[Paper Review] Algebraic Method in Tilings

Peter Horák, Dongryul Kim|arXiv (Cornell University)|Feb 29, 2016

Quasicrystal Structures and Properties4 references3 citations

TL;DR

This paper introduces a novel algebraic 'polynomial method' in tiling theory, combining it with Hilbert's Nullstellensatz to establish a necessary condition for tiling ℤⁿ by translates of a cluster V. The key result shows that if a cluster V of prime size q generates ℤⁿ, then any tiling of ℤⁿ by V must be lattice-tiling, supporting a conjecture that all such tilings by prime-sized clusters are lattice-tilings.

ABSTRACT

In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling $n$-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of $n$-space by translates of a cluster $V$ of prime size then there is a lattice tiling by $V$ as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster $V$ is lattice if $V$ generates $n$-space.

Motivation & Objective

• To develop a new algebraic method—'polynomial method'—for analyzing tilings of ℤⁿ by translates of a cluster V.
• To establish a necessary condition for tiling ℤⁿ by an arbitrary set V using Hilbert's Nullstellensatz.
• To prove that if a cluster V of prime size q generates ℤⁿ, then any tiling of ℤⁿ by V must be a lattice tiling.
• To provide supporting evidence for the conjecture that all tilings of ℤⁿ by a prime-sized cluster V are lattice tilings if V generates ℤⁿ.
• To investigate the uniqueness and periodicity of tilings by semi-crosses (n-crosses) of prime size.

Proposed method

• Introduce a polynomial method to model tiling conditions algebraically, treating tiling as a system of polynomial equations over a finite field.
• Apply Hilbert's Nullstellensatz to derive a necessary algebraic condition for the existence of a tiling of ℤⁿ by a set V.
• Use the structure of the group ring ℤ[V] and properties of roots of unity to analyze tiling symmetries and periodicity.
• Leverage the fact that for prime-sized V, the additive group structure of V allows the use of finite fields and cyclic group actions.
• Define a homomorphism φ: ℤⁿ → ℤ_q using a primitive root of ℤ_q*, mapping the tiling lattice to the kernel of φ.
• Show that the kernel of φ forms a lattice tiling, and that shifting codewords preserves the tiling structure, implying cyclicity and periodicity.

Experimental results

Research questions

• RQ1 Under what algebraic conditions does a set V ⊂ ℤⁿ tile ℤⁿ via translates?
• RQ2 Can the polynomial method be used to prove that all tilings of ℤⁿ by a prime-sized cluster V are lattice tilings?
• RQ3 Is there a unique tiling (up to congruence) of ℤⁿ by a semi-cross of prime size 2n+1?
• RQ4 Does the property of being cyclic (closed under shifts) characterize tilings by prime-sized clusters?
• RQ5 Can the polynomial method be extended to prove periodicity in tilings by non-prime-sized clusters?

Key findings

• For any set V ⊂ ℤⁿ of prime size q, if V tiles ℤⁿ, then the tiling must be periodic with period q(v−w) for any v,w ∈ V.
• If a cluster V of prime size q generates ℤⁿ, then any tiling of ℤⁿ by V is necessarily a lattice tiling.
• The tiling of ℤⁿ by a semi-cross of size 2n+1 is unique up to congruence if 2n+1 is prime, as shown for n=2,3,5.
• For prime q ≤ 7, any tiling of ℤ^{q−1} by a semi-cross of size q is unique up to congruence and is lattice-tiling.
• A cyclic tiling of ℤ^{q−1} by a semi-cross of prime size q exists for all odd primes q, constructed via a homomorphism to ℤ_q.
• The kernel of a homomorphism φ: ℤ^{q−1} → ℤ_q defined by a primitive root t generates a lattice tiling closed under cyclic shifts.