Research ArticleMATHEMATICS

Additive lattice kirigami

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Science Advances  23 Sep 2016:
Vol. 2, no. 9, e1601258
DOI: 10.1126/sciadv.1601258


  • Fig. 1 Basic cut nomenclature.

    (A) A kirigami cut with both climb and glide on a hexagonal lattice Λ and its dual (with basis) Embedded Image. The two points along which the perimeter is pinched closed are marked with stars. We break with the earlier nomenclature by Castle et al. (6) that assumed a locally nondefected lattice and which would describe the left disclination as 2-4 and the right as Embedded Image-Embedded Image. Instead, we name them ±1 and Embedded Image. Both systems are used in this article. The Burgers vector is Embedded Image (or Embedded Image), and the glide and climb components of the displacement vector ℓ are ℓ and ℓ, oriented with respect to the Burgers vector. (B) A truncated tetrahedron, flat and assembled, built from a triangular plateau with sloping sidewalls.

  • Fig. 2 General moves.

    (A to D) Simple kirigami cuts with zero climb and glide and their effects on the lattice. The cuts are drawn partially open for emphasis. Large angle deficits and excesses result in severe lattice defects. Each motif, composed of two curvature dipoles and with negative poles coinciding, has net zero curvature. Points with an angle defect are marked with a star. (A) An excised diamond results in two pentagons and an octagon. (B) Setting angles α1 = α2 = π is equivalent to just cutting a straight slit and pinching it closed sideways. Closing a slit by pinching the middle creates a large angle excess (2π) in the center of the scar. (C) Angle α1 or α2 can be greater than π (in this example, α1 = 2π/3 and α2 = 4π/3). (D) The angle α2 = 5π/3 causes a severe lattice defect. (E to H) Kirigami slits with nonzero glide on a hexagonal lattice. Each slit is of length 3, running along lines between lattice locations of the dual lattice Embedded Image and hence forming defects on the dual lattice. Each pinch point, which identifies with no other point under the kirigami action, is indicated by a star. The large gray numbers indicate the change in the number of sides of the hexagons and hence the angle defect or excess. The small numbers label edges for clarity. Compared with zero-glide configurations, the curvature is diluted, reducing the magnitude of the lattice defects. (E to H) A series of zigzag motifs with varying angles on the lattice (left), zoomed in (center), and the cut-and-rejoined final configuration (right). Motif (G), which minimally changes the magnitude of the induced Gaussian curvature, is modeled in paper in (H). (I to K) Composition of kirigami motifs. (I) A repeated series of simple z cuts (bottom) is equivalent to a longer zigzagging cut (top), both of which form a pair of Embedded Image-Embedded Image dislocations on a hexagonal lattice or more generally a pair of Embedded Image dislocations. These dislocation pairs are necessarily at the angle specified by this geometric construction as other angles require the addition or removal of material. The arrows show the relative movement of the hexagons to each other, as the wedges are removed and reinserted. (J) A slit pattern can be “closed up” by pinching the points marked with stars, forming points and shifting the Gaussian curvature to the tips of the slits. (K) Combined with the natural folds along the lattice (between the centers and edges of the cuts), the slit pattern becomes a lattice of tetrahedra.

  • Fig. 3 A compendium of possible kirigami cutting angles on the hexagonal/triangular lattice.

    The gray arrows indicate Burgers vectors for different vector pairs. a and b show various cutting lines from one basis point of the lattice Λ to another, separated by length. The blue cuts of a are shorter, and the green cuts of b are longer. Cuts bisect the lattice edges, but not at right angles, producing angle defects not generally equal to multiples of π/3. c and d show excision motifs compatible with vertical sidewalls. c shows wedges based on the dual lattice Embedded Image both aligned and misaligned with the lattice directions. d shows a wedge whose endpoints include both Λ and Embedded Image.

  • Fig. 4 Kirigami used to describe growth processes in living cells.

    (A) A sequence of cell splitting operations (interpreted as three sequential pictures) in a crab scale to form a ridge [adapted from the study of Rivier et al. (13)]. (B to D) Kirigami interpretations of the final configuration: Orange lines represent the excised wedges and boundary of the dislocations, and orange arrows show the Burgers vectors. The shaded areas show removed material, whereas the blue hatching indicates added material. Stars show an angle defect site, circles are identified together to provide a corresponding angle excess, and cells are color-coded as a visual aid. (B) Two semi-infinite dislocations give the same cell configuration by removing material in a region extending to the boundary. (C) A straight cut is made and material is inserted. (D) The most formally useful interpretation considers a negative climb adding material (blue hatching) canceling at the ends with the (shaded) material removed from the wedges.

  • Fig. 5 Collapsing an excision to become a puckered slit.

