From Tiles to Fractals: How Simple Rules Generate Infinite Complexity

Introduction: Can a Structure Build Itself?

Imagine having thousands of tiny square tiles laid out on a flat surface. Each tile has coloured edges that determine how it can connect to other tiles. These tiles have no knowledge of the final structure they are meant to form. They simply follow a local rule: whenever two edges with matching colours meet and their connection is strong enough, the tiles stick together. At first, small clusters of tiles form randomly. But as time passes, something surprising begins to appear. Groups of tiles form recognisable shapes. These shapes then combine with others, creating larger structures. The process continues, and assemblies of tiles gradually produce increasingly complex patterns.

Now, imagine zooming into a small portion of this growing structure. You notice something remarkable: the pattern looks similar to the entire structure, just smaller. Zoom in again, and the pattern repeats once more. The system is forming a fractal, a structure that repeats itself across scales. This phenomenon may seem almost impossible. How could simple pieces, with no global knowledge, create something so organised and precise? This is precisely the type of problem studied in algorithmic self-assembly, a field at the intersection of computer science, mathematics, and nanotechnology.

In the paper Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles, Jacob Hendricks, Meagan Olsen, Matthew J. Patitz, Trent A. Rogers, and Hadley Thomas explore how simple square tiles can assemble themselves into fractal structures using only local rules. By carefully designing how tiles attach, send signals, and interact, a system can grow automatically into an infinite fractal. The researchers focus particularly on the Sierpiński triangle, a well-known fractal that demonstrates perfect self-similarity. They show that hierarchical assembly, signal-passing, and geometric constraints work together to construct this fractal in a controlled and predictable manner. While the study is theoretical, it has important implications for future technologies, such as self-assembling materials, DNA nanostructures, and microscopic machines.

Fractals and Self-Similarity

A fractal is a geometric structure that repeats its pattern at different scales. If you zoom into a portion of a fractal, you often see a miniature copy of the entire shape. This property is called self-similarity. Fractals are common in nature. Snowflakes grow through repeated crystal patterns, river networks branch into smaller streams resembling the larger system, and fern leaves consist of smaller leaves that mirror the entire plant. Mathematically, fractals are often created using recursive rules, where a simple pattern repeats over and over to produce increasingly complex structures. The Sierpiński triangle begins with a single triangle. The next stage is created by placing three copies of that triangle together to form a larger triangular shape. This new shape then becomes the base for the next stage, and the process repeats indefinitely, generating an infinite number of smaller triangles arranged in the same pattern.

The never-ending repetition of this pattern makes fractals ideal for studying self-assembly. If a system can construct one stage of a fractal, the same rule can be applied repeatedly to build subsequent stages, allowing for potentially infinite growth while maintaining structure.

Discrete Self-Similar Fractals

While many mathematical fractals are continuous, tile assembly systems require fractals to exist on a grid. Discrete Self-Similar Fractals (DSSFs) consist of points with integer coordinates, allowing fractals to be represented computationally and physically. DSSFs grow in stages. The first stage, x1, is a small arrangement of points on the grid. Subsequent stages are constructed by placing scaled copies of previous stages at specific coordinates. These positions are determined by a generator, a small set of points that serves as a blueprint for the fractal’s growth. For the Sierpiński triangle, the generator consists of three points arranged in a triangular configuration. At each stage, copies of the previous stage are placed at the generator’s coordinates to produce the next stage.

Figure 1: Triangle copies placed at generator points 

By repeating this rule, the fractal grows indefinitely while preserving its self-similar structure. The generator demonstrates that complex global patterns can emerge from simple local rules applied recursively.

The Tile Assembly Model

To construct fractals through self-assembly, researchers use the Tile Assembly Model. Tiles are square, placed on a grid, and each side has a glue with a label (which edges can attach) and a strength (how strongly they stick). Tiles attach if the glue labels match and the total connection strength meets or exceeds a threshold called the temperature. This ensures only stable attachments survive. The study employs a variation called the Two-Handed Assembly Model (2HAM). 

Here, not only individual tiles, but entire assemblies of tiles can attach to other assemblies. This allows small clusters to form independently and then combine into larger structures, creating hierarchical assembly.

Figure 2: Two assemblies combining

Hierarchical assembly is particularly suitable for fractals because it mirrors their recursive structure. Multiple assemblies can grow simultaneously and later merge, improving efficiency and reflecting natural growth processes.

Signal-Passing Tiles

The Tile Assembly Model alone cannot ensure assemblies grow in the correct sequence. The researchers therefore introduce Signal-Passing Tile Assembly Model (STAM), in which tiles can send signals to neighbours when specific events occur. Signals activate or deactivate glues, controlling when connections are allowed. This temporal control prevents incorrect attachments. For example, a glue may remain inactive until a particular assembly forms, after which a signal activates it, allowing the next stage to attach.

This mechanism enables precise coordination, allowing complex structures to grow predictably. It also supports controlled detachment, removing temporary structures that are only needed during intermediate stages.

Frame Assemblies and Geometric Constraints

To further ensure accuracy, frame assemblies are constructed around fractal sections. These frames guide alignment and activate glues only when the correct structure is in place. Additionally, the tooth-and-gap mechanism prevents incorrect attachments. Assemblies grow protrusions called teeth, which fit into matching gaps. Misaligned or incorrect assemblies cannot attach, ensuring the fractal grows correctly.

Even if small junk assemblies form, they remain isolated and cannot interfere with the main structure, guaranteeing the unique growth of the infinite fractal.

Hierarchical Construction and Infinite Growth The fractal grows in stages: n produces n+1, n+1 produces n+2, and so on. Each stage uses the same generator rule, maintaining self-similarity. This allows infinite growth while preserving structure.

This demonstrates that simple local rules, when combined with hierarchical assembly and signal-passing, can reliably generate infinitely large, organised patterns without any centralised control.

Generalisation and Future Applications

While the Sierpiński triangle is the focus, these principles apply to a broad class of fractals. Any structure defined by a generator pattern can potentially be constructed using tile assembly systems.

Potential applications are far-reaching:

• DNA Nanotechnology: self-assembling DNA tiles forming nanoscale devices.

• Programmable Materials: materials that change shape autonomously.

• Molecular Computing: computation through assembly patterns.

• Microscopic Sensors: organised networks forming automatically.

These applications demonstrate that local rules can generate complex global order, reflecting patterns observed throughout nature.

Conclusion

The research in Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles shows that complexity can emerge from simplicity. Each tile follows only local rules, yet through hierarchical assembly, signal-passing, geometric constraints, and carefully designed interactions, an infinite fractal structure emerges reliably.

This work exemplifies a fundamental principle: intricate, scalable, and ordered structures can arise from simple components interacting under local rules. From natural fractals to advanced nanotechnologies, the same principle is evident: simple interactions, properly guided, can create extraordinary complexity. Reference: Hendricks, J., Olsen, M., Patiz, M. J., Rogers, T. A., & Thomas, H. (2016). Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles. 1–30. https://arxiv.org/pdf/1606.01856