Strict Self-Assembly of Discrete Self-Similar Fractals in the abstract Tile-Assembly Model

TL;DR

The paper resolves a long-standing question by showing that discrete self-similar fractals can be strictly self-assembled in the abstract Tile-Assembly Model (aTAM) via two distinct constructions: a quine-based approach embedded in an intrinsically universal (IU) standard aTAM system, and a self-describing circuit method that fixes a Sierpiński-carpet–like generator. A key theoretical advance is the Tree Pump Theorem, which characterizes when DSSFs can be assembled by relating productions to encircled squares or to ultimately periodic growth, and a polynomial-time procedure to decide, for any generator $G$, whether $G^{\infty}$ is producible as an aTAM DSSF. The two constructions provide complementary perspectives: one utilizes intrinsic universality and quines to achieve fractal self-similarity across scales, while the other encodes a self-describing computation into a circuit-on-a-tile framework, enabling fixed-point assembly for a broad class of DSSFs. Together, these results establish both existence and a practical decidability criterion for strict DSSFs in the aTAM, with implications for understanding the limits of geometric computation and fractal self-assembly in tiling systems.

Abstract

This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways, each taking advantage of a novel set of tools. One of our constructions shows that specializing the notion of a quine, a program which prints its own output, to the language of tile-assembly naturally induces a fractal structure. The other construction introduces self-describing circuits as a means to abstractly represent the information flow through a tile-assembly construction and shows that such circuits may be constructed for a relative of the Sierpinski carpet, and indeed many other DSSFs, through a process of fixed-point iteration. This later result, or more specifically the machinery used in its construction, further enable us to provide a polynomial time procedure for deciding whether any given subset of $\mathbb{Z}^2$ will generate an aTAM producible DSSF. To this end, we also introduce the Tree Pump Theorem, a result analogous to the important Window Movie Lemma, but with requirements on the set of productions rather than on the self-assembling system itself.

Figures (46)

• Figure 1: The substitution $\kappa$ of the Sierpinski's Cacarpet.
• Figure 2: (a) A schematic depiction of the construction of the quine system. (1) Subsets of tile types capable of various algorithmic functions (e.g. binary counting, rotation of values, etc.) make up the "functional" tileset $T_F$. (2) The glue labels and tile types of $T_F$ are encoded to generate the "seed row" tileset $T_S$ that self-assembles a row presenting an encoding of $T_F$ (plus some additional necessary information) via its northern glues. The seed tile is shown as green and the rest of the seed row as yellow. (3) Using the full tileset $Q = T_F \cup T_S$, the system grows from the seed tile to form the seed row, then the tiles of $T_F$ cause upward growth that computes the definitions of the seed row tiles (via their northern glues and locations), appending those definitions to those provided by the seed row. This results in an assembly with the full definition of $Q$ encoded in its northern glues. Further growth by the tiles of $T_F$ causes that assembly to grow into a terminal square macrotile that is consistent with the definition of a seed assembly representing the quine system's seed tile for the IU tileset $U$. (4) If the terminal assembly from (3) were used as the seed assembly for a system including the tile types from $U$, that system would simulate the quine system. That is, each tile of $Q$ would be simulated by a macrotile composed of tiles from $U$, resulting in a terminal assembly that is a macro-macrotile representing the seed tile of $Q$. (b) Schematic depiction of the formation of a macrotile square by the quine construction adapted for DSSFs. The dimension $X$, counted by binary counters, can be set to an arbitrary value as long as it is greater than the height of the grey rectangle (created by the earlier stages of the quine assembly growth), and the dimension $Y$ can be set to be any value between $0$ and $X/2$. Careful setting of $X$ and $Y$ values can allow for arbitrary percentages of the area contained within the macrotile to be empty space.
• Figure 3: Left, a $2$-digit by $1$-digit multiplier computing $32$ (left, written bottom-to-top) times $7$ (top) is $224$ (right, bottom-to-top). When a gate has two input wires, each of them bends towards the other wire. Right, the tile type corresponding to the $a$ gate at the bottom-right of the circuit, and a neighborhood where it can attach. The temperature $\tau \leq$ is the maximum in-degree of a gate in the circuit (here, 2).
• Figure 4: The three sets defined by \ref{['thm:tree_pump']}: $C_{m}[\mathcal{T}]$ is the assemblies which encircle an $m \times m$ square, $B_{F(n, m), \vec{d}}[\mathcal{T}]$ is the set of assemblies which do not reach further than $F(n,m)$ in direction $\vec{d}$, and $P_{\vec{d}}[\mathcal{T}]$ is the assemblies which contain a periodic path with period $\vec{p}$ such that $\vec{p}\cdot\vec{d} > 0$.
• Figure 5: The principle of $C_{\square}$. Each gate $G$ (top-left) is cut by $\kappa$ into a circuit (right). The messages are composed of two parts. On the border of $\kappa(G)$, their second part corresponds to the messages of $G$. When a gate is cut, some information is lost: the type of its input wires. This information is reconstructed from the output wires of the predecessor, which are given in the local part of the messages from the neighboring $\kappa(G')$. The seed gate's message (bottom-left) declares "I am the tile at $(0, 0)$ within a seed gate, at all scales".

Theorems & Definitions (143)

• Definition 1: Dicrete Self-Similar Fractal
• Theorem 1
• Definition 2
• Definition 3
• Theorem 2
• Theorem 3
• Theorem 4: Tree Pump
• Theorem 5
• Theorem 6
• Corollary 1