# Why lattice constant (unit cell) is always a parallelepiped?

Wikipedia defines lattice constant as physical dimension of unit cells in a crystal lattice.

Unit cell is defined as:

The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a parallelepiped

Why that elementary pattern that repeats itself in crystals is always a parallelepiped? In real life such elementary repeating unit/part/block is very often not a parallelepiped at all!

Is this some abstraction and a "bounding parallelepiped" is meant - the one into which definitely fits whatever real elementary repeating pattern is?

• Can you give a counterexample where the unit cell is not a parallelepiped? Jun 11, 2018 at 14:07
• @Floris Hexagonal lattices are a good example. (Yes, they also have a decomposition where the unit cell is a 2D parallelepiped, but that's really the point.) Jun 11, 2018 at 14:15
• Wigner-Seitz cells are typically hexagonal in 2D. And in 3D, there are the truncated octahedra of the bcc lattice etc
– user137289
Jun 11, 2018 at 17:38

The key point here is that if you have a tiling of the plane using

an elementary pattern that repeats itself,

then the pattern itself as originally given might indeed not be a parallelepiped, like the big-square-joined-with-a-smaller-square lumpy shape that's used to produce the following tiling, Image source

but, even in those cases, there always exists an alternative rearrangement of the pattern that can reproduce the entire tiling using a parallelepiped-shaped unit cell. This simpler unit cell might not be unique in its shape or contents, but the existence of a parallelepiped-shaped unit cell is always guaranteed by the periodicity of the tiling. In the case above, here are two viable options, Image source

and generally speaking, no matter how complicated an Escherian tiling pattern you're working with, if it is periodic, then you're always guaranteed the existence of at least one way to cut and trim and re-arrange your basic unit pattern into a box with parallel sides: Image source

• Very nice counterexamples! Jun 11, 2018 at 14:32
• Very nice examples and all, but I think the asker wanted to know the why of "there always exists an alternative rearrangement of the pattern that can reproduce the entire tiling using a parallelepiped-shaped unit cell". There's certainly a mathematical proof of that, I suppose. Jun 11, 2018 at 19:33
• @unreadableusername There is indeed a mathematical proof of this fact, but I don't read the question as inquiring about the details of the proof. I can of course expand this answer in that direction if OP confirms your interpretation of the question. Jun 11, 2018 at 20:49

Any shape that can be indefinitely extended along three axes (the definition of the "unit cell" is the thing that, when repeated, fills space) can be mapped to a parallelepiped.

Individual atoms may or may not be on the corners of the parallelepiped - but the "bounding box" that contains atoms at the same (relative) location on every instance must be a parallelepiped.

See this drawing (2D) for an example of why that must be so (and it does confirm your intuition of the "bounding parallelepiped"): Note - it is possible to have a "different" shape for certain specific cases (like a hexagonal lattice, certain "Escherian" shapes) - but that can ALSO be mapped to a parallelepiped (which is therefore a more sensible general shape, because it always works): • This is incorrect. See e.g. this image search for a bunch of examples where a non-parallelepiped shape can be used in a periodic tiling to cover the entire plane. Jun 11, 2018 at 14:27
• @EmilioPisanty fair point - made some updates / corrections. Jun 11, 2018 at 14:32

The key feature of a crystal lattice its its translational symmetry corresponding to a Bravais lattice. This means that a given pattern repeats in 3D-space with 3 (linearly independent) translational lattice basis vectors $\vec a_1$, $\vec a_2$, $\vec a_3$. These basis vectors are not unique but they always define a parallepiped which constitutes a unit cell of the lattice. Therefore the repeating pattern of a crystal is always connected with the parallelepiped of a unit cell.

• Extremely precious information, but I think the asker wants to know the why of "These basis vectors are not unique but they always define a parallepiped which constitutes a unit cell of the lattice.". There's almost certainly a mathematical proof of this statement, I suppose. Jun 11, 2018 at 19:34
• @ofhe_iAgDWolbuuTZO_5X1L6uuwfVP - The OP asks "Why is the elementary pattern that repeats itself in crystals always a parallelepiped". The answer is that three not coplanar basis vectors in 3D space always define the Bravais point lattice and the three basis vectors define the parallelepiped of a unit cell. As you suggest, why theses basis vectors of a Bravais lattice are not unique, and whether there exists a proof for it, is a mathematical question which might be posed on the mathematics stack exchange. Jun 12, 2018 at 0:53