Conventional unit cell for a hexagonal crystal system

[![enter image description here](https://i.stack.imgur.com/qJoEr.png)](https://i.stack.imgur.com/qJoEr.png)


The figure shows a unit cell of a hexagonal crystal system. Drawn in bold, is the unit cell. The lightly shaded one is a unit cell as well and has a six fold symmetry along an axis, hence is more symmetrical than the one in bold.
Why then do texts define the unit cell of a hexagonal system by the one drawn in bold? For example this this[![ A defining chart for Bravais structures](https://i.stack.imgur.com/n3X6K.jpg)](https://i.stack.imgur.com/n3X6K.jpg)




A crystal repeats contents of a unit cell in three dimensions. If you choose the unit cell to be a parallelepiped, the translation vectors are simply the edges of it. This is an easy way to describe a hexagonal crystal (and it is the conventional way given in the International Tables of Crystallography).


You could also choose a hexagonal prism (if you look at the figure in the question, you will realize its volume is 3 times bigger, so there is translational symmetry within the cell now). You can also repeat that in three dimensions to fill up the entire space. However, the translation vectors will be different.


[![enter image description here](https://i.stack.imgur.com/iooS0m.png)](https://i.stack.imgur.com/iooS0m.png)


The smaller unit cell is more common, probably because the math is easier and more closely related to that of other crystal systems.


The relationship between the two ways of dividing up space into repeating units is shown below. Note that two of the green unit cells complete fit into a hexagon, with the remainder of the space taken up by two halves of green unit cells.


[![enter image description here](https://i.stack.imgur.com/VnhLIm.png)](https://i.stack.imgur.com/VnhLIm.png)


For face-centered or body-centered lattices, we do (conventionally) choose the larger unit cell because it makes many things easier. In this case, choosing the hexagonal prism would make a lot of things harder.


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Interestingly enough, the reason why the cell in bold is the unit-cell for the crystal is actually *because* the hexagonal cell has this higher symmetry you mention.


To understand why, we have to scratch at the true meaning of symmetry: not just in terms of crystal cells, but more broadly what symmetry is as a *mathematical idea*. When we say something is symmetric, we think of it as "looking exactly the same" in some way (maybe specifically a reflection or rotation, perhaps.) A reflection makes things look the same, but backwards in a way; a rotation makes things look the same, but just spun around in a way. In mathematics, symmetry is defined as **any action** you can perform to some object where—after the action—the object remains fundamentally unchanged. Translations are types of symmetries; we can move things around, and it doesn't fundamentally change what they are. Sounds simple enough.


This idea is extremely powerful when we start analyzing systems of crystals—as we add more atoms to what we are trying to study, it becomes exponentially more difficult to keep track of what everything is doing. Chemists and physicists, in order to cope with all the excess information, have to encode that information into something that is possible to understand: in quantum chemistry, it's the wavefunction; in gas chemistry, it's pressure-volume-temperature coordinates; and in solid-state chemistry, chemists leverage the symmetry of crystals to encode that information **in the unit cell**.


How do we do this? Instead of focusing on all $6.022 \* 10^{20}$ atoms or whatever in our sample, we instead have two things we focus on: we have to find the arrangement of atoms in the smallest volume of atoms we can possibly can without losing information, and then we have to describe all of the symmetries of the unit cell (a.k.a. all of the reflections, translations and rotations) that allow the unit cell to fit with itself and—in the mathematician speak—"tile the plane."


Because the atoms are arranged in a 3D-lattice, you can use the mathematics of matrices and linear algebra to prove that the smallest amount of information you need for atomic positions in a crystal unit-cell is three vectors that describe a parallelepiped (hence how all the forms are parallelepipeds.) The "hexagonal" in hexagonal crystal packing is not a statement about the shape of the unit-cell necessarily, but is instead a description of the **types of symmetries** necessary to know how unit-cells can be sewn together to make the entire crystal. The unit-cell describes all the atomic positions you need to know, and the type of unit-cell it is implies some of (but usually not all) of the information you need to know about the symmetries.


You can tile a crystal with larger boundaries than a unit-cell (a hexagonal boundary, e.g.,) and perhaps you could use an expanded definition of unit-cell to encapsulate cells larger than the smallest necessary... but there really is no need to conflate the unit-cell with anything larger than it needs to be—it would get muddy and confusing, and I would opt to call it a different word (I'm sure there is one in crystallography that I am just not aware of at the moment.)


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