Lattice Triangle Problem

Updated 2 July 2026

Lattice triangles are the convex hulls of three affinely independent integer points, defined by boundary and interior lattice points and computed via Pick’s theorem.
They are classified using invariants such as lattice widths and normal forms, with constraints given by Scott-type inequalities and forbidden (b, i) regions.
Special configurations include Heronian and amicable triangles, which have applications in combinatorial geometry, coloring problems, and the theory of translation surfaces.

A lattice triangle is the convex hull of three affinely independent points of the integer lattice $\mathbb{Z}^2$ . The study of such triangles connects discrete geometry, combinatorics, number theory, and algebraic geometry, with links to classifying congruence types, lattice-point enumeration, Ehrhart theory, triangle centers, coloring problems, and connections with translation surfaces and Teichmüller theory.

1. Fundamental Structures and Invariants

Lattice triangles are characterized by integer-coordinate vertices, giving rise to invariants accessible via discrete and algebraic techniques. The fundamental combinatorial quantities for a lattice triangle $T$ are:

Boundary points $b(T)$ : The number of lattice points lying on the edges of $T$ .
Interior points $i(T)$ : The number of lattice points strictly inside $T$ .
Area: By Pick’s theorem,

$\operatorname{Area}(T) = i(T) + \frac{b(T)}{2} - 1,$

which uniquely determines the area of the triangle given $b(T)$ and $i(T)$ (Hofscheier et al., 2016).

There is a complete classification of the pairs $(b(T), i(T))$ that arise for lattice triangles, governed by Scott's inequalities and refined by infinitely many Scott-type "spike" inequalities, giving a polygonal and conical forbidden-region structure in the $T$ 0-plane (Hofscheier et al., 2016). The full set of possible $T$ 1 is sharply restricted, and extremal triangles realizing boundary cases are explicitly constructed.

2. Classification up to Lattice Equivalence: Widths and Normal Forms

The affine-unimodular classification reduces the lattice-triangle problem to the analysis of lattice widths:

Width in direction $T$ 2: $T$ 3 for nonzero primitive $T$ 4.
First and second lattice widths $T$ 5: The minimal lexicographical pair for two linearly independent directions.

A key result states that two lattice triangles are equivalent if and only if their ordered pair $T$ 6 of widths coincide (Hamm, 2023). Consequently, there is a finite list of normal forms for each possible $T$ 7, and explicit combinatorial formulas enumerate the number of inequivalent lattice triangles in dilations of the unit square, recovered in the generating function

$T$ 8

with ties to algebraic geometry via the Hilbert series of a degree-8 hypersurface in weighted projective space (Hamm, 2023).

3. Embeddings, Heronian Triangles, and Amicability

Any triangle with integer side lengths and integer area—a Heronian triangle—can be embedded, up to lattice isometry, in $T$ 9 (Lunnon, 2012). The explicit embedding employs a Gaussian integer GCD routine acting on an "axial" rational pose, followed by a rotation that clears denominators. For primitive Heronian triangles, such embeddings are unique modulo the 8-element lattice isometry group.

A notable problem is to find all amicable pairs of lattice triangles: pairs $b(T)$ 0 such that the perimeter of $b(T)$ 1 equals the area of $b(T)$ 2 and vice versa. There is exactly one such pair up to rigid motion: a 9-12-15 triangle and a 3-25-26 triangle, both with explicit lattice embeddings and with perimeters and areas swapped ( $b(T)$ 3, $b(T)$ 4) (Praton et al., 26 Mar 2025). This uniqueness is a corollary of the classification of Heronian triangles and exhaustive Diophantine analysis.

Triangle	Vertices	Sides	Perimeter	Area
$b(T)$ 5	(0,0), (0,9), (12,0)	9,12,15	36	54
$b(T)$ 6	(0,0), (24,7), (24,10)	3,25,26	54	36

4. Centers, Symmetries, and Lattice Constraints

The geometric centers (centroid $b(T)$ 7, circumcenter $b(T)$ 8, orthocenter $b(T)$ 9) of a lattice triangle exhibit notable lattice-theoretic obstructions.

