every point is a₀ + a₁d + a₂d² + a₃d³ with d = cos30° + i·sin30°, integers aᵢ. no floats, so two vertices are the same iff the tuples match. click the canvas to walk one unit step at a time, or use the buttons.
d⁴ = d² − 1, so multiplying by d is a rotation by 30° that stays in the lattice. that's why every vertex has integer coords.