FloaTiles

Self-assembly based on the Cheerios effect and aperiodic monotiles.

A small experimental project by Georgii Karelin at the Embodied Cognitive Science Unit (ECSU), Okinawa Institute of Science and Technology Graduate University (OIST).

https://karegeo.github.io/floatiles/

Last updated: 2026-05-15 11:43 UTC

About

FloaTiles is a simple, affordable tabletop experiment. Identical 3D-printed polygonal tiles float on a shallow water surface and slowly come together because of the Cheerios effect — the familiar capillary attraction that makes breakfast cereal clump in a bowl of milk. Adding gentle vibration or airflow changes how the tiles aggregate and break apart, and the shape of the tile (we mostly use the recently discovered Hat and Spectre aperiodic monotiles) influences the patterns that emerge.

The goal of the project is modest: to see how much interesting collective behaviour can come out of very simple ingredients — geometry, surface tension, and a bit of noise — without any electronics or active components in the tiles themselves. Further analysis of the recordings is ongoing.

Status (May 2026): paper in preparation for the AROB / ICAROB conference series; in-browser simulation v1 live; Hat- and Spectre-tile experiments ongoing.

Interactive simulation

A small in-browser simulation of the on-lattice cluster–cluster aggregation model that we use to compare against the experiment:

➡ Open the simulation

Sliders let you change the surface density φ, the sticking probability pstick, the evaporation rate ε, and the diffusion exponent γ. Live stats show the number of clusters, the largest cluster, and the mean cluster size. The algorithm is a faithful port of the C reference (lcca_v18): 50×50 grid, 8-direction isotropic diffusion with size-dependent mobility, per-face stochastic sticking with union-find + BFS relabel, surface evaporation with short-range teleport that conserves the total tile count.

ALIFE 2024

Karelin, G. (2024). Floatiles: Self-Assembly Based On Cheerios Effect and Aperiodic Monotiles. ALIFE 2024 (extended abstract).

Videos

Latest experiments with the Spectre monotile in a medium container, with vibrational automatic perturbations:

Earlier experiments with the Hat monotile in a small container with manual perturbations:

Experiments with the Hat monotile in a large container with automatic perturbations:

Previous experiments with the Spectre monotile in a small container:

photo of the experiment photo of the experiment photo of the experiment

Background

The Cheerios effect

Floating objects on a liquid surface come together due to the deformation of the air–water interface around each particle. Surface tension and buoyancy combine to produce capillary attraction — a phenomenon named “the Cheerios effect” by Vella & Mahadevan in 2005, though its physics had been studied much earlier (Gifford & Scriven 1971; Hosokawa et al. 1994–1996; Bowden, Whitesides and collaborators 1997–1999).

Three physical principles govern the effect:

Aperiodic monotiles

Aperiodic monotiles are single shapes that tile the plane without producing a repeating pattern. The 2023 discoveries of the Hat and Spectre tiles (Smith, Myers, Kaplan & Goodman-Strauss) made the einstein problem concrete with shapes that can be fabricated and studied physically. In FloaTiles we use 3D-printed tiles inspired by these shapes; small spikes and notches on the edges act as a primitive key-and-lock system that increases the effective capillary contact between adjacent tiles.

Why it might be interesting

Self-organisation shows up everywhere in nature, from cereal in a bowl to mosquito eggs and fire-ant rafts. FloaTiles is a small attempt to reproduce a tiny corner of that in a controlled, easy-to-rebuild setup, and to see what happens when the tile shape is unusual.

A theoretical anchor: the Smoluchowski equation

The classical starting point for thinking about aggregation kinetics is Smoluchowski’s coagulation equation (Smoluchowski, 1916). If $n_k(t)$ is the number density of clusters of size $k$ at time $t$, and $K_{ij}$ is the rate at which clusters of size $i$ and $j$ meet and merge, then

\[\frac{\partial n_k}{\partial t} \;=\; \tfrac{1}{2}\sum_{i+j=k} K_{ij}\,n_i\,n_j \;-\; n_k \sum_{j=1}^{\infty} K_{kj}\,n_j .\]

The first term creates clusters of size $k$ by merging smaller ones; the second removes them by merging with anything else. In FloaTiles, where agitation (vibration, airflow) breaks clusters apart, the natural extension is the aggregation–fragmentation version, with an additional fragmentation kernel $F_{ij}$:

\[\frac{\partial n_k}{\partial t} \;=\; \tfrac{1}{2}\!\!\sum_{i+j=k}\!\! K_{ij}\,n_i\,n_j \;-\; n_k\!\sum_{j\ge 1}\! K_{kj}\,n_j \;-\; n_k \,\Gamma_k \;+\!\!\sum_{i+j=k}\!\! F_{ij}\,n_{i+j} ,\]

where $\Gamma_k = \sum_{i+j=k} F_{ij}$ is the total fragmentation rate of clusters of size $k$. The shape of $K_{ij}$ and $F_{ij}$ encodes the physics: the geometry of the tiles, the strength of the capillary bond, and the level of agitation. These coarse-grained rates are exactly what FloaTiles, viewed as an aggregation-kinetics experiment, can be used to probe.

