# Growing Isotropic Neural Cellular Automata

arXiv: https://arxiv.org/abs/2205.01681 · HTML: https://mlsanity.com/paper/2205.01681/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 17 references that are in the map.
- First posted on arXiv 3 May 2022.

## Builds on (most cited first, 2 of 2)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [DiffusionNet: Discretization Agnostic Learning on Surfaces](https://mlsanity.com/paper/2012.00888/) (2020)

## Built on by (most cited first, 19 of 19)

- [E(n)-equivariant Graph Neural Cellular Automata](https://mlsanity.com/paper/2301.10497/) (2023)
- [Biomaker CA: a Biome Maker project using Cellular Automata](https://mlsanity.com/paper/2307.09320/) (2023)
- [Learning spatio-temporal patterns with Neural Cellular Automata](https://mlsanity.com/paper/2310.14809/) (2023)
- [EngramNCA: a Neural Cellular Automaton Model of Memory Transfer](https://mlsanity.com/paper/2504.11855/) (2025)
- [Growing Steerable Neural Cellular Automata](https://mlsanity.com/paper/2302.10197/) (2023)
- [Heuristically Adaptive Diffusion-Model Evolutionary Strategy](https://mlsanity.com/paper/2411.13420/) (2024)
- [Neural Cellular Automata Can Respond to Signals](https://mlsanity.com/paper/2305.12971/) (2023)
- [Neural Cellular Automata for Lightweight, Robust and Explainable Classification of White Blood Cell Images](https://mlsanity.com/paper/2404.05584/) (2024)
- [Identity Increases Stability in Neural Cellular Automata](https://mlsanity.com/paper/2508.06389/) (2025)
- [Neural Particle Automata: Learning Self-Organizing Particle Dynamics](https://mlsanity.com/paper/2601.16096/) (2026)
- [EINCASM: Emergent Intelligence in Neural Cellular Automaton Slime Molds](https://mlsanity.com/paper/2305.13425/) (2023)
- [Multi-Texture Synthesis through Signal Responsive Neural Cellular Automata](https://mlsanity.com/paper/2407.05991/) (2024)
- [Remapping and navigation of an embedding space via error minimization: a fundamental organizational principle of cognition in natural and artificial systems](https://mlsanity.com/paper/2601.14096/) (2026)
- [Coralai: Intrinsic Evolution of Embodied Neural Cellular Automata Ecosystems](https://mlsanity.com/paper/2406.09654/) (2024)
- [Missing Data Imputation using Neural Cellular Automata](https://mlsanity.com/paper/2509.00651/) (2025)
- [LOGOS-CA: A Cellular Automaton Using Natural Language as State and Rule](https://mlsanity.com/paper/2602.00036/) (2026)
- [Communication Heterogeneity and Collective Consensus in Neural Cellular Automata](https://mlsanity.com/paper/2606.21202/) (2026)
- [Applications of Neural Cellular Automata: State of the Art, Challenges and Opportunities](https://mlsanity.com/paper/2609.24595/) (2026)
- [Where Does Randomness Matter in Neural Cellular Automata?](https://mlsanity.com/paper/2609.36797/) (2026)

## Abstract (arXiv)

Modeling the ability of multicellular organisms to build and maintain their bodies through local interactions between individual cells (morphogenesis) is a long-standing challenge of developmental biology. Recently, the Neural Cellular Automata (NCA) model was proposed as a way to find local system rules that produce a desired global behaviour, such as growing and persisting a predefined target pattern, by repeatedly applying the same rule over a grid starting from a single cell. In this work, we argue that the original Growing NCA model has an important limitation: anisotropy of the learned update rule. This implies the presence of an external factor that orients the cells in a particular direction. In other words, "physical" rules of the underlying system are not invariant to rotation, thus prohibiting the existence of differently oriented instances of the target pattern on the same grid. We propose a modified Isotropic NCA (IsoNCA) model that does not have this limitation. We demonstrate that such cell systems can be trained to grow accurate asymmetrical patterns through either of two methods: (1) by breaking symmetries using structured seeds or (2) by introducing a rotation-reflection invariant training objective and relying on symmetry-breaking caused by asynchronous cell updates.