Exploring Ancient Architectural Designs with Cellular Automata

The paper discusses the utilization of three-dimensional cellular automata employing the two-dimensional totalistic cellular automata to simulate how simple rules could emerge a highly complex architectural designs of some Indonesian heritages. A detailed discussion is brought to see the simple rules applied in Borobudur Temple, the largest ancient Buddhist temple in the country with very complex detailed designs within. The simulation confirms some previous findings related to measurement of the temple as well as some other ancient buildings in Indonesia. This happens to open further exploitation of the explanatory power presented by cellular automata for complex architectural designs built by civilization not having any supporting sophisticated tools, even standard measurement systems.

💡 Research Summary

The paper investigates how the intricate architecture of ancient Indonesian temples, especially the world‑famous Borobudur, could have emerged from very simple generative principles. The author adopts a three‑dimensional cellular automaton (CA) built on a two‑dimensional totalistic CA with nine‑cell neighborhoods. Each cell can be in one of two states (0 or 1) and the update rule is defined by the sum of the states of the nine neighboring cells. Out of the 2¹⁰ = 1024 possible totalistic rules, the study focuses on rule C = 816, whose binary representation is 1100110000. This rule can be expressed as: a new cube is added in the next layer whenever the underlying cell has 4, 5, 8, or all 9 neighboring cells occupied in the previous layer.

The initial condition for the simulation is a digitised plan of Borobudur’s ground level. At each iteration the CA generates a new horizontal layer, but the horizontal extent of each layer is constrained to the size of the original ground plan, preventing unrealistic lateral expansion. The evolution proceeds through discrete time steps (t = 0, 2, 4, 6, 8), producing a stacked three‑dimensional structure.

The resulting shape matches several key empirical observations about Borobudur. First, the vertical cross‑section exhibits the 4 : 6 : 9 proportional relationship previously reported by Atmadi (1998) and confirmed by Situngkir (2010). Second, the overall geometry displays a fractal dimension of approximately 2.325, consistent with earlier fractal analyses of the monument. The author further applies the same rule to other Indonesian heritage sites—Prambanan, Sukuh, Mendut, and Cangkuang—and finds comparable structural similarities, suggesting that a single totalistic rule can capture a family of temple forms.

In the discussion, the paper situates these findings within Stephen Wolfram’s classification of cellular automata. The ancient temple patterns correspond to Class I behavior (convergence to a fixed point) rather than the chaotic Class III or the complex Class IV dynamics often observed in two‑dimensional CA. This classification aligns with the notion that ancient builders produced stable, ordered structures rather than designs that oscillate or exhibit unpredictable growth.

The conclusions emphasize that cellular automata provide a powerful explanatory framework for understanding how societies lacking sophisticated measurement tools could still achieve highly ordered, large‑scale architecture. Simple local interaction rules can give rise to global self‑similarity, proportional ratios, and fractal characteristics observed in the archaeological record. The study acknowledges limitations: dependence on the chosen initial condition, omission of physical constraints such as material strength or labor logistics, and the fact that rule C = 816 is not proven to be unique. Future work is suggested to explore combinations of multiple rules, integrate physics‑based constraints, and examine whether similar generative mechanisms can inform contemporary generative architecture.