This piece almost passed the test for Katie.
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At this time of year, everything seems to be decorated with some kind of seasonal ornament – trees, hollies, jolly Santa Clauses and so on. One of the figures you often see is snowflake. Yes, they are beautiful and complex, but their large number deeply irritates me.
The shape of snowflakes is the result of the chemical structure of ice, and although (as they say) each snowflake is unique, in fact there is a surprisingly correct mathematical pattern here too. We often use language symmetry describe forms. If something has mirror symmetry, we can draw a line through 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 along the path that result in the same shape is called the order of symmetry: a shape like a square has fourth-order rotational symmetry, while an equilateral triangle has third-order symmetry.
Some figures simply have rotational symmetry (for example, the three-legged Isle of Man emblem), and some simply have mirror symmetry (for example, a figure that has a single line of reflection in the middle).
Regular polygons have both rotational and reflection symmetries, called dihedral symmetries, and we can combine these symmetries to get others. For example, flipping a square vertically and then horizontally equals rotating it 180 degrees. Just like we add numbers, there are ways of “adding” symmetry to describe what happens when you combine them – this is part of a branch of mathematics called group theory.
A snowflake is a perfect example: it has a hexagonal structure that can be reflected along six different lines through the center of the shape and rotated 60 degrees six times. This symmetry arises from the chemical structure of water and ice. The angle between the bonds is such that when water freezes, the molecules held together by hydrogen bonds form a rigid hexagonal lattice.
This chemistry means that the vast majority of ice structures, including snowflakes, are hexagonal in shape. The exact shape of a snowflake depends on the conditions under which it forms, including temperature, humidity and pressure. This means that they all have slight differences but the same basic structure.
As a mathematician, I find it very pleasant in the winter to be surrounded by figures with such elegant structure, even if they are too small to be seen. But I'm also deeply upset by the decorations (not the ones shown!) that depict snowflakes with eight (boo) or five (ugh) branches. Readers beware: beware of seasonal snow!
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Katie Steckles is a mathematician, teacher, YouTuber and author from Manchester, UK. She is also a consultant for New scientistBrainTwister puzzle column. Follow her @stecks
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