Regular Triangular Tilings

• Regular triangular tilings are structured subdivisions of equilateral triangles that satisfy strict combinatorial, geometric, and algebraic constraints.
• Tile configurations combine unit rhombi and two types of trapezoids, incorporating no-overpacking conditions for precise enumeration and algorithmic verifications.
• Matroid-theoretic frameworks connect tiling properties to independence, circuits, and flats, offering fresh insights for research in discrete geometry and combinatorial optimization.

A regular triangular tiling refers to the subdivision of an equilateral triangle into smaller geometric tiles in a regular, patterned manner that satisfies combinatorial, geometric, and algebraic constraints. The study of these tilings encompasses discrete geometry, algebraic combinatorics, and matroid theory, with applications in enumerative combinatorics and geometric algorithms. The classification and properties of such tilings have been developed through rigorous research, including results on matroid cryptomorphisms and complete enumerative catalogs of possible tessellations.

1. Foundational Definitions and Geometric Structures

A central object is the equilateral triangle

$$S_n = \{ (x_1, x_2, x_3) \in \mathbb{R}_{\ge 0}^3 : x_1 + x_2 + x_3 = n-1 \}$$

whose set of integer lattice points is

$P_n = S_n \cap \mathbb{Z}^3.$

This construction yields a regular subdivision of $S_n$ into $n^2$ unit equilateral triangles, with $\left(i, j, k\right) \in P_n$ corresponding to the barycenter of a unit upward-pointing triangle in the subdivision. The ground set $u(T_n)$ of the matroid $T_n$ consists of all upward-pointing unit triangles, while the complement $d(T_n)$ contains all downward-pointing triangles interleaved among the upward ones. Lattice regions are unions of such triangles, with "holey regions" denoted by $T_n \setminus A(s)$ for a subcollection $s \subseteq u(T_n)$, where $A(s) = \bigcup_{X \in s} X$ (Gotti et al., 2018).

2. Tile Types and Packing Configurations

Regular triangular tilings are constrained to three principal tile types:

• Unit Rhombus (Lozenge): Formed by the union of one adjacent upward and one downward triangle. These rhombi present in three orientations (one vertical, two horizontal).
• Type-1 Trapezoid: Consists of two upward and one downward unit triangle—appearing in three orientations (horizontal and two slanted).
• Type-2 Trapezoid: Mirror image of Type-1; comprises one upward and two downward triangles.

Any region formed by removing upward triangles from $T_n$ can be tiled by combinations of these three tile types, subject to several packing inequalities. One checks that all lattice regions admit such tilings if and only if certain combinatorial (no-overpacking) conditions are met (Gotti et al., 2018).

3. Matroid-Theoretic Framework and Cryptomorphisms

The subdivision induces the matroid $T_n$ on the set of upward triangles. The collection of independent sets $\mathcal{I}_{n,3}$ of $T_n$ is given by

$$\mathcal{I}_{n,3} = \left\{ s \subseteq P_n : \forall k \le n, \ \forall \text{ translate } P \text{ of } P_k,\ |s \cap P| \le k \right\}$$

—encapsulating the "no-overpacking" axiom initially described by Ardila and Billey.

Key Theorems and Correspondences

• Independence–Tilings Correspondence: $s \subseteq u(T_n)$ is independent in $T_n$ if and only if the holey region $T_n \setminus A(s)$ can be tiled with unit rhombi and exactly $n - |s|$ type-1 trapezoids.
• Rank–Maximal Rhombi Correspondence: For $s \subseteq u(T_n)$, any tiling that maximizes the number of rhombi leaves exactly $|s| - r_{T_n}(s)$ unit downward triangles in the holey region; equivalently, the maximal number of rhombi is ${n \choose 2} - (|s| - r_{T_n}(s))$.
• Circuits–Type-2 Trapezoid Characterization: A subset $c \subseteq u(T_n)$ with $|c| \ge 4$ is a circuit if and only if there is exactly one strictly over-saturated triangle in its hull and every valid tiling of the holey region requires precisely one type-2 trapezoid.
• Flats–Saturation Criterion: $f \subseteq u(T_n)$ is a flat if for any lattice triangle $T \subseteq T_n$, $|f \cap u(T)| \ge \text{size}(T)$ implies $u(T) \subseteq f$. All over-saturated sub-triangles are completely covered by $f$ (Gotti et al., 2018).

