There are exactly fourteen ways points can fill space periodically — no more, no fewer. Bravais proved it in 1848. Down the center, data flows: pick a cubic cell in, the engine counts its atoms and how tightly identical spheres pack it, and the proven fractions come out — 0.52 simple, 0.68 body-centred, 0.74 face-centred. The blue team builds and defends it; the red team tries to break it.
source A. Bravais, Mémoire sur les systèmes formés par des points distribués régulièrement sur un plan ou dans l'espace, first presented 1848; pub. J. de l'École Polytechnique 19 (1850) 1–128 — no stable open-access scan, cited author/title/year AMBER: xray-exhibit.scs.illinois.edu/books/bravais.php. Rendered, not quoted.
Atoms per conventional cubic cell is not counted, it is shared: a corner atom sits in 8 cells so it contributes 1/8; a face atom in 2, so 1/2; a body-centre belongs wholly to 1.
Live sharing arithmetic for the current cell:
| site | count | share | = atoms |
|---|
SC = 8·⅛ = 1 · BCC = 8·⅛+1 = 2 · FCC = 8·⅛+6·½ = 4.
Seven crystal systems, centred in the allowed ways, give 14 Bravais lattices — Bravais 1848, correcting Frankenheim's miscounted 15. Three are cubic: SC / BCC / FCC, packing 0.52 / 0.68 / 0.74.
This periodic scaffold is what its neighbours act on: the-miller-indices labels its planes (h k l), and the-braggs-law probes their spacing d = a/√(h²+k²+l²). Each sphere is the next one's lattice.
The blue team's live check: recompute the three packing fractions and coordinations from scratch, and confirm FCC is the densest. If red tampers, this badge is where it shows.
A cubic Bravais lattice is one edge length a plus a centring rule. Identical hard spheres are grown until they touch — the contact geometry fixes the radius as a fraction of a:
| cell | touch along | radius r |
|---|---|---|
| SC | cube edge | a/2 |
| BCC | body diagonal | a·√3/4 |
| FCC | face diagonal | a·√2/4 |
Give the engine a cell and a value of a; that is all it needs — the packing fraction does not depend on a, only on the centring.
APF = (atoms · ⁴⁄₃πr³) / a³, r from the touching geometry. Coordination is counted live by generating the lattice and finding the nearest-neighbour shell — never looked up.
What the machine proves: SC = π/6 = 0.5236 (Z=6), BCC = π√3/8 = 0.6802 (Z=8), FCC = π/(3√2) = 0.7405 (Z=12). FCC ties HCP for the densest possible packing of equal spheres.
The blue team's witness (left) recomputes these live; the red team (right) tries to make them wrong.
APF also assumes identical hard spheres touching. Real bonds are directional: diamond-cubic silicon packs at only 0.34, not because its lattice is loose but because covalent bonds forbid close contact. The number is exact; its premise is a fiction.
"There are 14 crystal systems." Cut. There are 7 crystal systems; the 14 are Bravais lattices (systems × allowed centrings).
"HCP is a Bravais lattice." Cut. Hexagonal close-packing is not a Bravais lattice — it is a simple-hexagonal lattice with a two-atom basis. It still reaches 0.7405.
"Frankenheim found the 14 first." Kept, corrected. Frankenheim (1842) found 15 — two of his were identical. Bravais fixed the count to 14.
The red team's move: model FCC's spheres as touching along the cube edge (SC contact, r = a/2) instead of the face diagonal. Wrong geometry — the blue team's witness (window 7) is watching.
Use SC's corner-only contact for the face-centred cell and its packing collapses to π/6 ≈ 0.52 — below BCC. FCC stops being densest, the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.