a Bravais lattice
/ Bravais -> bruh-VAY /
Combine the 7 crystal systems with the allowed centering types (P, I, F, C, R) and you do not get 7 times 5 possibilities — many combinations are redundant or would break the system's symmetry. Remove the duplicates and exactly 14 genuinely distinct lattices remain. These are the Bravais lattices, named after Auguste Bravais, and they exhaust every possible way to arrange lattice points periodically in three-dimensional space.
The 14 are: cubic P, I, F (3); tetragonal P, I (2); orthorhombic P, C, I, F (4); hexagonal P (1); trigonal/rhombohedral R (1); monoclinic P, C (2); triclinic P (1). An example of why some are missing: face-centered tetragonal is not listed because it can always be redrawn as a smaller body-centered tetragonal cell — it is not a new arrangement, just a bigger description of one already counted.
Fourteen is a complete, proven enumeration — not 14 found so far. Every crystalline material on Earth, from ice to steel to protein crystals, has a lattice that is one of these 14 (the motif supplies the endless variety). Combined with the 32 point groups, they build up to the 230 space groups.
Why 14 and not 7 times 5 = 35? Many system-plus-centering combinations either duplicate a smaller lattice or destroy the system's symmetry. Face-centered tetragonal, for example, redraws as body-centered tetragonal; base-centered cubic is not cubic at all. Strike out the redundancies and exactly 14 remain.
The 14 come from removing all redundant system-plus-centering combinations.
The 14 Bravais lattices, 32 crystallographic point groups, and 230 space groups are exact, closed enumerations proved by group theory — not empirical tallies that might grow. A newly discovered crystal always fits one of them (quasicrystals are the aperiodic exception that forced crystal to be redefined).