← No. 013 MON 12 OCT 2026

One shape tiles forever without repeating.

Aperiodic monotile

In 2023, four researchers showed a single 13‑sided shape, nicknamed the hat, can cover the plane endlessly yet never fall into a repeating wallpaper. Their proof allows mirror‑image copies. Weeks later they introduced a cousin, the spectre, that forces the same non‑repeating behavior without using mirrors. Both results, first posted as preprints, have since been peer‑reviewed. A decades‑long hunt for a lone non‑repeating tile was finally closed.

Why it’s interesting

Fig. 013 Thirteen tiles in the cluster echo the hat’s 13 sides; the black outline marks a chiral spectre.

Why it’s interesting

A single outline can dictate endless, never‑repeating order. It also shows how letting in—or banning—mirror images quietly changes what symmetry can do.

The longer version

Mathematicians have long known how to make endlessly varied tilings with sets of shapes, but a single shape that forces non‑repetition had remained missing since the 1960s. On March 20, 2023, David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman‑Strauss unveiled the hat: a 13‑sided polygon that admits tilings of the plane yet forbids periodic ones, via a built‑in hierarchical structure and a geometric incommensurability argument.

On May 28, 2023 the team showed variants that remove the need for mirror images. A weakly chiral relative works only if mirrors are banned, and strictly chiral spectre shapes remain aperiodic even when mirrors are allowed. Both papers were later peer‑reviewed in Combinatorial Theory in 2024.

Edition: No. 013
Date: Monday 12 October 2026
Subject: Aperiodic monotile
Period: 2023
Field: mathematics
Evidence: 4 sources, 2 primary

See how we got here

Found after 33 research paths · Selected from 13 candidates

01 Wallpaper patterns Periodic tilings repeat with shifts.
02 Aperiodic tilings Penrose used a small set, not one.
03 Monotiles One shape to tile them all.
04 The hat 13 sides; non‑repeating with mirrors allowed.
05 Chiral monotiles Spectre tiles aperiodically without mirrors.
→ One shape tiles forever without repeating.

Sources

Nothing publishes without independent sources. How we verify →

1
An aperiodic monotile
arXiv · 2023 · primary document Primary

Defines and proves the ‘hat’ as an aperiodic monotile; establishes no periodic tilings and gives methods of proof.

arxiv.org
2
A chiral aperiodic monotile
arXiv · 2023 · primary document Primary

States that hat mixes reflections; proves a weakly chiral monotile (Tile(1,1)) and strictly chiral Spectre monotiles.

arxiv.org
3
An aperiodic monotile (peer‑reviewed publication)
Combinatorial Theory / eScholarship (DOI) · 2024 · peer reviewed

Peer‑reviewed journal version confirming the hat result.

doi.org
4
A chiral aperiodic monotile (peer‑reviewed publication)
Combinatorial Theory / eScholarship · 2024 · peer reviewed

Peer‑reviewed journal version confirming weakly chiral Tile(1,1) and strictly chiral Spectres.

escholarship.org

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