Polygon · Rational polygons

Vertices	3
Interior lattice points	0
Boundary lattice points	6
Lattice points	6
Denominator ?The smallest k such that kP has integral vertices.	1
Euclidean area	2
Lattice width ?The minimum of max⟨w, x⟩ − min⟨w, x⟩ over x ∈ P, taken over all non-zero w ∈ ℤ².	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.	D3
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.	yes
Thin ?Can be realized in ℝ × [0, 1].	no

Vertices

(1, -1), (-1, 1), (-1, -1)

2 t² + 3 t + 1

All lattice polygons with 0 interior lattice points: [Kho97], [Koe91], [Sch03], [BS26, Theorem 1.1], [SB24a]
All maximal lattice polygons with 0 interior lattice points: [BS26, Classification 4.5], [SB24a]
All lattice polygons with at most 50 lattice points: [Koe91], [SB24e], [OEIS, A371917]
All lattice polygons in a square of side length at most 8: [BK13], [BS26, Table 2.13], [OEIS, A374975], [SB24f]
All k-maximal polygons, for k = 2–20, with 0 interior lattice points: [BS26, Classification 4.5], [SB24a]
All 21-maximal polygons with 0 interior lattice points: [SB24a]

Has the maximal number of vertices, 3, among all lattice polygons with no interior lattice points that cannot be realized in ℝ × [0, 1], and is the only one.
Listed among the small polygons of the Graded Ring Database [BK13]; this search finds it.
Drawn in [BS26, Figure 3], the first polygon from the left.
Drawn in [BS26, Figure 4], first row, the first polygon from the left.
Drawn in [BS26, Figure 5], first row, the first polygon from the left.
Drawn in [Koe91, Figure 4.1.3c].

[BK13] G. Brown, A. M. Kasprzyk: Small polygons and toric codes. J. Symbolic Comput. 51, 55–62 (2013). https://doi.org/10.1016/j.jsc.2012.07.001
[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
[Kho97] A. G. Khovanskiĭ: Newton polytopes, curves on toric surfaces, and inversion of Weil's theorem. Uspekhi Mat. Nauk 52(6), 113–142 (1997). https://doi.org/10.1070/RM1997v052n06ABEH002156
[Koe91] R. J. Koelman: The number of moduli of families of curves on toric surfaces. PhD thesis, University of Nijmegen (1991).
[OEIS] OEIS Foundation Inc.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org
[SB24a] J. Springer, M. Bohnert: Rational polygons with no interior lattice points. Zenodo (2024). https://doi.org/10.5281/zenodo.13838991
[SB24e] J. Springer, M. Bohnert: Lattice polygons with at most 70 lattice points. Zenodo (2024). https://doi.org/10.5281/zenodo.13959996
[SB24f] J. Springer, M. Bohnert: Lattice subpolygons of a square with given sidelength. Zenodo (2024). https://doi.org/10.5281/zenodo.13838476
[Sch03] J. Schicho: Simplification of surface parametrizations — a lattice polygon approach. J. Symbolic Comput. 36(3–4), 535–554 (2003). https://doi.org/10.1016/S0747-7171(03)00094-4