4. Complete Classification of Tiling Numbers and Tile Shapes

Tilings of an equilateral triangle $ABC$ by congruent smaller triangles $T$ admit a full classification with respect to possible values of $N$ (the number of congruent tiles) and the tile's shape.

• Quadratic Family: If $T$ is equilateral, then $N = m^2$ for $m \ge 1$ ("quadratic tiling").
• Hexagonal and Triple-Square Families: If $T$ is a $30$–$60$–$90$ right triangle, then $N = 3m^2$ ("hexagonal" tiling) or $N = 6m^2$ ("double-half" tiling), $m \ge 1$.
• No Other Possibilities: Galois-theoretic and eigenvalue arguments show that only these three cases are possible for tilings of equilateral triangles; no other values or tile shapes yield a tiling (Beeson, 2012).

A catalog of $N$ values and tile types is summarized in the following table:

Tile Shape	Possible $N$ Values	Construction Type
Equilateral ($T \sim ABC$)	$m^2$	Quadratic grid
$30$–$60$–$90$ triangle	$3m^2,\, 6m^2$	Hexagonal, double-half

The construction for $N=m^2$ involves subdividing each side of $ABC$ into $m$ equal segments and generating an $m \times m$ grid; for $N=3m^2$, one sub-divides $ABC$ into three smaller equilateral triangles and applies the grid method recursively (Beeson, 2012).

5. Enumerative and Algorithmic Consequences

Each basis of $T_n$—i.e., a maximal independent set of size $n$—is in bijection with pure lozenge tilings of $T_n$. More generally, independent sets of size $k$ map to mixed tilings containing exactly $(n-k)$ type-1 trapezoids. Enumeration of tilings for triangular regions with holes, or with mixed trapezoidal configurations, remains an open challenge. For hexagonal regions, closed formulas such as MacMahon’s for lozenge tilings exist, but not for general triangular regions with holes (Gotti et al., 2018).

Algorithmic implications arise from the concrete characterizations:

• Matroid rank can be computed algorithmically by examining maximal-rhombi tilings.
• Dependence, closure, and flat identification reduce to local geometric checks of saturation and tile combinatorics in sub-triangles.
• Circuit and flat membership can be checked via inspection of over-packed or saturated triangles and the necessity of type-2 trapezoids. This suggests particularly efficient algorithmic implementations for independence and closure testing, with applications in discrete geometry and combinatorial optimization.

6. Algebraic Constraints and Classification Proofs

Algebraic methods underpin the impossibility of alternative tilings. The integer "d-matrix" records boundary assignments and induces eigenvalue conditions. For similar tiles $T$, the eigenvalue equation yields

$$(D \cdot [a\ b\ c]^T) = \sqrt{N} \cdot [a\ b\ c]^T,$$

forcing $\sqrt{N}$ to be either $m$, $m\sqrt{3}$, or $m\sqrt{6}$, corresponding to the three distinct $N$ families. Galois-theoretic restrictions exclude other tile shapes, recovered as corollaries from the vanishing diagonal of $D$ and rationality arguments. Only the $30$–$60$–$90$ right triangle survives as an admissible shape in non-quadratic cases, as algebraic closures are realizable solely for $\alpha = \pi/6,\, \beta = \pi/3,\, \gamma = \pi/2$ (Beeson, 2012).

7. Context, Open Problems, and Research Directions

The explicit enumeration and geometric-matroid correspondence in regular triangular tilings clarify the combinatorial structure and offer discrete algorithmic approaches for a variety of matroid concepts. A plausible implication is that the correspondence between tiling configurations and matroid invariants could lead to new insights in both combinatorial theory and computational geometry.

Open problems remain, particularly for the enumeration of tilings in triangular regions with holes and general mixed trapezoidal arrangements, where closed product formulas analogous to MacMahon’s for hexagons do not yet exist. The packing and saturation conditions that govern valid tiling configurations are central to further investigation of these enumeration challenges.

Furthermore, the tiling–matroid correspondence reduces questions of independence, rank, circuits, and flats in $T_n$ to geometric considerations involving packing and saturation of unit triangles, rhombi, and trapezoids on a fixed triangular board. This synthesis motivates research on the connections between discrete geometry, algebraic combinatorics, and computational algorithms (Gotti et al., 2018, Beeson, 2012).