Lattice and Fundamental Domain

Consider a 2-dimensional lattice represented by a grid of equally spaced points. Each point has integer coordinates (x, y).

Your task is to find the fundamental domain of this lattice. The fundamental domain is the smallest possible region that, when translated throughout the lattice, covers the entire space without overlaps.


  • Draw a lattice grid on a piece of paper, e.g., a 3x3 grid or larger.
  • Choose any lattice point as the origin (0, 0) on your grid.
  • Explore the lattice by moving horizontally and vertically from the origin.
  • Mark each lattice point you visit on your map.
  • Continue exploring until you find the smallest region that includes at least one representative point from each equivalence class under lattice translations.
  • Identify and describe the shape of this region, and count the number of unique lattice points it contains.

💡 Hint: The fundamental domain is often a polygon with a specific number of lattice points inside it.