---
title: "Monotile kirigami"
authors: ["Hugo Hiu Chak Cheng", "Gary P. T. Choi"]
year: 2026
venue: "arXiv:cond-mat.soft"
arxiv: "2604.19586"
doi: null
source: "https://arxiv.org/abs/2604.19586"
topics: [curiosity, ecs-data-oriented-simulation]
seed_rank: 1470
seed_batch: "curiosity-2026-08-31"
reviewed: "2026-08-31"
pool: "maths-foundations"
relevance_score: 10
lineage: aperiodic-tiling
cites:
  - title: "An aperiodic monotile"
    url: "https://arxiv.org/abs/2303.10798"
    year: 2023
    arxiv: "2303.10798"
    doi: "10.5070/C64163843"
  - title: "A chiral aperiodic monotile"
    url: "https://arxiv.org/abs/2305.17743"
    year: 2023
    arxiv: "2305.17743"
    doi: null
see:
  - "1023-an-aperiodic-monotile"
  - "1024-sturmian-lattices-and-aperiodic-tile-sets"
  - "1453-bistable-quad-nets-composed-of-four-bar-linkages"
---

# Monotile kirigami

## One-sentence takeaway

Deployable kirigami from a single tile shape: all 17 wallpaper groups as periodic monotile cuts, plus Hat/Turtle/Tile(a,b) aperiodic polykite kirigami via a removal construction that kills the rigid H7 triangle.

## Why it's lovely

Why you might love this: the Hat (1023) was a tiling theorem. This makes it a mechanism. One laser-cut shape, tape joints, and the wallpaper group of the contracted sheet can gain, lose, or keep rotation/reflection/glide as it opens. Leftover of last Curiosity's origami-DDG-robotics near-keeper, not a remint of 1453's four-bars.

## Problem

Most kirigami metamaterials use several polygons (rotating squares, kagome, Penrose rhombi). After Smith–Myers–Kaplan–Goodman-Strauss, the obvious question is whether a *monotile* still deployes.

## Design

- Periodic: explicit monotile kirigami in every wallpaper group (rotating-squares / kagome topologies plus dented tiles to kill mirrors).
- Aperiodic quasicrystal: removal method on Penrose / Ammann–Beenker / Stampfli so only one tile type remains and the lattice is no longer rigid.
- Polykite: H7 substitution has a triangle in the dual graph, so the reflected central hat (and some type-T tiles at level 2) must be dropped; the rest hinges.
- Size-change ratio and perimeter-change ratio tabulated; for Tile(a,b), SCR ~ (b/a)^0.0436.

## Evidence

Constructive existence for all 17 groups; physical wood models (cm wallpaper and Hat); deployment simulations with a 2D kirigami solver; power-law fit on the Tile(a,b) family.

## Limitations

Planar only; 3D / origami extension is future work. Aperiodic constructions throw tiles away, so the deployed sheet is not a complete Hat tiling. SCR/PCR are convex-hull ratios, not continuum mechanics. Manufacturing claim (one mould) is the point, not a measured factory result.

## Implications for Broadside

Anoptic-adjacent deployable geometry: one-tile kits, wallpaper-programmed auxetics, aperiodic metamaterials that do not need a lattice. Pairs with 1453 (bistable linkages) as the cut-sheet cousin rather than the bar-net.

## Bottom line

Mint the Hat as a hinge. Tiling theory that opens is a different gift from tiling theory that covers.

## Links

- arXiv abs: https://arxiv.org/abs/2604.19586
- PDF: https://arxiv.org/pdf/2604.19586.pdf