrystals and snowfall and shows us that our traditional imagining of six-sided flakes falling from the sky needs some updating!

Visit http://www.snowcrystals.com/ to learn more about real snowflakes. According to their guestimate, over a typical year, about a million billion snowflakes fall on the earth each second – enough to make one snowman for every person on earth every ten minutes. Looks like I better get busy!