Snowflake Crystal Symmetry or: My Rant on Bedsheet Design

In celebration of winter weather (or disdain of it, your choice), I thought I'd do a brief post on beds, but not the geological kind. This post is mainly in response to a bedsheet set made by Cannon that I own. The following image is a photograph of the snowflake pattern on my bed sheets.

For those of you who have taken a mineralogy class (especially my mineralogy labs), what kind of rotational symmetry do you see in the three different snowflake patterns?

I see five-fold, six-fold, and eight-fold rotational symmetry:

In crystallography, there are only five rotational symmetry patterns: one-fold, two-fold, three-fold (triangular), four-fold (square), and six-fold (hexagonal). You can repeat patterns with these symmetries across a two-dimensional plane in some arrangement to fill the entire plane. Five-fold, seven-fold, and anything higher is forbidden in classical symmetry; you cannot repeat these patterns in any arrangement without leaving gaps between the pattern. 

In the above image, the pattern on the left is a five-fold rotational symmetry pattern that is repeated, but leaves gaps. This is forbidden in classical symmetry. On the right is a series of six-fold rotational symmetry patterns that fill the area without gaps (Cornelius and Klein, 2001).

There are special circumstances where it is possible to have the "forbidden symmetries", but not in the simple, classic symmetry sense. There are arrangements of semi-periodical patterns in crystals, called quasicrystals. Within quasicrystals the symmetry rules are a bit different and can allow for these "forbidden" types of symmetry. Two studies from 2009 and 2015 on a meteorite that fell in Russia was found to have quasicrystals with seven- and ten-fold rotational symmetry (see: icosahedrite). Until then, quasicrystals were only known to be produced in laboratories since the 1960s (Bindi et al., 2009, 2015). 

The Khatyrka meteorite contains the first known natural samples of quasicrystals (Photo courtesy: Luca Bindi).

The ten-fold rotational symmetry of quasicrystals in the Khatyrka meteorite (Bindi et al., 2015).

Snowflakes are water crystals that form with six-fold rotational symmetry, unless they have some sort of crystalline defect or have melted - or broken - portions of it's crystal. The six-fold arrangement is based on the angle of the atoms in the H₂O molecule (120°) which bond to other H₂O molecules forming a plane of hexagons. Interestingly, the temperature of crystallization can not only affect the unique design of the snowflakes, but also whether or not they will form the typical, thin, wide platy crystals, or elongated needle or prismatic crystals, where the hexagons extend down the c-axis of the crystal (see Libbrecht [2007] for more on this). The crystal can extend along the axis where water vapor is most favorable, attaching additional water molecules to the crystal, thereby growing the crystal. Additional information regarding the formation of snowflakes can be found in publications by Kenneth G. Libbrecht, such as this 2007 paper titled "The Formation of Snow Crystals" or his other publications, including books.

Variety of snowflake crystals (Bentley, 1902).

So, only one of the three snowflake patterns on the bedsheets have the correct symmetry and the other two are unnatural snowflakes. Other than the snowflake patterns, these bedsheets are quite comfortable and warm for the winter. This concludes my geological review/rant on bedsheets. Maybe someday, I'll rant on napkins or something.

Sources:

Bentley, W. A., 1902, Studies among the snow crystals during the winter of 1901-2 with additional data collected during previous winters: Monthly Weather Review, 9 p. 22 pls.

Bindi, L., Steinhardt, P. J., Yao, N. & Lu, P. J., 2009, Natural Quasicrystals: Science, 324, p. 1306–1309.

Bindi, L. and others, 2015, Natural quasicrystal with decagonal symmetry: Scientific Reports, 5:9111, p. 1-5, doi: 10.1038/srep09111.

Klein, C., 2001, Manual of Mineral Science, 22nd ed., John Wiley & Sons: New York, 656 p.

Libbrecht, K. G., 2007, The formation of snow crystals: American Scientist, v. 95, p. 52-59.