---
title: "Gliders on Aperiodic Monotilings: Cellular Automata on the Hat and Spectre"
authors:
  - "James Edmond"
year: 2026
venue: "arXiv:nlin.CG"
arxiv: "2609.22579"
doi: null
source: "https://arxiv.org/abs/2609.22579"
topics:
  - "curiosity"
  - "cellular-automata"
  - "aperiodic-tilings"
  - "evolutionary-search"
seed_rank: 1843
seed_batch: "curiosity-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 10
lineage: aperiodic-monotiles
cites:
  - title: "Gliders on Aperiodic Monotilings: Cellular Automata on the Hat and Spectre"
    url: "https://arxiv.org/abs/2609.22579"
    year: 2026
    arxiv: "2609.22579"
    doi: null
see:
  - "1023-an-aperiodic-monotile"
  - "1778-chasing-rotational-symmetry-in-the-spectre-tiling"
  - "1370-matching-rules-as-cocycle-conditions-discrete-potentials-on-penrose"
  - "1807-probabilistic-cellular-automata-for-granular-media-in-video-games"
---

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

## One-sentence takeaway

First cellular-automaton dynamics on the hat and spectre monotiles: simple edge-neighbour rules give only mortal travellers, but widening the rule space lets evolutionary search find true gliders that fly a million rings at constant speed on a six-spoke compass.

## Why it's lovely

Why you might love this: it is Game of Life meets the 2023 einstein tile, done like a reproducible engineering project. Patches are generated by finite-state transducers so every experiment regenerates from a tiny record, the glider is tracked by a sliding window that regenerates the tiling along its flight, and every result replays live in an interactive essay. The surprise is the compass: on a tiling with no translational symmetry, glider headings still snap (to millidegrees) onto six directions, the fast axes of the tiling's graph metric. Lovely next to the monotile (1023), spectre (1778) and Penrose matching-rule (1370) cards, and a nudge for anyone running CA sims on irregular grids (1807).

## Key ideas

- Hat/spectre patches generated deterministically by finite-state transducers; experiments are records, not snapshots.
- Edge-adjacency semi-totalistic rules: exhaustive plus evolutionary search finds only "mortal travellers", no gliders.
- Guided by reproducing the known Penrose-tiling glider, the rule space is extended to vertex neighbourhoods and priority-table rules whose non-quiescent states are visible to neighbours.
- Evolutionary search then finds gliders on both monotilings; tracked over one million rings at constant speed and heading.
- Headings quantize onto a six-spoke compass (the graph metric's fast axes); an ablation shows both rule-space extensions are individually necessary.

## Caveats

Computational/empirical study by a single author; gliders are found by search, not classified or proved to exist in a given rule family. The million-ring flights are tracked inside a regenerated window, so the claim is about the generated tilings the transducers produce.

## Links

- arXiv abs: https://arxiv.org/abs/2609.22579
- PDF: https://arxiv.org/pdf/2609.22579
- HTML: https://arxiv.org/html/2609.22579
- Code and records (Zenodo): https://doi.org/10.5281/zenodo.22836247