Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Combinatorial design of textured mechanical metamaterials


The structural complexity of metamaterials is limitless, but, in practice, most designs comprise periodic architectures that lead to materials with spatially homogeneous features1,2,3,4,5,6,7,8,9,10,11. More advanced applications in soft robotics, prosthetics and wearable technology involve spatially textured mechanical functionality, which requires aperiodic architectures. However, a naive implementation of such structural complexity invariably leads to geometrical frustration (whereby local constraints cannot be satisfied everywhere), which prevents coherent operation and impedes functionality. Here we introduce a combinatorial strategy for the design of aperiodic, yet frustration-free, mechanical metamaterials that exhibit spatially textured functionalities. We implement this strategy using cubic building blocks—voxels—that deform anisotropically, a local stacking rule that allows cooperative shape changes by guaranteeing that deformed building blocks fit together as in a three-dimensional jigsaw puzzle, and three-dimensional printing. These aperiodic metamaterials exhibit long-range holographic order, whereby the two-dimensional pixelated surface texture dictates the three-dimensional interior voxel arrangement. They also act as programmable shape-shifters, morphing into spatially complex, but predictable and designable, shapes when uniaxially compressed. Finally, their mechanical response to compression by a textured surface reveals their ability to perform sensing and pattern analysis. Combinatorial design thus opens up a new avenue towards mechanical metamaterials with unusual order and machine-like functionalities.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Figure 1: Voxelated mechanical metamaterials.
Figure 2: Combinatorial design.
Figure 3: Pattern recognition and pattern analysis.


  1. Lakes, R. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987)

    Article  ADS  CAS  Google Scholar 

  2. Mullin, T., Deschanel, S., Bertoldi, K. & Boyce, M. C. Pattern transformation triggered by deformation. Phys. Rev. Lett. 99, 084301 (2007)

    Article  ADS  CAS  Google Scholar 

  3. Grima, J. N. & Evans, K. E. Auxetic behavior from rotating squares. J. Mater. Sci. Lett. 19, 1563–1565 (2000)

    Article  CAS  Google Scholar 

  4. Schaedler, T. A. et al. Ultralight metallic microlattices. Science 334, 962–965 (2011)

    Article  ADS  CAS  Google Scholar 

  5. Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608–613 (2012)

    Article  ADS  CAS  Google Scholar 

  6. Babaee, S., Shim, J., Weaver, J. C., Patel, N. & Bertoldi, K. 3D soft metamaterials with negative Poisson’s ratio. Adv. Mater. 25, 5044–5049 (2013)

    Article  CAS  Google Scholar 

  7. Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013)

    Article  ADS  Google Scholar 

  8. Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014)

    Article  ADS  Google Scholar 

  9. Coulais, C., Overvelde, J. T. B., Lubbers, L. A., Bertoldi, K. & van Hecke, M. Discontinuous buckling of wide beams and metabeams. Phys. Rev. Lett. 115, 044301 (2015)

    Article  ADS  Google Scholar 

  10. Schenk, M. & Guest, S. D. Geometry of Miura-folded metamaterials. Proc. Natl Acad. Sci. USA 110, 3276–3281 (2013)

    Article  ADS  CAS  Google Scholar 

  11. Waitukaitis, S., Menaut, R., Chen, B. G. & van Hecke, M. Origami multistability: from single vertices to metasheets. Phys. Rev. Lett. 114, 055503 (2015)

    Article  ADS  Google Scholar 

  12. Silverberg, J. L. et al. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647–650 (2014)

    Article  ADS  CAS  Google Scholar 

  13. Bückmann, T., Thiel, M., Kadic, M., Schittny, R. & Wegener, M. An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat. Commun. 5, 4130 (2014)

    Article  ADS  Google Scholar 

  14. Paulose, J., Meeussen, A. S. & Vitelli, V. Selective buckling via states of self-stress in topological metamaterials. Proc. Natl Acad. Sci. USA 112, 7639–7644 (2015)

    Article  ADS  CAS  Google Scholar 

  15. Liu, Z. et al. Locally resonant sonic materials. Science 289, 1734–1736 (2000)

    Article  ADS  CAS  Google Scholar 

  16. Wannier, G. H. Antiferromagnetism: the triangular Ising net. Phys. Rev. 79, 357–364 (1950)

    Article  ADS  MathSciNet  Google Scholar 

  17. Sadoc, J. F. & Mosseri, R. Geometrical Frustration (Cambridge Univ. Press, 1999)

  18. Zykov, V., Mytilinaios, E., Adams, B. & Lipson, H. Robotics: Self-reproducing machines. Nature 435, 163–164 (2005)

    Article  ADS  CAS  Google Scholar 

  19. Ware, T. H., McConney, M. E., Wie, J. J., Tondiglia, V. P. & White, T. J. Voxelated liquid crystal elastomers. Science 347, 982–984 (2015)

    Article  ADS  CAS  Google Scholar 

  20. Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7 . Phys. Rev. Lett. 79, 2554–2557 (1997)

    Article  ADS  CAS  Google Scholar 

  21. Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006)

    Article  ADS  CAS  Google Scholar 

  22. Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008)

