pith. sign in

arxiv: 2604.19586 · v1 · submitted 2026-04-21 · ❄️ cond-mat.soft · cond-mat.mtrl-sci· cs.CG

Monotile kirigami

Hugo Hiu Chak Cheng , Gary P. T. Choi This is my paper

Pith reviewed 2026-05-10 01:24 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.mtrl-scics.CG
keywords kirigamimonotilemetamaterialstilingwallpaper groupsquasicrystalspolykitesshape morphing
0
0 comments X

The pith

Monotile patterns can form the basis for deployable kirigami metamaterials across all periodic symmetries and many aperiodic ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that kirigami structures, created by cutting a single sheet to allow deployment through expansion, can be designed from monotiles—single shapes that tile the plane by themselves. Explicit constructions are given for periodic cases that realize every one of the 17 wallpaper groups and for aperiodic cases that produce quasicrystal patterns and polykite tilings. Theoretical and computational studies track how these patterns change in shape and size when deployed. The approach reduces the building blocks needed for shape-morphing metamaterials to the simplest tiling units. This broadens the possible designs for materials that transform under stretching or other stimuli.

Core claim

We prove the existence of periodic and aperiodic monotile kirigami structures via explicit constructions. In particular, we present a comprehensive collection of periodic monotile kirigami structures covering all 17 wallpaper groups and aperiodic monotile kirigami structures covering various quasicrystal patterns as well as polykite tilings. We further perform theoretical and computational analyses of monotile kirigami patterns in terms of their shape and size changes under deployment.

What carries the argument

Monotile kirigami pattern, a deployable cut-sheet structure derived from a single tile that tiles the plane periodically or aperiodically, which carries the deployment motion without needing multiple distinct tile types.

Load-bearing premise

The explicit constructions can be physically realized as deployable kirigami without violating material constraints or requiring additional cuts that break the monotile property.

What would settle it

A physical prototype built from one of the described monotile patterns that either fails to deploy as a single connected sheet or requires extra cuts to function, breaking the monotile condition.

Figures

Figures reproduced from arXiv: 2604.19586 by Gary P. T. Choi, Hugo Hiu Chak Cheng.

Figure 1
Figure 1. Figure 1: Existence of deployable periodic monotile kirigami patterns for all 17 wallpaper groups. Here, every red dot represents a rotation center, every red dashed line represents a reflectional axis, and every blue dashed line represents a glide reflectional axis. We then consider the cases with 4-fold rotational symmetry (p4m, p4g, p4). For the p4m case (with mirrors at 45◦ ), it is easy to see that the standard… view at source ↗
Figure 2
Figure 2. Figure 2: Aperiodic monotile kirigami patterns constructed using quasicrystal tilings. (a) For the 5-fold Penrose tiling, by removing the thinner rhombi (highlighted in grey) as described in [20], we can achieve deployable monotile kirigami patterns with different resolutions. (b) For the 8-fold Ammann–Beenker tiling, by removing the square tiles (highlighted in grey) as described in [20], we can achieve deployable … view at source ↗
Figure 3
Figure 3. Figure 3: Aperiodic monotile kirigami patterns constructed using polykite monotiles. (a) An aperiodic tiling with the “Hat” monotile generated using the first level of the H7 substitution rule [28]. The red lines show the lattice representation of the kirigami structure if we consider adding a connection between all tiles that share an edge. The existence of triangles in the lattice representation indicates that we … view at source ↗
Figure 4
Figure 4. Figure 4: Additional examples for gain, loss, and preservation of symmetry in periodic monotile kirigami structures. (a) A pg ↔ pmg pattern. (b) A p1 ↔ p3m1 pattern. (c) A p1 ↔ p4m pattern. (d) A p1 ↔ p6m pattern. (e) A pmm ↔ p6m pattern. (f) A p3 ↔ p6m pattern. (g) A p4 ↔ p4m pattern. (h) A p6 ↔ p6m pattern. (i) A p3m1 ↔ cm pattern. Here, every red dot represents a rotation center, every red dashed line represents … view at source ↗
Figure 5
Figure 5. Figure 5: Analyzing the relationship between the tile geometry and the size change of aperiodic polykite monotile kirigami patterns. Here, we plot log(r) (where r is the size change ratio (SCR) or perimeter change ratio (PCR)) against log(b/a) for different choices of tile geometry parameters (a, b). The best-fit least-squares straight lines are also displayed. target outcomes. More broadly, the monotile kirigami pa… view at source ↗
read the original abstract