Apparatus

A shallow rectangular pool sits on a slab supported by a modular perforated-steel-strut frame. A small DC motor with an eccentric mass is mounted underneath and couples vibration into the slab and the water; a household speed controller (JDT-001) lets us adjust amplitude and frequency. Optional small fans and an aquarium bubble generator provide alternative ways of adding noise. A USB webcam above the pool records the experiment.

The frame and pool are described by a parametric OpenSCAD model. The model is a conceptual rendering made after the table was already built from available materials — it is close to the real apparatus but is not a literal blueprint of this particular table.

Source: apparatus.scad. Open it in OpenSCAD to inspect, modify, or render the geometry.

🖱️ Open the 3D viewer in its own page (Three.js, ~2 MB STL)

Tiles

Poster (CCS 2023)

Cheerios Effect: Floating objects on a liquid surface come together due to surface tension and buoyancy. This phenomenon can be harnessed to control and manipulate the assembly of specific structures. The light reflection visible in our images effectively demonstrates the deformation of the water surface around the floating objects — this deformation results in the force that pulls objects together, allowing for controlled aggregation of structures.

Aperiodic tiling: covering a whole plane without a repeating pattern using a single tile type. The 2023 discovery of the Hat and Spectre offers unique geometric shapes that can be incorporated into the experiment. Tiles can be created using 3D printing or laser cutting and are introduced to a water surface to see how they aggregate based on the Cheerios effect.

The FloaTiles project combines these two ideas to demonstrate how individual tiles interact through the Cheerios effect to produce emergent behaviour. The system can produce more complex patterns by adding tiles and stationary elements (engines) that have specific effects like attraction and repulsion, and patterns can affect the level of perturbation in real time via a video-feedback loop from the camera.

Full poster (PDF)

