# Why are there no crystals with 5-fold symmetry?

They say you can't pack objects like pentagons or octagons such that they fill up space entirely and that that's one reason there is no 5-fold or 8-fold rotational axis.

Are you telling me you can fill up space stacking dodecahedrons or hexakis octahedrons? I don't think so. Please explain this to me. I thought that crystals are built by stacking unit cells on one another.

• There are crystals with five-fold symmetry, it's just that they don't perfectly repeat. The mathematics of it is quite interesting - see en.wikipedia.org/wiki/Quasicrystal . (Unfortunately I don't think I really understand your question though.) May 26 '14 at 7:45
• If you repeat dodecahedrons or other crystal forms from the cubic system there will be gap between them, they don't fill up space. Only the cube does. Why then we use them as models of unit cells? May 26 '14 at 8:16
• Dec 4 '14 at 5:07

All unit cells are parallel-sided hexahedra. These are six sided shapes with parallel opposite sides. Their three principle angles may or may not be 90 degrees. And the three side lengths may or may not be equal. All of these unit cells can be uniformly stacked.

Using these building blocks it is only possible to produce planes of reflection, diads (axis of rotational symmetry order 2), triads, tetrads, and hexads.

For example, a cube (all sides the same length, all angles 90 degrees) has diads, triads, and tetrads; plus planes of reflection. A hexagonal symmetry can be used with 60 / 120 degree angles.

Note that it is impossible to produce a regular arrangement of unit cells to produce a pentad (order 5 symmetry). As @Nathaniel says, this can be almost achieved using Penrose Tiles (2d mathematical constructions), and quasicrystals (real 3d materials). Quasicrystals will produce an x-ray diffraction pattern with a pentad, but the actual atoms do not follow a true 5-fold symmetry.

I suspect your confusion is over the final (macroscopic) crystal shape and the unit cell shape. These are rarely the same. (Salt being an example where they are; but diamonds, garnets, do not)

• "I suspect your confusion is over the final (macroscopic) crystal shape and the unit cell shape." <- exactly. How do I end up by stacking simple shapes of the 7 crystal system with shapes complex shapes? How does the crystal 'know' where to leave a row unfilled so that it makes a new face? And what is the difference between coordination numbers and Bravais lattices. Do I use symmetry operations to construct the 32 point groups? I could really use a textbook or some online resource. Please help me out! May 26 '14 at 18:23
• I'm afraid I don't know Bravais lattices. Rows are filled according to energy - new atoms will be attached at the lowest energy location. I don't have a copy but the answer you're looking for is probably in Andrew Putnes 1992 "An Introduction to Mineral Sciences" amazon.com/Introduction-Mineral-Sciences-Andrew-Putnis/dp/… (I had Dr Putnes as a supervisor way back in the early 1990s) May 26 '14 at 19:25
• I'm quite sure you are incorrect that the atoms do not follow a 5-fold symmetry in a 5-fold quasicrystal. They do follow it in some area, the larger the area is, the smaller the number of centers of the symmetry has to be.
– yo'
May 31 '14 at 10:17
• Can't unit cells also be tetrahedral? (4 faces) And Octohedral (8 faces, as well as a hexagon with flat ends?) Jun 18 '18 at 12:19

Your question is much more complicated from mathematical point of view than it seems to be. First, I'll start with a nice photo:

(source: Wikipedia). What you see is really a photo and it is almost a mono-crystal. The only problem is that we all know that it cannot be a monocrystal since it cannot tile the space. So what is it? A quasicrystal -- matter with a well-defined structure, i.e., position of atoms in its quasigrid, yet such one that these positions are not periodic.

Speaking of basic cells: This notion is much more complicated for quasicrystals since their basic cells do live in 4-dimensional space, at least from the mathematical point of view. The reason is that a quasicrystal can be modelled as a cut-and-project set. For the one depicted in the figure, this set lives in 4 dimensions. If you still insisted on having basic cells, you would need at least two of them, put together in a way similar to Penrose tiling, which is formed by two different quadrilaterals. For more details on mathematical quasicrystals, see for instance Z. Masáková, J. Patera, J. Zich, Classification of Voronoi and Delone tiles in quasicrystals: I. General method, J. Phys. A: Math. Gen. 36 (2003) pp. 1869-1894.

To conclude: Your idea that 5-fold symmetry is impossible for a crystal is correct: basically only 3,4 and 6-fold symmetries are possible. For more details, see Wikipedia: Crystal families. Still, matters with 5-fold symmetries do exist.