Kirigami, the art of paper cutting, has been widely used in the modern design of mechanical metamaterials. In recent years, many kirigami-based metamaterials have been designed based on different planar tiling patterns and applied to different science and engineering problems. However, it is natural to ask whether one can create deployable kirigami structures based on the simplest forms of tilings, namely the monotile patterns. In this work, we answer this question by proving the existence of periodic and aperiodic monotile kirigami structures via explicit constructions. In particular, we present a comprehensive collection of periodic monotile kirigami structures covering all 17 wallpaper groups and aperiodic monotile kirigami structures covering various quasicrystal patterns as well as polykite tilings. We further perform theoretical and computational analyses of monotile kirigami patterns in terms of their shape and size changes under deployment. Altogether, our work paves a new way for the design and analysis of a wider range of shape-morphing metamaterials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper claims to prove the existence of periodic monotile kirigami structures covering all 17 wallpaper groups and aperiodic versions covering quasicrystal patterns and polykite tilings, achieved via explicit constructions. It further provides theoretical and computational analyses of shape and size changes under deployment, positioning these as a basis for new shape-morphing metamaterials.

Significance. If the explicit constructions are valid and the deployment analyses hold, the work establishes that the simplest (monotile) tilings suffice for kirigami metamaterials with full symmetry coverage and aperiodic variants. This is a notable expansion of the design space beyond multi-tile patterns, with potential for broader applications in deployable materials; the mathematical/computational nature of the existence proof is a strength.

major comments (1)
  1. The central existence claim rests on the explicit constructions; however, without detailed verification in the main text (e.g., via enumerated tile placements or symmetry checks for each wallpaper group), it is difficult to confirm that no auxiliary cuts or tile variants are inadvertently introduced in the periodic cases.
minor comments (3)
  1. The abstract states the coverage of all 17 wallpaper groups but does not indicate where in the manuscript the explicit constructions for each are presented or tabulated.
  2. Computational deployment analysis is mentioned but lacks reference to specific methods, software, or boundary conditions used for the shape/size change calculations.
  3. Figure captions and legends should explicitly label which wallpaper group or quasicrystal pattern each panel corresponds to for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The central existence claim rests on the explicit constructions; however, without detailed verification in the main text (e.g., via enumerated tile placements or symmetry checks for each wallpaper group), it is difficult to confirm that no auxiliary cuts or tile variants are inadvertently introduced in the periodic cases.

    Authors: We thank the referee for this constructive observation. Our explicit constructions are generated from a single monotile shape per pattern whose boundaries define the kirigami cuts exactly, with no auxiliary cuts or additional tile variants; the periodicity and symmetry follow directly from the standard wallpaper-group lattice. To remove any ambiguity and allow direct verification without consulting the supplementary material, we will add a concise table (or short subsection) in the revised main text that enumerates the unit-cell tile placements and lists the symmetry operations confirming each of the 17 wallpaper groups. This addition will be accompanied by a brief statement that the constructions remain strictly monotile. revision: yes

Circularity Check

0 steps flagged

No significant circularity; existence via explicit constructions

full rationale

The paper establishes its claims by presenting explicit constructions of periodic monotile kirigami patterns for all 17 wallpaper groups and aperiodic versions for quasicrystals and polykites. These are geometric designs shown directly, followed by independent theoretical and computational analyses of shape/size changes under deployment. No equations, fitted parameters, self-definitional loops, or load-bearing self-citations appear in the provided abstract or described structure; the result does not reduce to its inputs by construction and remains self-contained as a catalog of patterns with deployment analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that monotile patterns can be adapted to kirigami cuts while preserving deployability and tiling properties; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Monotiles exist and admit kirigami modifications that allow deployment.
    The abstract presupposes that recent monotile discoveries can be directly translated into cut patterns without loss of the monotile property.

pith-pipeline@v0.9.0 · 5486 in / 1133 out tokens · 26691 ms · 2026-05-10T01:24:48.262301+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Mechanical metamaterials based on origami and kirigami,

    Z. Zhai, L. Wu, and H. Jiang, “Mechanical metamaterials based on origami and kirigami,” Applied Physics Reviews, vol. 8, no. 4, 2021

    work page 2021

  2. [2]

    Mechanical metamaterials and beyond,

    P. Jiao, J. Mueller, J. R. Raney, X. Zheng, and A. H. Alavi, “Mechanical metamaterials and beyond,”Nature Communications, vol. 14, no. 1, p. 6004, 2023. 19

    work page 2023

  3. [3]

    Engineering kirigami frameworks toward real-world applications,

    L. Jin and S. Yang, “Engineering kirigami frameworks toward real-world applications,” Advanced Materials, vol. 36, no. 9, p. 2308560, 2024

    work page 2024

  4. [4]

    Computational design of art-inspired metamaterials,

    G. P. T. Choi, “Computational design of art-inspired metamaterials,”Nature Computa- tional Science, vol. 4, no. 8, pp. 549–552, 2024

    work page 2024

  5. [5]

