Gliders on Aperiodic Monotilings: Cellular Automata on the Hat and Spectre

• Published in 2026
• Added on 2026-10-14

The hat and spectre monotiles, discovered in 2023, tile the plane only aperiodically; no cellular automaton dynamics on these tilings has previously been reported. Cellular automata are studied here on patches generated by finite-state transducers, so that every experiment regenerates deterministically from a small record. Within edge-adjacency semi-totalistic rules, exhaustive and evolutionary searches find only mortal travelers: gliders are absent. Guided by a reproduction of the known Penrose-tiling glider, the rule space is extended to vertex neighborhoods and to priority-table rules whose non-quiescent states are visible to neighbors. Evolutionary search then discovers gliders on both monotilings; tracked by a sliding window that regenerates the patch along the flight, they travel one million rings at constant speed and heading. All headings are quantized, to millidegrees, onto a six-spoke compass - the fast axes of the tiling's graph metric. An ablation shows both rule-space extensions are individually necessary. All results replay exactly in an accompanying interactive essay.

Links

• https://arxiv.org/abs/2609.22579v1
• https://arxiv.org/pdf/2609.22579v1

Other information

BibTeX entry

@article{GlidersonAperiodicMonotilingsCellularAutomataontheHatandSpectre,
	key = {GlidersonAperiodicMonotilingsCellularAutomataontheHatandSpectre},
	type = {article},
	title = {Gliders on Aperiodic Monotilings: Cellular Automata on the Hat and Spectre},
	author = {James Edmond},
	abstract = {The hat and spectre monotiles, discovered in 2023, tile the plane only aperiodically; no cellular automaton dynamics on these tilings has previously been reported. Cellular automata are studied here on patches generated by finite-state transducers, so that every experiment regenerates deterministically from a small record. Within edge-adjacency semi-totalistic rules, exhaustive and evolutionary searches find only mortal travelers: gliders are absent. Guided by a reproduction of the known Penrose-tiling glider, the rule space is extended to vertex neighborhoods and to priority-table rules whose non-quiescent states are visible to neighbors. Evolutionary search then discovers gliders on both monotilings; tracked by a sliding window that regenerates the patch along the flight, they travel one million rings at constant speed and heading. All headings are quantized, to millidegrees, onto a six-spoke compass - the fast axes of the tiling's graph metric. An ablation shows both rule-space extensions are individually necessary. All results replay exactly in an accompanying interactive essay.},
	comment = {},
	date_added = {2026-10-14},
	date_published = {2026-10-14},
	urls = {https://arxiv.org/abs/2609.22579v1,https://arxiv.org/pdf/2609.22579v1},
	collections = {easily-explained,geometry},
	url = {https://arxiv.org/abs/2609.22579v1 https://arxiv.org/pdf/2609.22579v1},
	year = 2026,
	urldate = {2026-10-14},
	archivePrefix = {arXiv},
	eprint = {2609.22579},
	primaryClass = {nlin.CG}
}