The Geometry of Space: Bravais Lattices

In 1848, Auguste Bravais demonstrated that there are exactly 14 ways to arrange points in 3D space such that every point has an identical environment.

These 14 lattices are categorized into 7 Crystal Systems based on the symmetry of the unit cell (lengths $a, b, c$ and angles $\alpha, \beta, \gamma$).

The Atomic Packing Factor (APF)

The APF is a measure of how "densely" atoms are packed into a unit cell. It is the ratio: $\text{APF} = \frac{N*{atoms} \times V*{atom}}{V\_{unit_cell}}$

In the gallery below, pay attention to how the APF jumps from 0.52 in a Simple Cubic lattice to 0.74 in an FCC lattice. This jump is the reason why most metals prefer close-packed structures—they are simply more energy-efficient.

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Challenge: Can you find the system with the lowest symmetry? (Hint: Look for the one where $a \neq b \neq c$ and none of the angles are $90^\circ$).