During this festive season, it’s hard to miss seasonal designs like trees, holly, and joyful Santa figures. Among the most recognizable motifs are snowflakes. Although famously intricate, they can also be quite bothersome.

The snowflake’s unique structure is influenced by ice’s chemical makeup. While each snowflake is touted as one-of-a-kind, they exhibit intriguing mathematical patterns. Symmetry is a term we often use for shapes, where reflection symmetry means that one side mirrors the other when a line divides it.

Shapes can also showcase rotational symmetry, enabling partial rotations to maintain identical appearances. The count of distinct positions where the shape looks the same is known as the symmetry order. For instance, a square has a degree of 4 rotational symmetry, while an equilateral triangle exhibits a degree of 3.

Some shapes possess only rotational symmetry (like the Isle of Man’s emblem), while others exhibit only reflection symmetry (similar to a stick figure split down the middle).

Regular polygons combine both rotational and reflection symmetries, referred to as dihedral symmetries, allowing us to achieve additional symmetries. For example, reflecting a square vertically followed by horizontally results in a 180-degree rotation. Much like numerical addition, we can “add” symmetries, a concept rooted in group theory.

Snowflakes beautifully embody this concept. With a hexagonal formation, they reflect across six distinct lines through the center and can be rotated six times every 60 degrees. This symmetry arises from the chemical structure of water and ice, where hydrogen bonds form a rigid hexagonal lattice as water freezes.

This unique chemistry leads to the hexagonal foundation of most ice formations, including snowflakes. Variations in temperature, humidity, and pressure impact the specific shape of each snowflake, ensuring that while no two are alike, their basic form remains consistent.

As a mathematician, I am delighted to see such elegant shapes gracing winter, though I find the decorations (excluding the ones shown!) displaying snowflakes with 5 (ugh) or 8 (boo) branches incredibly irritating. Reader beware of seasonal snow fakes!

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Katie Steckles is a mathematician, lecturer, YouTuber, and author based in Manchester, UK. She is also an advisor for New Scientist‘s puzzle column “BrainTwister”. Follow her on Twitter @stex