The twelve-fold way (Science Advent 2)

(Every day until December 25, I’m posting a science-related image and description.)

Day 2

A 12-fold quasicrystal: a two-dimensional arrangement of atoms that can never fully repeat itself. The arrangement can be duplicated using three types of tiles, shaped like squares, flattened rhomboids (figures with four equal sides), and equilateral triangles. [Credit: Wolf Widdra]

A 12-fold quasicrystal: a two-dimensional arrangement of atoms that can never fully repeat itself. The arrangement can be duplicated using three types of tiles, shaped like squares, flattened rhomboids (figures with four equal sides), and equilateral triangles. [Credit: Wolf Widdra]

Many solids are crystalline: the atoms are arranged in repeating patterns, such as cubes (with several variants), hexagons,  and other regular structures. Others are irregular, and we call that type “glass”. Between those are the quasicrystals: solids that have regular structures, but ones that never repeat.

Think of it this way: if you have a set of square tiles of a single color, you can lay them out on a flat surface with no gaps, and the same pattern repeats forever, like a checkerboard. You can do the same with equilateral triangle tiles or hexagonal tiles (which are commonly used in board games as well). Square tiles exhibit four-fold symmetry; hexagonal tiles exhibit six-fold symmetry.

However, five-fold symmetry doesn’t work if you want an endlessly repeating pattern: if you try to tile a surface with pentagons of the same size, you’ll leave gaps. The solution is to create five-fold symmetry without repetition; you can do it most simply with two sizes of rhombus, though other options also exist. You’ll fill an entire surface with these shapes, but the pattern will never repeat, even if you had an infinite number of tiles.

That’s the basic idea of a quasicrystal as well: a regular but non-repeating arrangement of atoms in a solid. A meteorite discovered in Russia contains quasicrystalline structures, and the 2011 Nobel Prize in chemistry was awarded to Dan Schechtman for his discovery of quasicrystals in the lab. These quasicrystals were both variants on five-fold symmetry, but in a three-dimensional crystal rather than on a flat surface.

This brings us back to the image for today’s post: Stefan Förster, Klaus Meinel, René Hammer, Martin Trautmann, and Wolf Widdra discovered a type of two-dimensional quasicrystal with twelve-fold symmetry. (A two-dimensional lattice is a single layer of atoms; the most famous of these is graphene, a hexagonal arrangement of carbon atoms.) The particular symmetry could be achieved with three types of tile: equilateral triangles, squares, and squashed rhomboids. While the tile choice might seem arbitrary, the spacing of the atoms as measured using X-ray diffraction corresponds to the vertices of these shapes!

Notes

  • Stefan Förster, Klaus Meinel, René Hammer, Martin Trautmann, and Wolf Widdra, “Quasicrystalline structure formation in a classical crystalline thin-film system”. Nature 502, 215–218 (10 October 2013). DOI: 10.1038/nature12514. Alas, this paper is not freely available, but you can all read Nadia Drake’s coverage at Wired.
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