| Vertices | 3 |
|---|---|
| Interior lattice points | 0 |
| Boundary lattice points | 3 |
| Lattice points | 3 |
| Denominator ?The smallest k such that kP has integral vertices. | 2 |
| Euclidean area | 3/2 |
| Boundary k-rational points ?The number of points of (1/k)ℤ² on the boundary of P, where k is the denominator of P. | 6 |
| Lattice width ?The minimum of max⟨w, x⟩ − min⟨w, x⟩ over x ∈ P, taken over all non-zero w ∈ ℤ². | 3/2 |
| Integer hull dimension ?The dimension of the convex hull of the lattice points of P, or −1 if P contains none. | 2 |
| Interior integer hull dimension ?The dimension of the convex hull of the interior lattice points of P, or −1 if there are none. | −1 |
| Square side length ?The smallest m such that P is equivalent to a subpolygon of [0, m]². | 2 |
| Automorphism group ?The group of affine unimodular transformations mapping P to itself: Zn is cyclic of order n, Dn is dihedral of order 2n. | Z1 |
| Maximal ?No strictly larger polygon with vertices in (1/k)ℤ², where k is the denominator of P, has the same number of interior lattice points. | no |
| Thin ?Can be realized in ℝ × [0, 1]. | no |
Vertices
(-1, -1/2), (1, 1/2), (1, -1)
Ehrhart quasipolynomial ?The number of lattice points in tP for integers t ≥ 1. Its coefficients are periodic in t, with a period dividing the denominator, so there is one polynomial for each residue of t modulo the period.
t ≡ 0 mod 23/2 t² + 3/2 t + 1
t ≡ 1 mod 23/2 t² + 3/2 t
Sources
- All polygons with denominator 2 and 0 interior lattice points: [BS26, Theorem 1.1], [SB24a]
References
- [BS26] M. Bohnert, J. Springer: Classifying rational polygons with small denominator and few interior lattice points. Discrete Comput. Geom. 76, 1238–1265 (2026). https://doi.org/10.1007/s00454-026-00822-0
- [SB24a] J. Springer, M. Bohnert: Rational polygons with no interior lattice points. Zenodo (2024). https://doi.org/10.5281/zenodo.13838991