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Polski2D Crystal Lattice: Bravais Grids, Packing & Lennard-Jones Energy
Switch between square, triangular and honeycomb Bravais lattices built from real primitive vectors, tune spacing, thermal vibration amplitude and vacancy-defect rate, and watch coordination number, packing fraction and Lennard-Jones bond energy update live as bonds stretch and compress.
This 2D companion builds real crystallography instead of the 3D version's decorative gem scene: pick square, triangular or honeycomb, and the engine constructs the lattice from genuine primitive (Bravais) vectors plus a basis — a two-atom basis on a triangular net for honeycomb, matching how graphene's structure is actually derived — then finds nearest-neighbour bonds geometrically rather than hard-coding them. Raising temperature adds a √T-scaled sinusoidal jitter to every atom, visibly straining bonds away from their equilibrium length; a Lennard-Jones potential (minimum pinned at the lattice constant) turns that live bond length into a running energy total, while a vacancy-defect slider removes atoms at random and the coordination-number and packing-fraction readouts recompute against what's actually left on screen, not the ideal defect-free structure.
2D crystal-lattice lab: switch between square, triangular and honeycomb Bravais lattices, tune spacing, thermal vibration and vacancy defects, and read live coordination number, packing fraction and Lennard-Jones bond energy.
2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install
A single Bravais lattice point can't reproduce honeycomb's alternating up/down hexagon pattern on its own — you need a triangular Bravais net plus a two-atom basis (like graphene's A and B sublattices) so every atom still ends up with exactly 3 nearest neighbours at the same bond length.
Packing fraction is the fraction of the plane covered by touching circles centred on every site. Triangular packs tightest (φ ≈ 0.9069, the 2D close-packing maximum), square is looser (φ ≈ 0.7854), and honeycomb is sparsest (φ ≈ 0.6046) because each site has only 3 neighbours instead of 6 or 4.
Each bond's energy follows U(r) = 4ε[(σ/r)¹² − (σ/r)⁶], with σ chosen so the minimum sits exactly at the lattice constant. At T = 0 bonds sit at equilibrium and the total is near its most negative; raising temperature stretches and compresses bonds away from that minimum, pushing the summed energy up.