This decoration just about passes muster for Katie
Andrii Oleksiienko/Shutterstock
At this time of year, everything seems to be decorated with some kind of seasonal design – trees, holly, jolly Santas and so on. One shape you often see is a snowflake. Yes, they are beautiful and intricate, but I can find their proliferation deeply annoying.
The shape of snowflakes is an artefact of the chemical structure of ice, and even though (as they say) every snowflake is unique, there is actually a surprisingly regular mathematical pattern in there too. We often use the language of symmetry to describe shapes. If something has reflection symmetry, we can draw a line across it, and the shapes on each side will be mirror images of each other.
A shape can also have rotational symmetry – we can partially rotate it and get the same shape. The number of different positions on the way round that result in the same shape is called the order of the symmetry: a shape like a square has order 4 rotational symmetry, while an equilateral triangle has order 3.
Some shapes just have rotational symmetry (like the three-legged emblem of the Isle of Man) and some just have reflection symmetry (like a stick figure, which has a single line of reflection down the middle).
Regular polygons have both rotation and reflection symmetries – called dihedral symmetry – and we can combine these symmetries to get others. For example, reflecting a square vertically then horizontally equals a rotation by 180 degrees. In the same way we add together numbers, there are also ways to “add” symmetries to describe what happens when you combine them – part of an area of maths called group theory.
The snowflake is a perfect example: it has the structure of a hexagon, which can be reflected along six different lines passing through the centre of the shape, and rotated by 60 degrees six times. This symmetry arises due to the chemical structure of water and ice. The angle between the bonds is such that when the water freezes, the molecules – held together by hydrogen bonds – arrange themselves into a rigid hexagonal lattice.
This chemistry means that the vast majority of ice structures, including snowflakes, have an underlying hexagonal shape. The exact form of the snowflake depends on the conditions under which it forms, including temperature, humidity and pressure – meaning they all have tiny differences, but the same underlying structure.
As a mathematician, I am very pleased in the winter to be surrounded by shapes with such an elegant structure, even if it’s too small to see. But I am also deeply frustrated by decorations (not the one shown!) depicting snowflakes with eight (boo) or five (ugh) branches. Be vigilant, readers: beware the seasonal snow-fake!
These articles are posted each week at
newscientist.com/maker
Katie Steckles is a mathematician, lecturer, YouTuber and author based in Manchester, UK. She is also adviser for New Scientist‘s puzzle column, BrainTwister. Follow her @stecks
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