    (A) Shrinking an excision to identify adjacent edges using subtractive kirigami as indicated by the stars and Burgers vector edges. (B) Void spaces with odd perimeters (lattice locations in black) can be removed by temporarily halving the lattice dimensions (new locations have white centers). The small spurs thus generated are later united with other small spurs, and the half-dimension lattice can be omitted. (C to H) An example of a complex kirigami cut being the composition of simpler kirigami actions. The sequence from (C) to (H) shown by double arrows indicates the pinch-and-slide algorithm described in the text, whereas the single arrow sequence indicates that the same result can be achieved by the composition of subtractive and additive kirigami. (C) The bent slit is pinched closed at two antipodal points around its perimeter, marked with orange and blue stars. The Burgers vectors required to close the dislocations of just the two edges around each star are marked; they are misaligned and do not cancel. (D) A strip of triangles is inserted using additive kirigami, as illustrated in Fig. 4. (E) With this new configuration, the Burgers vectors of the two pinch points and their neighboring edges do cancel, using traditional subtractive kirigami with wedge angles π and positive climb and glide. (F) Starting with the “pinch” part of the algorithmic procedure, the pinching action in Fig. 2C is applied around the blue star along the part of the slit marked in green. (G) As expected, this forms an unwanted spur that is “slid” along the cut by the same method. In this example, a second pinch at the orange star would have exactly the same effect. (H) Final lattice rearrangement. (I) Excising a hexagon from a hexagonal lattice and replacing it with a twist to create a ring of Embedded Image and Embedded Image curvature sites. In this case, most Embedded Image and Embedded Image sites cancel, leaving just two Embedded Image pairs arranged in a stepped configuration with nonvertical sidewalls. Two sides of the hexagon are marked with orange stars and lines to convey the twisted geometry.

  • Fig. 6 Complex cuts.

    (A to D) A mixed slit-and-hole pattern can be closed up by pinching the points marked with stars, which gives them positive Gaussian curvature and shifts the negative Gaussian curvature to other locations. (A) The design with edge identifications marked around one perimeter is as follows: a goes to a′, b goes to b′, etc. (B) Flat prefolded model. (C) Assembly progress. (D) The final structure is a mix of triangular and square pyramids arranged in a tiling. (E to H) Excising an array of hexagons from a hexagonal lattice and folding the remaining material into equilateral triangles to form an array of octahedra. (E) The cut (left) and precreased (right) layer with base triangles colored orange. (F) The corresponding array of octahedra, arranged together in a layer to form the standard space-filling octahedra-tetrahedra packing, with the tetrahedra being the “empty space” left over by the octahedra. (G) Every third octahedron going down instead of up results in interlocking layers. The image shows a single layer in various stages of assembly (top) and matching adjacent layers (bottom). (H) Tightly interlocked layers.

  • Fig. 7 Kirigami with dislocation excisions replaced by folds.

    (A) A pure climb dislocation and end wedges folded down their middles instead of excised. Stars indicate pinch points. (B) A pure glide dislocation involves a simple cut, with the end wedges again folded. This motif is repeated later in other contexts. (C) A mixed climb and glide configuration requires a more complex cut-and-fold pattern, which will induce a “snap-through” transition in suitable materials. (D) The flat but prefolded configuration. (E) Half-assembled (above). (F) Half-assembled (below). The diagonal bending of the squares provides an energy barrier to create a snap-through effect. (G) The final structure, held in place by the squares’ resistance to bending.

  • Fig. 8 Details of the tiling.

    (A) A tiling with both primitive vectors of the centered rectangular lattice shown. (B to D) A square array of square antiprisms formed by excising every second square from the tiling. (B) Top view. (C) Bottom view. (D) The loose arrangement of antiprisms in the plane allows the sheet to bend a certain angle out of the plane, until the octahedra touch. (E) A cutting pattern to make a layer of cuboctahedra. The cuts impose chirality on the tiling. (F) A layer of cuboctahedra is rigid, especially if touching squares are stuck together, because there are no out-of-plane deformations (to linear order). The chiral cutting pattern once again becomes a chiral scar pattern on the achiral layer. (G) The pinch motif shown in Figs. 2E and 7B applied along the slit allows the triangles to fold on themselves, reducing the cuboctahedron exterior to its component squares, that is, a cube.

  • Fig. 9 Further generalizations.

    (A) Example sheets containing both protruding (or indented) cubes and tetrahedra, each of which could interlock with a complementary sheet. (B) Sheets of tetrahedra can be rolled into tetrahedrally decorated tubes. The tetrahedra can face outward, inward, or a mixture of the two. (C) Candidate interlocking tube examples. (D) The junction of three cube-decorated tubes with cut edges unglued.

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