---
title: "Chasing rotational symmetry in the spectre tiling"
authors:
  - "Marianne Imperor-Clerc"
  - "Jean-Francois Sadoc"
year: 2026
venue: "arXiv:math.MG"
arxiv: "2609.07592"
doi: null
source: "https://arxiv.org/abs/2609.07592"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1778
seed_batch: "curiosity-2026-09-11"
reviewed: "2026-09-11"
pool: "maths-foundations"
relevance_score: 10
lineage: aperiodic-monotile
cites:
  - title: "Chasing rotational symmetry in the spectre tiling"
    url: "https://arxiv.org/abs/2609.07592"
    year: 2026
    arxiv: "2609.07592"
    doi: null
see:
  - "1023-an-aperiodic-monotile"
  - "1024-sturmian-lattices-and-aperiodic-tile-sets"
  - "1370-matching-rules-as-cocycle-conditions-discrete-potentials-on-penrose"
---

# Chasing rotational symmetry in the spectre tiling

## One-sentence takeaway

The spectre monotile's tiling secretly hosts coexisting 2-fold and 3-fold rotational centres; three finite "diabolo / 2-arms / 3-arms" motifs make them visible and constructible.

## Why it's lovely

Why you might love this: after the hat/spectre aperiodic monotile shock, the next delight is finding the hidden rotational choreography inside an aperiodic tiling. Local vs global symmetry, substitution schemes, and three named finite patterns you can actually draw — pure curiosity sparkle with a straight line back to the monotile shelf.

## Key ideas

- Identifies coexisting 2-fold and 3-fold rotational symmetry centres in the spectre tiling.
- Distinguishes local from global symmetry operations.
- Links substitution schemes to rotational symmetry.
- Introduces diabolo, 2-arms, and 3-arms finite patterns as visual witnesses.
- Gives geometric constructions for spectre patches with prescribed global 2-fold or 3-fold symmetry.

## Caveats

Geometry/visualization paper; not a new aperiodicity proof. Complements the hat/spectre cards (1023–1024) rather than replacing them.

## Links

- arXiv abs: https://arxiv.org/abs/2609.07592
- PDF: https://arxiv.org/pdf/2609.07592.pdf