What are the dual polyhedra of the face-centered cubic lattice?

Question

For a given lattice $L$ we could define the set of points closest to one point more than any other.

$$ \{ x \} = \min_{\ell \in L} \|x - \ell \| \in \mathbb{R}^3$$

This generalizes the "fractional part" in the case $L = \mathbb{Z}$ the integer lattice $\{ x\} = \min_{n \in \mathbb{Z}} |x-n| $

For any lattice point (such as the origin) I would like to know which points are closest to $\ell$ than any other point? In otherworse, which points have $x - \{ x\} = \ell$ ?

There are two cases $\ell = (0,0,0)$ and $ \ell = (\frac{1}{2}, \frac{1}{2}, 0)$. All the other atoms can b found by shifting the lattice a bit.

So I am asking what the voronoi cells (or Wigner-Seitz cell) are for the Face-centered cubic lattice?

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Can we simplify this expression for the Face-Centered Cubic lattice?

$$ \mathbb{Z}^3 \cup \Big((\frac{1}{2}, \frac{1}{2}, 0) + \mathbb{Z}^3 \Big) \cup \Big((\frac{1}{2}, 0,\frac{1}{2}) + \mathbb{Z}^3 \Big) \cup \Big((0, \frac{1}{2}, \frac{1}{2}) + \mathbb{Z}^3 \Big)$$

I suspsect it can just be written as all integer linear combinations of three appropriately chosen vectors.

Answer 1

this is all in SPLAG by Conway and Sloane. The lattice is usually doubled, so that the lattice points are integer points such that $x+y+z \equiv 0 \pmod 2.$ The Voronoi cell is a rhombic dodecahedron

The rhombic dodecahedron can be used to tessellate three-dimensional space. It can be stacked to fill a space much like hexagons fill a plane.

This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. Some minerals such as garnet form

https://en.wikipedia.org/wiki/Tessellation#Voronoi_tilings