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Slab tilings, flips and the triple twist

George L. D. Alencar, Nicolau C. Saldanha, Arthur M. M. Vieira

TL;DR

The paper introduces slab tilings and a triple twist TTw as a robust invariant under local flips, extending ideas from domino tilings to the 2×2×1 slab setting. It develops a four-color slab-type coloring to translate slab and mixed tilings into domino tilings, enabling a three-axis twist construction and a TTw in $\mathbb{Z}^3$. The authors prove both upper and lower bounds on the number of possible TTw values in large boxes, showing that TTw values can be as large as order $N^{12}$ in $N$-sized cubes and can realize a wide range of vectors, while also demonstrating that no fixed finite set of local moves can guarantee flip-connectivity for all slab tilings. They establish flip-invariance of the twists, analyze symmetry effects, and discuss connectivity, explicit constructions, and open questions, thereby laying groundwork for understanding the combinatorial structure of slab tilings and their invariants. The results have implications for the study of three-dimensional tilings and their local-move connectivity, revealing rich invariant-based geometry beyond the domino case.

Abstract

A \textit{domino} is a $2\times 1\times 1$ parallelepiped formed by the union of two unit cubes and a \textit{slab} is a $2\times 2\times 1$ parallelepiped formed by the union of four unit cubes. We are interested in tiling regions formed by the finite union of unit cubes. Domino tilings have been studied before; here we investigate \textit{slab tilings}. As for domino tilings, a flip in a slab tiling is a local move: two neighboring parallel slabs are removed and placed back in a different position. Inspired by the twist for domino tilings, we construct a flip invariant for slab tilings: the \textit{triple twist}, assuming values in $\mathbb{Z}^3$. We show that if the region is a large box then the triple twist assumes a large number of possible values, roughly proportional to the fourth power of the volume. We also give examples of smaller regions for which the set of tilings is connected under flips, so that the triple twist assumes only one value.

Slab tilings, flips and the triple twist

TL;DR

The paper introduces slab tilings and a triple twist TTw as a robust invariant under local flips, extending ideas from domino tilings to the 2×2×1 slab setting. It develops a four-color slab-type coloring to translate slab and mixed tilings into domino tilings, enabling a three-axis twist construction and a TTw in . The authors prove both upper and lower bounds on the number of possible TTw values in large boxes, showing that TTw values can be as large as order in -sized cubes and can realize a wide range of vectors, while also demonstrating that no fixed finite set of local moves can guarantee flip-connectivity for all slab tilings. They establish flip-invariance of the twists, analyze symmetry effects, and discuss connectivity, explicit constructions, and open questions, thereby laying groundwork for understanding the combinatorial structure of slab tilings and their invariants. The results have implications for the study of three-dimensional tilings and their local-move connectivity, revealing rich invariant-based geometry beyond the domino case.

Abstract

A \textit{domino} is a parallelepiped formed by the union of two unit cubes and a \textit{slab} is a parallelepiped formed by the union of four unit cubes. We are interested in tiling regions formed by the finite union of unit cubes. Domino tilings have been studied before; here we investigate \textit{slab tilings}. As for domino tilings, a flip in a slab tiling is a local move: two neighboring parallel slabs are removed and placed back in a different position. Inspired by the twist for domino tilings, we construct a flip invariant for slab tilings: the \textit{triple twist}, assuming values in . We show that if the region is a large box then the triple twist assumes a large number of possible values, roughly proportional to the fourth power of the volume. We also give examples of smaller regions for which the set of tilings is connected under flips, so that the triple twist assumes only one value.
Paper Structure (10 sections, 14 theorems, 11 equations, 30 figures)

This paper contains 10 sections, 14 theorems, 11 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Construction of the twist
  3. Mixed tilings
  4. Coloring and twist
  5. Properties of the Triple Twist
  6. Connectivity under local moves
  7. Possible values of the triple twist
  8. Final remarks
  9. Declarations
  10. Funding and Acknowledgements

Key Result

Theorem 1

Consider a cubiculated box ${\cal R}$ of size $N \times N \times N$, $N \in 2{\mathbb{N}}$. If $N$ is sufficiently large then there exists positive constants $C_{-} < C_{+}$ with the following properties. If ${\mathbf t}$ is a slab tiling of ${\cal R}$ then $|\operatorname{TTw}({\mathbf t})| < C_{+}

Figures (30)

  • Figure 1: Domino tiling of a $3\times 3\times 2$ box
  • Figure 2: A slab tiling of a $6\times 6\times 6$ box
  • Figure 3: Examples of effects for pairs of dominoes
  • Figure 4: Two examples of tilings of cylinders; in both cases, $\operatorname{Tw} = 0$
  • Figure 5: The $x$-view of the two tilings in Figure \ref{['fig:415']}, i.e., the same tilings after a rotation; we still have $\operatorname{Tw}=0$
  • ...and 25 more figures

Theorems & Definitions (34)

  • Theorem 1
  • Example 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Definition 4.4: Mixed twist
  • Definition 4.5: Triple twist
  • Lemma 5.1
  • proof
  • Proposition 5.2
  • ...and 24 more