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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?

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    $\begingroup$ What cases do you think at not described by a parallelepipied? Are you confusing the lattice and the basis? $\endgroup$
    – Ian Bush
    Jun 11, 2018 at 17:10
  • $\begingroup$ It is not the only one but the norm for chemists/biochemists, solid state physicists define the primitive cell differently as the Wigner-Seitz cell. This has a rather polygonal look. Choose a victim point to be the centre. Connect this point to other nearby points with lines, then draw perpendicular lines at the midpoints. These latter intersect to form the primitive cell. $\endgroup$
    – porphyrin
    Jun 11, 2018 at 17:26

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A unit cell is always a parallelepiped because this is the definition of a unit cell. In case when you don't have a repeating unit of this kind (which might indeed be the case in real life), you don't have a crystal, also by definition. You might be dealing with a quasicrystal or something.

3D periodicity is what defines a crystal. If you have it, this means you have three non-coplanar periodic vectors, and these inevitably define a parallelepiped.

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For a 3D-lattice, due to the periodicity of the system all the information necessary to encode it can be encoded in a space described by three vectors and a set of operations/symmetries that tells you how those cells get sewn together to tile the plane.

In linear algebra speak, the rank of the matrix made from composing the lattice vectors is equivalent to the dimension of the space they span. Any fewer vectors, and there isn't enough information to map to a specific system (does not span the lattice.) Any more vectors, and there is excess information (each vector can be written as a linear combination of the other vectors, and thus encodes superfluous information that can be trimmed down.) The symmetries encode the rest of the information and just describe any other restrictions to the configuration.

This is generalizable to lattices and parallelepipeds in n-dimensions, as lattices are just an example of a free abelian group over some topological space (here, a vector space) and parallelepipeds are the "fundamental regions" of lattice groups acting on Euclidean topological spaces.

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