    Shape-morphing metamaterials,

    K. K. Dudek, M. Kadic, C. Coulais, and K. Bertoldi, “Shape-morphing metamaterials,” Nature Reviews Materials, vol. 10, pp. 783–798, 2025

    work page 2025

  6. [6]

    Graphene kirigami,

    M. K. Blees, A. W. Barnard, P. A. Rose, S. P. Roberts, K. L. McGill, P. Y. Huang, A. R. Ruyack, J. W. Kevek, B. Kobrin, D. A. Muller,et al., “Graphene kirigami,”Nature, vol. 524, no. 7564, p. 204, 2015

    work page 2015

  7. [7]

    Kirigami skins make a simple soft actuator crawl,

    A. Rafsanjani, Y. Zhang, B. Liu, S. M. Rubinstein, and K. Bertoldi, “Kirigami skins make a simple soft actuator crawl,”Sci. Robot., vol. 3, no. 15, p. eaar7555, 2018

    work page 2018

  8. [8]

    Physics-aware differentiable design of magnetically actuated kirigami for shape morphing,

    L. Wang, Y. Chang, S. Wu, R. R. Zhao, and W. Chen, “Physics-aware differentiable design of magnetically actuated kirigami for shape morphing,”Nature Communications, vol. 14, no. 1, p. 8516, 2023

    work page 2023

  9. [9]

    A new class of trans- formable kirigami metamaterials for reconfigurable electromagnetic systems,

    Y. Yang, A. Vallecchi, E. Shamonina, C. J. Stevens, and Z. You, “A new class of trans- formable kirigami metamaterials for reconfigurable electromagnetic systems,”Scientific Reports, vol. 13, no. 1, p. 1219, 2023

    work page 2023

  10. [10]

    Gr¨ unbaum and G

    B. Gr¨ unbaum and G. C. Shephard,Tilings and patterns. WH Freeman & Co., 1986

    work page 1986

  11. [11]

    Auxetic behavior from rotating triangles,

    J. N. Grima and K. E. Evans, “Auxetic behavior from rotating triangles,”Journal of Materials Science, vol. 41, no. 10, pp. 3193–3196, 2006

    work page 2006

  12. [12]

    Auxetic behavior from rotating squares,

    J. N. Grima and K. E. Evans, “Auxetic behavior from rotating squares,”Journal of Materials Science Letters, vol. 19, no. 17, pp. 1563–1565, 2000

    work page 2000

  13. [13]

    Negative poisson’s ratios from rotating rectangles,

    J. N. Grima, A. Alderson, and K. E. Evans, “Negative poisson’s ratios from rotating rectangles,”Computational Methods in Science and Technology, vol. 10, no. 2, pp. 137– 145, 2004

    work page 2004

  14. [14]

    Auxetic behaviour from rotating rhombi,

    D. Attard and J. N. Grima, “Auxetic behaviour from rotating rhombi,”Physica Status Solidi B, vol. 245, no. 11, pp. 2395–2404, 2008

    work page 2008

  15. [15]

    Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs,

    A. Rafsanjani and D. Pasini, “Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs,”Extreme Mechanics Letters, vol. 9, pp. 291–296, 2016

    work page 2016

  16. [16]

    Auxetic behaviour from connected different- sized squares and rectangles,

    J. N. Grima, E. Manicaro, and D. Attard, “Auxetic behaviour from connected different- sized squares and rectangles,”Proceedings of the Royal Society A, vol. 467, no. 2126, pp. 439–458, 2011

    work page 2011

  17. [17]

    Geometrical elaboration of auxetic structures,

    M. Stavric and A. Wiltsche, “Geometrical elaboration of auxetic structures,”Nexus Network Journal, vol. 21, no. 1, pp. 79–90, 2019. 20

    work page 2019

  18. [18]

    Wallpaper group kirigami,

    L. Liu, G. P. T. Choi, and L. Mahadevan, “Wallpaper group kirigami,”Proceedings of the Royal Society A, vol. 477, no. 2252, p. 20210161, 2021

    work page 2021

  19. [19]

    Auxetic dihedral escher tessellations,

    X. Liu, L. Lu, L. Cao, O. Deussen, and C. Tu, “Auxetic dihedral escher tessellations,” Graphical Models, vol. 133, p. 101215, 2024

    work page 2024

  20. [20]

    Quasicrystal kirigami,

    L. Liu, G. P. T. Choi, and L. Mahadevan, “Quasicrystal kirigami,”Physical Review Research, vol. 4, no. 3, p. 033114, 2022

    work page 2022

  21. [21]

    Programming shape using kirigami tessellations,

    G. P. T. Choi, L. H. Dudte, and L. Mahadevan, “Programming shape using kirigami tessellations,”Nature Materials, vol. 18, no. 9, pp. 999–1004, 2019

    work page 2019

  22. [22]

