A Local Characterization of Combinatorial Multihedrality in Tilings

A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity.

💡 Research Summary

The paper investigates face‑to‑face tilings of Euclidean d‑space by convex polytopes and asks when such a tiling is “combinatorially multihedral”, i.e. when the combinatorial automorphism group Aut_c(T) partitions the set of tiles into only finitely many orbits. This property is weaker than geometric multihedrality (finite orbits under the full Euclidean isometry group) but still imposes a strong global regularity on the tiling.
The authors introduce the notion of a centered corona C_k(P), the subcomplex consisting of a tile P together with all tiles whose combinatorial distance from P is at most k (normally k = 1). The corona records the full incidence structure of P, its faces, and the way neighboring tiles attach to those faces. The central observation is that if two tiles have isomorphic centered coronas, then they belong to the same Aut_c(T)‑orbit, provided a modest additional condition holds.
The main result, called the Local Characterization of Combinatorial Multihedrality, consists of two equivalent conditions:

1. Finite Corona Types (Condition A). The set of isomorphism classes of centered coronas C_1(P) occurring in the tiling is finite. In other words, there are only finitely many distinct local neighbourhood patterns up to combinatorial equivalence.

2. Non‑trivial Corona Symmetry (Condition B). For each corona type C, the combinatorial automorphism group of the corona, Aut_c(C), contains at least one element that does not fix the central tile. Equivalently, each corona possesses a non‑identity symmetry that moves the centre to one of its neighbours while preserving the whole local incidence structure.

The theorem states that a tiling T is combinatorially multihedral iff both (A) and (B) are satisfied. The proof proceeds by showing that (A) guarantees only finitely many “local models” to consider, while (B) prevents each model from splitting further into infinitely many orbits. The authors develop a corona‑extension technique: an isomorphism between two centered coronas can be uniquely prolonged to an isomorphism between the corresponding 2‑coronas, then to 3‑coronas, and ultimately to the whole tiling. This inductive extension demonstrates that local isomorphism propagates globally, yielding a global automorphism that maps one central tile to the other. Conversely, if Aut_c(T) has finitely many orbits, any two tiles in the same orbit must have isomorphic coronas, and the existence of a non‑trivial orbit‑preserving symmetry follows from the finiteness of the orbit set.

The paper also revisits the earlier Local Theorem for Monotypic Tilings, which dealt with the special case where there is only one corona type (i.e., the tiling is combinatorially tile‑transitive). The new theorem subsumes that result as the case where Condition A holds trivially (single type) and Condition B reduces to the existence of any non‑identity automorphism of the corona.

To illustrate the theory, the authors present several examples:

• A 2‑dimensional checkerboard of squares coloured black and white. There are exactly two corona types, each possessing a 180° rotational symmetry that swaps the central square with its opposite colour. Hence the tiling is combinatorially 2‑hedral.

• A three‑dimensional mixed tiling of cubes and regular tetrahedra (the “tetrahedral‑cubic honeycomb”). Three distinct corona types appear, each admitting rotational symmetries that move the centre tile to a neighbour. The combinatorial automorphism group splits the tiles into three orbits, confirming multihedrality.

• A spiral‑like tiling in the plane where each successive tile introduces a new local pattern, leading to infinitely many corona types. Condition A fails, and indeed the combinatorial automorphism group has infinitely many orbits.

The authors discuss algorithmic implications: centered coronas can be encoded as labeled graphs, and graph‑isomorphism software can be employed to test Condition A. Once the finite set of corona types is known, checking Condition B reduces to computing the automorphism group of each finite graph and verifying the existence of a non‑trivial element moving the distinguished central vertex. This provides a practical procedure for deciding combinatorial multihedrality in concrete tilings.

In the concluding section, the paper outlines possible extensions. The current framework assumes convex tiles and strict face‑to‑face adjacency; relaxing these constraints (e.g., allowing non‑convex tiles or non‑face‑to‑face contacts) would require refined notions of local neighbourhoods. Moreover, the authors suggest that similar local characterizations might be developed for other symmetry‑related properties, such as combinatorial chirality or colour‑symmetry groups.

Overall, the work delivers a clean, purely combinatorial local criterion that exactly captures the global multi‑orbit structure of tilings. By reducing a potentially intractable global symmetry problem to a finite set of local checks, it opens the door to systematic classification of highly regular tilings in any dimension and provides a solid theoretical foundation for computational investigations in discrete geometry, crystallography, and related fields.