Rep-Tiles

TL;DR

The paper proves that every compact smooth $n$-manifold with connected boundary embedded in $\,\mathbb{R}^n$ is topologically isotopic to a rep-tile, i.e., a polycube that $2^n$-tiles the cube. It introduces cube swapping and taloned boundary patterns to construct rep-tiles and establishes an explicit toolkit for realizing rep-tiles in all dimensions, including ones with the homotopy types $S^n\times D^2$ and finite bouquets of spheres, via stacks and suspensions. It also shows how rep-tiles induce self-similar tilings of Euclidean space and interact with Golomb's tiling hierarchy, while proving that not every manifold in an isotopy class is a rep-tile, demonstrating the limits of the geometric side. Overall, the work completes isotopy classifications of manifolds that tile the cube and provides explicit constructions of rep-tiles with rich topological structure, enabling new self-similar tilings with controlled topology.

Abstract

An $n$-dimensional rep-tile is a compact, connected submanifold of $\mathbb{R}^n$ with non-empty interior which can be decomposed into pairwise isometric rescaled copies of itself whose interiors are disjoint. We show that every smooth compact $n$-dimensional submanifold of $\mathbb{R}^n$ with connected boundary is topologically isotopic to a polycube that tiles the $n$-cube, and hence is topologically isotopic to a rep-tile. It follows that there is a rep-tile in the homotopy type of any finite CW complex. In addition to classifying rep-tiles in all dimensions up to isotopy, we also give new explicit constructions of rep-tiles, namely examples in the homotopy type of any finite bouquet of spheres.

Figures (16)

• Figure 1: The "chair" rep-tile.
• Figure 2: A stack of cubes (left) and its labeled footprint (right). This polycube and its image under rotation by $\pi$ about $P_0$ tile $[0,4]^2$. Thus, the polycube is a rep-tile.
• Figure 3: Top : Footprint of a 3-dimensional rep-tile homeomorphic to $S^1\times D^2$ (left), and its corresponding stack of cubes (right), rotated by 90 degrees for visualization. Bottom: Footprint of a 4-dimensional rep-tile homeomorphic to $S^2\times D^2$ (left), and space to imagine the corresponding stack of cubes (right).
• Figure 4: Cube swaps in a 3-dimensional rep-tile. The swap effectuates the suspension of a polycube representation of $S^0\times D^2$ to obtain a polycube representation of $S^1\times D^2$. Top: $S^0\times D^2\times [0, 4]\cong S^0\times D^3$. Middle: A cube swap which ensures that the first and last slices become disks. Bottom: the union of the four layers is a rep-tile homeomorphic to $S^1\times D^2$, the result of the suspension. A further cube swap between the same pairs of columns would result in the $S^1\times D^2$ rep-tile given in Section \ref{['sec:spheres']}.
• Figure 5: The left and right columns represent the labeled footprints of $4$-dimensional stacks of cubes. Taken together, the three columns depict the process of suspension from $S^1\times D^2$ to $S^2\times D^2$. Left column: four layers of $S^1\times D^2\times [i, i+1]$, combining to form $S^1\times D^2\times [0, 4]$. Middle column: cube swaps occur in the first and fourth slices. Right column: bottom slice: $D^3\times [0,1]$, second slice: $S^1\times D^2\times[1,2]$; third slice: $S^1\times D^2\times[2,3]$; fourth slice: $D^3\times[3,4]$. The union of the four slices is the suspended rep-tile. Note that by a further cube swap we could replace all 3's and all 1's by 2's. This would produce another rep-tile homeomorphic to $S^2\times D^2$, namely the one described in Section \ref{['sec:spheres']}.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.2.1
• Corollary 1.2.1
• proof
• Proposition 2.1
• proof : Proof of Proposition \ref{['prop:Richard']}
• Definition 3.1
• Definition 3.2
• Lemma 3.1