    Compact reconfigurable kirigami,

    G. P. T. Choi, L. H. Dudte, and L. Mahadevan, “Compact reconfigurable kirigami,” Physical Review Research, vol. 3, no. 4, p. 043030, 2021

    work page 2021

  23. [23]

    An additive framework for kirigami design,

    L. H. Dudte, G. P. T. Choi, K. P. Becker, and L. Mahadevan, “An additive framework for kirigami design,”Nature Computational Science, vol. 3, no. 5, pp. 443–454, 2023

    work page 2023

  24. [24]

    Excess floppy modes and multibranched mechanisms in metamaterials with symmetries,

    L. A. Lubbers and M. van Hecke, “Excess floppy modes and multibranched mechanisms in metamaterials with symmetries,”Physical Review E, vol. 100, no. 2, p. 021001, 2019

    work page 2019

  25. [25]

    Programmable hierarchical kirigami,

    N. An, A. G. Domel, J. Zhou, A. Rafsanjani, and K. Bertoldi, “Programmable hierarchical kirigami,”Advanced Functional Materials, vol. 30, no. 6, p. 1906711, 2020

    work page 2020

  26. [26]

    Deterministic and stochastic control of kirigami topology,

    S. Chen, G. P. T. Choi, and L. Mahadevan, “Deterministic and stochastic control of kirigami topology,”Proceedings of the National Academy of Sciences, vol. 117, no. 9, pp. 4511–4517, 2020

    work page 2020

  27. [27]

    Explosive rigidity percolation in kirigami,

    G. P. T. Choi, L. Liu, and L. Mahadevan, “Explosive rigidity percolation in kirigami,” Proceedings of the Royal Society A, vol. 479, no. 2271, p. 20220798, 2023

    work page 2023

  28. [28]

    An aperiodic monotile,

    D. Smith, J. S. Myers, C. S. Kaplan, and C. Goodman-Strauss, “An aperiodic monotile,” Combinatorial Theory, vol. 4, no. 1, p. 6, 2024

    work page 2024

  29. [29]

    A chiral aperiodic monotile,

    D. Smith, J. S. Myers, C. S. Kaplan, and C. Goodman-Strauss, “A chiral aperiodic monotile,”Combinatorial Theory, vol. 4, no. 2, p. 13, 2024

    work page 2024

  30. [30]

    An isotropic zero poisson’s ratio metamaterial based on the aperiodic ‘hat’monotile,

    D. J. Clarke, F. Carter, I. Jowers, and R. J. Moat, “An isotropic zero poisson’s ratio metamaterial based on the aperiodic ‘hat’monotile,”Applied Materials Today, vol. 35, p. 101959, 2023

    work page 2023

  31. [31]

    Aperiodicity is all you need: Aperiodic monotiles for high-performance composites,

    J. Jung, A. Chen, and G. X. Gu, “Aperiodicity is all you need: Aperiodic monotiles for high-performance composites,”Materials Today, vol. 73, pp. 1–8, 2024

    work page 2024

  32. [32]

    Effective elastic properties of novel aperiodic monotile- based lattice metamaterials,

    M. M. Naji and R. K. A. Al-Rub, “Effective elastic properties of novel aperiodic monotile- based lattice metamaterials,”Materials & Design, vol. 244, p. 113102, 2024. 21

    work page 2024

  33. [33]

    Physical properties of an aperiodic monotile with graphene-like features, chirality, and zero modes,

    J. Schirmann, S. Franca, F. Flicker, and A. G. Grushin, “Physical properties of an aperiodic monotile with graphene-like features, chirality, and zero modes,”Physical Review Letters, vol. 132, no. 8, p. 086402, 2024

    work page 2024

  34. [34]

    Reconfigurable modular origami for tunable 2D symmetry groups,

    W. Liu, Q. Ren, Y. Wang, Z. Zhang, X. Wang, Y. Liang, H. Huang, J. Ma, H. Jiang, and Y. Chen, “Reconfigurable modular origami for tunable 2D symmetry groups,”Science Advances, vol. 11, no. 46, p. eady3812, 2025

    work page 2025

  35. [35]

    Control of connectivity and rigidity in prismatic assemblies,

    G. P. T. Choi, S. Chen, and L. Mahadevan, “Control of connectivity and rigidity in prismatic assemblies,”Proceedings of the Royal Society A, vol. 476, no. 2244, p. 20200485, 2020

    work page 2020

  36. [36]

    Designing metamaterials with programmable nonlinear responses and geometric constraints in graph space,

    M. Maurizi, D. Xu, Y.-T. Wang, D. Yao, D. Hahn, M. Oudich, A. Satpati, M. Bauchy, W. Wang, Y. Sun,et al., “Designing metamaterials with programmable nonlinear responses and geometric constraints in graph space,”Nature Machine Intelligence, vol. 7, no. 7, pp. 1023–1036, 2025. 22

    work page 2025