For a triangle to have all three classical centers (Euler line) in $T$ 0, it must be a non-primitive (i.e., dilated by 3) triangle; no primitive triangle possesses this property (Aebi et al., 24 Feb 2026).
Detailed necessary and sufficient criteria are established for the existence of acute lattice triangles of given lattice perimeter with orthocenter, circumcenter, or centroid at a lattice point. For the acute case, the results are sharp:
- Orthocenter: Only for $T$ 1 or $T$ 2.
- Circumcenter: Only for even $T$ 3 or $T$ 4.
- Centroid: For all $T$ 5, excluding $T$ 6 (Aebi et al., 27 Apr 2026).
For obtuse and right triangles, the perimeter restrictions relax accordingly.

Additionally, the automorphism group of a lattice triangle is explicitly described in terms of its normal form (cyclic, trivial, $T$ 7, etc.), and the Ehrhart polynomial counts are parametrized by boundary and interior lattice points (Hamm, 2023).

5. Lattice Triangles in Algebraic and Combinatorial Geometries

Connections to more global aspects:

Enumeration by Ehrhart theory: The Ehrhart polynomial of a lattice triangle is determined by $T$ 8, with explicit dependence via Pick’s theorem. The set of all $T$ 9 feasible pairs is cut out by new infinite families of inequalities (Scott-type cones), giving rise to a distinctive “spiky” region in parameter space (Hofscheier et al., 2016).
Coloring and triple systems: The proper coloring problem for points in a triangular lattice $i(T)$ 0 with respect to forbidden monochromatic equilateral triangles gives rise to the coloring function $i(T)$ 1, with established bounds ( $i(T)$ 2) and explicit structural counting for pairs extending to triangles. These coloring numbers are related to generalizations of Steiner triple systems and to extremal combinatorics (Brouwer et al., 2024).
List coloring: Every triangle-free induced subgraph of the infinite triangular lattice is shown to be $i(T)$ 3-choosable, with structurally optimized extensions along handles and Hall-type combinatorics, contributing toward the McDiarmid–Reed conjecture in topological graph theory (Aubry et al., 2011).

6. Special Lattice-Triangle Configurations and Number-Theoretic Constraints

Collinearity of interior lattice points: The classification of possible $i(T)$ 4 for which every lattice triangle with exactly 3 boundary points and $i(T)$ 5 interior points has all interior points collinear is complete: only $i(T)$ 6 (Li et al., 25 Jan 2025).
Dissections into integer-area subtriangles: A convex lattice polygon can be dissected into lattice triangles of integer area if and only if a contractibility condition applies to the parity-color boundary word; this is algorithmically decidable in linear time and substantially strengthens mere parity-based area constraints (Abrams et al., 2024).
Lattice-point enumeration in nonstandard lattices: The enumeration of lattice points within a triangle for sublattices such as $i(T)$ 7 is governed by formulae involving Hardy sums and their geometric interpretations as intersection numbers with certain geodesic nets in the upper half-plane, and satisfies extended reciprocity laws (Lägeler, 2022).
Optimization and fractal phenomena: The problem of maximizing the number of positive-integer lattice points in a right triangle of fixed area exhibits non-uniqueness in the optimal slope, with an infinite, fractal set of optimal slopes, and the absence of a unique limiting shape, in contrast to other convex-figure scenarios (Marshall et al., 2017).

7. Rational Angles, Translation Surfaces, and the Veech Lattice-Triangle Problem

The lattice triangle problem in the context of translation surfaces is to classify those rational triangles $i(T)$ 8 (angles rational multiples of $i(T)$ 9) for which the unfolded translation surface $T$ 0 is a Veech (lattice) surface, i.e., with a closed $T$ 1-orbit. The classification is complete in most regimes—two infinite families and one sporadic example—but in the window where the largest angle lies in $T$ 2 ("hard obtuse window"), it is conjectured that no such lattice triangles exist. This has recently been confirmed up to a density-zero set via number-theoretic modular obstructions (Mirzakhani–Wright rank), Fourier/Ramanujan-sum analysis, and automated Lean formalization (Angdinata et al., 25 Mar 2026). Explicit modular tests using triangle angle data provide a practical criterion for checking the Veech property.

Lattice triangles thus lie at a crossroads of discrete, algebraic, and geometric theories, with their classification reflecting deep arithmetic, combinatorial, and geometric phenomena. Ongoing research investigates sharpening enumeration bounds, classifying possible geometric and combinatorial properties, and extending techniques to higher-dimensional lattice polytopes and translation-surface analogues.