References

  1. Vella, D. & Mahadevan, L. (2005). The “Cheerios effect.” American Journal of Physics 73(9), 817–825. doi:10.1119/1.1898523
  2. Gifford, W. A. & Scriven, L. E. (1971). On the attraction of floating particles. Chemical Engineering Science 26(3), 287–297. doi:10.1016/0009-2509(71)83003-8
  3. Hosokawa, K., Shimoyama, I. & Miura, H. (1994). Dynamics of self-assembling systems: analogy with chemical kinetics. Artificial Life 1(4), 413–427. doi:10.1162/artl.1994.1.4.413
  4. Hosokawa, K., Shimoyama, I. & Miura, H. (1996). Two-dimensional micro-self-assembly using the surface tension of water. Sensors and Actuators A 57(2), 117–125. doi:10.1016/S0924-4247(97)80102-1
  5. Bowden, N., Terfort, A., Carbeck, J. & Whitesides, G. M. (1997). Self-assembly of mesoscale objects into ordered two-dimensional arrays. Science 276, 233–235. doi:10.1126/science.276.5310.233
  6. Bowden, N., Choi, I. S., Grzybowski, B. A. & Whitesides, G. M. (1999). Mesoscale self-assembly of hexagonal plates using lateral capillary forces. JACS 121(23), 5373–5391. doi:10.1021/ja983882z
  7. Whitesides, G. M. & Grzybowski, B. (2002). Self-assembly at all scales. Science 295, 2418–2421. doi:10.1126/science.1070821
  8. Smith, D., Myers, J. S., Kaplan, C. S. & Goodman-Strauss, C. (2024). An aperiodic monotile. Combinatorial Theory 4(1). doi:10.5070/C64163843
  9. Smith, D., Myers, J. S., Kaplan, C. S. & Goodman-Strauss, C. (2023). A chiral aperiodic monotile. arXiv:2305.17743
  10. Hooshanginejad, A. et al. (2024). Interactions and pattern formation in a macroscopic magnetocapillary SALR system of mermaid cereal. Nature Communications 15, 5466. doi:10.1038/s41467-024-49754-4
  11. Wilt, J. K., Schramma, N., Bottermans, J.-W. & Jalaal, M. (2024). ActiveCheerios: 3D-printed Marangoni-driven active particles at an interface. arXiv:2411.16011
  12. Eatson, J. L., Morgan, S. O., Horozov, T. S. & Buzza, D. M. A. (2024). Programmable 2D materials through shape-controlled capillary forces. PNAS 121(35). doi:10.1073/pnas.2401134121
  13. Smoluchowski, M. (1916). Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physikalische Zeitschrift 17, 557–585.
  14. Brilliantov, N. V. et al. (2018). Steady oscillations in aggregation–fragmentation processes. Physical Review E 98, 012109. doi:10.1103/PhysRevE.98.012109
  15. Witten, T. A. & Sander, L. M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon. Physical Review Letters 47(19), 1400–1403. doi:10.1103/PhysRevLett.47.1400
  16. Meakin, P. (1984). The effects of rotational diffusion on the fractal dimensionality of structures formed by cluster–cluster aggregation. J. Chem. Phys. 81(10), 4637–4639. doi:10.1063/1.447398
  17. Karelin, G. (2024). Floatiles: Self-Assembly Based On Cheerios Effect and Aperiodic Monotiles. ALIFE 2024: Proceedings of the 2024 Artificial Life Conference, MIT Press.
  18. Vassileva, N. D., van den Ende, D., Mugele, F. & Mellema, J. (2005). Capillary forces between spherical particles floating at a liquid–liquid interface. Langmuir 21(24), 11190–11200. doi:10.1021/la051186o
  19. Ginot, F., Theurkauff, I., Detcheverry, F., Ybert, C. & Cottin-Bizonne, C. (2018). Aggregation–fragmentation and individual dynamics of active clusters. Nature Communications 9, 696. doi:10.1038/s41467-017-02625-7
  20. Miyashita, S., Nagy, Z., Nelson, B. J. & Pfeifer, R. (2009). The influence of shape on parallel self-assembly. Entropy 11(4), 643–666. doi:10.3390/e11040643
  21. Ko, H., Hadgu, M., Komilian, K. & Hu, D. L. (2022). Small fire ant rafts are unstable. Physical Review Fluids 7(9), 090501. doi:10.1103/PhysRevFluids.7.090501
  22. Haghighat, B., Droz, E. & Martinoli, A. (2015). Lily: a miniature floating robotic platform for programmable stochastic self-assembly. IEEE ICRA 2015. doi:10.1109/ICRA.2015.7139452
  23. Sayama, H. (2025). Swarm systems as a platform for open-ended evolutionary dynamics. Phil. Trans. R. Soc. A 383(2289), 20240143. doi:10.1098/rsta.2024.0143
  24. Zhao, L., Jiang, Y., She, C.-Y., Li, A. Q., Chen, M. & Balkcom, D. (2026). SoftRafts: floating and adaptive soft modular robots. npj Robotics 4, 8. doi:10.1038/s44182-025-00070-z
  25. Voigt, J. et al. (2025). An aperiodic chiral tiling by topological molecular self-assembly. Nature Communications 16, 83. doi:10.1038/s41467-024-55405-5
  26. Lattuada, M. (2012). Predictive model for diffusion-limited aggregation kinetics of nanocolloids under high concentration. J. Phys. Chem. B 116(1), 120–129. doi:10.1021/jp2097839

The full BibTeX file for these references is available in the repository as references.bib.

Acknowledgments

This project would not have been possible without the help of the OIST Embodied Cognitive Science Unit (ECSU): Stephen Estelle for 3D printing and laser cutting, Brian Morrissey for the vibrational platform, and Tom Froese for supervision and support. Roman Mukhin kindly helped with the first 3D print. The first pilot experiments were carried out during an educational visit to the Takashi Ikegami Laboratory at the University of Tokyo, with the help of johnsmith.

Tile-design files originally drew on https://github.com/christianp/aperiodic-monotile and Craig Kaplan’s spectre resource at https://cs.uwaterloo.ca/~csk/spectre/. QR code generated with https://stablediffusion.fr.

This web page was assembled with the help of Anthropic’s Claude (Claude Code), working from drafts and source material provided by the author. All scientific content, decisions, and any remaining errors are the author’s own.

Background reading that the FloaTiles project sits next to. Most of these are general overview pages — useful starting points if you want to dig into the physics, mathematics, or artificial-life context.

Physics and self-organisation

Aggregation and pattern formation

Self-assembly as a manufacturing approach

Aperiodic order

Artificial life context

Institutional

Contributing & feedback

If you’re interested in extending the experiment, reproducing the apparatus, or have observations of your own, please open an issue or pull request.