    Article  ADS  CAS  Google Scholar 

  23. Nisoli, C., Moessner, R. & Schiffer, P. Colloquium: Artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013)

    Article  ADS  CAS  Google Scholar 

  24. Grünbaum, B. & Shephard, G. C. Tilings and Patterns (Freeman, 1987)

  25. Kirkpatrick, S., Gelatt, C. D. Jr & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  26. Mezard, M., Parisi, G. & Virasoro, M. A. Spin Glass Theory and Beyond (World Scientific, 1987)

  27. Cho, Y. et al. Engineering the shape and structure of materials by fractal cut. Proc. Natl Acad. Sci. USA 111, 17390–17395 (2014)

    Article  ADS  CAS  Google Scholar 

  28. Leong, T. G. et al. Tetherless thermobiochemically actuated microgrippers. Proc. Natl Acad. Sci. USA 106, 703–708 (2009)

    Article  ADS  CAS  Google Scholar 

  29. Shepherd, R. F. et al. Multigait soft robot. Proc. Natl Acad. Sci. USA 108, 20400–20403 (2011)

    Article  ADS  CAS  Google Scholar 

  30. Overvelde, J. T., Kloek, T., Dhaen, J. J. & Bertoldi, K. Amplifying the response of soft actuators by harnessing snap-through instabilities. Proc. Natl Acad. Sci. USA 112, 10863–10868 (2015)

    Article  ADS  CAS  Google Scholar 

Download references


We are grateful to J. Mesman for technical support. We thank R. Golkov, Y. Kamir, G. Kosa, K. Kuipers, F. Leoni, W. Noorduin and V. Vitelli for discussions. We acknowledge funding from the Netherlands Organisation for Scientific Research through grants VICI No. NWO-680-47-609 (M.v.H. and C.C.) and VENI No. NWO-680-47-445 (C.C.) and from the Israel Science Foundation through grant numbers 617/12 and 1730/12 (E.T and Y.S.).

Author information

Authors and Affiliations



C.C. and M.v.H. conceived the main concepts. C.C., E.T., Y.S. and M.v.H. formulated the spin problem. E.T. and Y.S. solved the spin problem. C.C and K.d.R. performed the experiments and simulations with inputs from E.T., Y.S. and M.v.H. C.C. and M.v.H wrote the manuscript with contributions from all authors.

Corresponding author

Correspondence to Corentin Coulais.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Motif-based design.

a, 2D representation of the six bricks (x, x+, y, y+, z, z+), and illustration of complex motifs. All complex motifs can be generated by defining two binary vectors with elements ci (column) and rj (row) that govern the placement of z bricks at location (i, j). The remaining sites are then filled with x and y bricks. Respecting parity, this generates all motifs for given c and r. b, The six motifs (A+, …, F+) that are compatible with a 5 × 5 smiley texture. c, A total of 65 smiley metacubes can be designed by varying the stacking order; here A denotes the same motif as A+ but with inversed spins. The x and y spins follow from the choice of motifs.

Extended Data Figure 2 Implementation.

a, b, Computer assisted design of the geometry of the unit cell (a) and a 5 × 5 × 5 cube (b). All our samples were 3D printed with the dimensions a = 11.46 mm, D = 10.92 mm and w = 3.6 mm. To make the wall thickness outside the cube equal to the internal wall thickness, the outer walls are thickened by 0.27 mm.

Extended Data Figure 3 Lock-and-key experiment.

a, Picture of the textured clamp. b, Side view of the experiment.

Extended Data Figure 4 10 × 10 × 10 metacube under uniaxial compression.

a, Motif A+—the cube is designed by stacking motifs A+ and A. b, Opposite face of the one shown in Fig. 1e showing the inverted pattern. c, One of the transverse faces showing a checkerboard pattern.

Extended Data Figure 5 Complex sensory properties of a complex 5 × 5 × 5 metacube with internal smiley texture.

a, Force–compression (Fu) curves for five experiments (thick solid lines), where the colour indicates the external texture shown in Fig. 3c. The black thin lines show fits with a quadratic function F(u) = ku + ηu2 performed in the shaded region 0.8 mm ≤ u ≤ 2.5 mm. b, Corresponding numerical results. c, Scatter plot showing very good correspondence between the stiffness obtained from simulations (ks) and experiments (ke).

Extended Data Table 1 The exact value of Ω for L × L × L metacubes up to L = 14

Supplementary information

Supplementary Information

This file contains Supplementary Text and Data, Supplementary Figures 1-14 and Supplementary Table 1. (PDF 679 kb)

5x5x5 metacube, uniaxially compressed along its minor axis by flat clamps.

As discussed in the main text, it exhibits a pattern transformation, where its building blocks suddenly morph in to alternated bricks of elongated and flattened shape. (MP4 1718 kb)

10x10x10 metacube decorated with square pedestals, which is uniaxially compressed along its minor axis by clamps textured in a checker board pattern (seemethods).

As discussed in the main text, its surface texture morphs into an exactingly designed ”smiley” pattern. (MP4 875 kb)

The opposite face of the same 10x10x10 metacube during a similar experiment.

As discussed in the methods, its surface texture morphs into the inverted ”smiley” pattern. (MP4 822 kb)

A side face of the same 10x10x10 metacube during a similar experiment.

As discussed in the methods, its surface texture morphs into a checkerboard pattern. (MP4 824 kb)

PowerPoint slides

Source data

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coulais, C., Teomy, E., de Reus, K. et al. Combinatorial design of textured mechanical metamaterials. Nature 535, 529–532 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing