Source-linked AI summary
Algorithmic Lattice Kirigami: A Route to Pluripotent Materials
Daniel M. Sussman, Yigil Cho, Toen Castle, Xingting Gong, Euiyeon Jung, Shu Yang, Randall D. Kamien
TL;DR
The paper addresses how to algorithmically fold flat sheets into complex target surfaces without requiring a new intricate pattern for every design. It develops lattice-kirigami methods using disclination pairs and a fixed sixon lattice, showing that local fold reassignment can program many stepped surfaces within a gradient constraint.
Problem
Origami inverse-design patterns can be complex, require specific folding sequences, and generally need a new cut or crease pattern for each target surface.
Method
The paper arranges ˜5-˜7 disclination pairs on a honeycomb super-lattice and uses triangular lattices of sixons as fixed kirigami cut templates whose mountain and valley folds encode target heights.
Results
Triangular sixon lattices can accommodate arbitrary stepped triangulated target structures subject to a gradient constraint, while fold ordering need not be precisely controlled.
Takeaways & Limitations
A single sixon lattice can be reconfigured among many surface shapes by changing fold types, supporting robust and potentially self-assembling kirigami designs.
Takeaways & Limitations
The sixon approach remains constrained by the target-surface gradient, and complete dynamic control over every fold may be difficult to implement.
Abstract
from arXiv · showhide
We use a regular arrangement of kirigami elements to demonstrate an inverse design paradigm for folding a flat surface into complex target configurations. We first present a scheme using arrays of disclination defect pairs on the dual to the honeycomb lattice; by arranging these defect pairs properly with respect to each other and choosing an appropriate fold pattern a target stepped surface can be designed. We then present a more general method that specifies a fixed lattice of kirigami cuts to be performed on a flat sheet. This single "pluripotent" lattice of cuts permits a wide variety of target surfaces to be programmed into the sheet by changing the folding directions.
INTRODUCTION
The paper extends origami-based inverse design with lattice kirigami, targeting complex surfaces while addressing fold-pattern complexity and assembly constraints. It introduces arrangements of defect motifs and a pluripotent sixon lattice for algorithmically designing stepped and reconfigurable surfaces.
- INTRODUCTION: Origami inverse-design methods can match target surfaces, but their intricate crease patterns and prescribed folding sequences hinder generic and self-assembling structures.Some designs lack foldable crease subsets, require repetitive folding sequences, and waste material through hidden folds and pleats.
- INTRODUCTION: Lattice kirigami supplements origami folds with cuts and edge re-gluing, creating localized Gaussian curvature that helps define three-dimensional surfaces.The constructions use honeycomb and dual-lattice motifs, including disclination pairs and sixons.
- INTRODUCTION: Arranged kirigami motifs algorithmically solve inverse design for target surfaces with a specified maximum gradient.The paper focuses on surfaces whose cut-and-folded states match desired stepped configurations.
- INTRODUCTION: The proposed designs are simple enough to design by hand and robust to folding order and relative folding rates.The paper presents both a defect-pair method for stepped surfaces and a sixon-based pluripotent method for many target shapes.
DESIGN USING ˜5-˜7 ELEMENTS
The ˜5-˜7 climb-pair method arranges compatible kirigami elements on a larger honeycomb super-lattice to reverse engineer stepped target surfaces. Its main limitation is that each new target requires a newly programmed cut pattern.
- DESIGN USING ˜5-˜7 ELEMENTS: A ˜5-˜7 climb-pair element has four allowed configurations because P regions can change height independently while shared-edge R regions must remain level.The element is formed by excising an extended hexagon, identifying its edges, and adding perpendicular mountain and valley folds.
- DESIGN USING ˜5-˜7 ELEMENTS: Super-lattice placement enumerates connected building blocks, including Unit n structures and special units where three dislocation paths converge.Special units simplify target design but may be less stable in real materials.
- DESIGN USING ˜5-˜7 ELEMENTS: A reduced fold-and-cut map permits reverse engineering of stepped target surfaces, subject to a junction rule allowing zero or two folding lines at each vertex.The Supporting Information demonstrates the approach with an experimentally realized kirigami ziggurat.
- DESIGN USING ˜5-˜7 ELEMENTS: For every desired target surface, the ˜5-˜7 approach requires programming an entirely new cut pattern before folding.This motivates a single pluripotent blueprint accessed through local fold reassignment.
DEPLOYING SIXONS FOR SURFACE DESIGN
A triangular lattice of sixons provides a fixed kirigami cut pattern whose plateau heights can be changed through local fold assignments. Within a maximum-gradient constraint, the resulting stepped triangulated surfaces can be designed by projecting and rounding target heights.
- DEPLOYING SIXONS FOR SURFACE DESIGN: A triangular sixon lattice supports stepped surfaces whose adjacent triangular plateaus differ in height by exactly one.The ground state consists of height-zero triangular basins, while local fold changes create higher plateaus.
- DEPLOYING SIXONS FOR SURFACE DESIGN: Target surfaces are limited by a maximum gradient determined by the ratio of plateau height to plateau width.Cut sizes can be selected to match the maximum gradient expected among target surfaces.
- DEPLOYING SIXONS FOR SURFACE DESIGN: Inverse design reduces to projecting target heights onto a triangular lattice and rounding each triangle to an even or odd integer so edge-sharing triangles differ by one.The corresponding mountain and valley fold pattern follows immediately from the resulting height map.
CONCLUSION
Kirigami cutting motifs enable algorithmic design of target structures, while a triangular sixon lattice supports many stepped surfaces by changing fold assignments. The designs are robust to folding order and may support dynamically reconfigurable sheets, although further work is needed on gradient and geometric limits.
- CONCLUSION: Triangular lattices of sixons accommodate arbitrary stepped triangulated target structures subject to a maximum-gradient constraint, with different surfaces requiring only changed mountain and valley assignments.The framework uses a fixed underlying triangular lattice; the target surface determines the folding template.
- CONCLUSION: The framework may inform two-dimensional mechanical metamaterials, while current work seeks to remove the gradient limitation and other geometric and topological limits.The paper also notes that square plateaus provide a similar but more restrictive design paradigm.
- CONCLUSION: Kirigami designs are robust to fold ordering and relative folding rates, and their fold-pattern complexity remains constant per unit surface area.An experimental ziggurat realization used heat-shrink tape and did not require detailed control over fold sequencing.
- CONCLUSION: A sixon lattice has more fold lines than degrees of freedom, enabling disjoint fold subsets to encode multiple stimulus-responsive target surfaces.The proposed duopotent example assigns separate fold subsets to a half-cylinder and a Mexican-hat potential.
SUPPORTING INFORMATION
The supporting information covers dynamic control of sixon folds, additional ˜5-˜7 kirigami design details including a self-folding ziggurat, and square-lattice design rules.
- SUPPORTING INFORMATION: The supporting information addresses duopotent sixon sheets, a self-folding ziggurat demonstration, and the behavior of design rules on square lattices.These topics are organized into three supporting-information sections.
PLURIPOTENT SIXONS
A triangular lattice of sixons can be folded into target configurations satisfying a maximum-gradient constraint by reprogramming fold directions. Because complete dynamic control may be difficult, disjoint fold subsets offer a route to multipotent sheets.
- PLURIPOTENT SIXONS: A sixon lattice can represent any target conformation satisfying the maximum-gradient constraint, with fold lines determining the final configuration after the hexagons are excised.Dynamic fold reprogramming could allow transitions between configurations through the flat state.
- PLURIPOTENT SIXONS: Complete dynamic control over every sixon fold may be difficult, but excess fold lines allow multiple disjoint subsets to encode different stimulus-responsive target surfaces.Each fold line can be assigned to one external stimulus in the proposed scheme.
- PLURIPOTENT SIXONS: A duopotent sixon sheet can control one set of folds around upward-facing triangles while complementary folds complete the target configuration as holes close.This provides a robust implementation of separate fold responses within the coupled lattice.
Design rules
The ˜5-˜7 climb-pair method projects target heights onto a honeycomb super-lattice and converts the projection into cuts and folds. Resolution is constrained by one-step height differences and junction rules.
- Design rules: Target-surface heights are projected onto super-lattice faces, which become hexagonal plateaus in a coarse-grained kirigami surface.The projected pattern is then converted into a cut-and-fold design.
- Design rules: Adjacent hexagons may differ by no more than one step, and every hexagon must touch at least one hexagon at a different height.These conditions set the achievable resolution of the design.
- Design rules: Junction rules can impose non-local restrictions, so projected heights may need modification before compatible kirigami units are joined.The ziggurat example demonstrates projection onto the super-lattice followed by conversion into a cut-and-fold pattern.
Experimental realization
The kirigami design was experimentally realized by cutting Tyvek, bonding heat-shrinkable polyolefin beneath the cuts, and using heating to self-fold a ziggurat structure.
- Experimental realization: Tyvek and heat-shrinkable polyolefin were combined to implement the kirigami pattern experimentally.Tyvek was selected for its strength and tear resistance relative to standard paper.
- Experimental realization: The resulting ziggurat kirigami demonstrated autonomous self-assembly of a three-dimensional target structure.
- Experimental realization: Heating the unpressed assembly at 95°C for one minute curled the polyolefin and folded the Tyvek into the target structure.
- Experimental realization: Pre-creasing the folds produced sharper steps than applying polyolefin directly to the fold lines.
SQUARE CUT DESIGN
Square-lattice kirigami uses a tiled cut unit whose folded states produce square plateaus and sidewalls, allowing stepped target surfaces to be assembled from stitched units.
- SQUARE CUT DESIGN: Square-lattice kirigami tiles the plane with a base unit whose colored patches become square plateaus, vertical sidewalls, or excised regions after folding.
- SQUARE CUT DESIGN: Different edge identifications of the excised square yield two symmetrically distinct folded configurations, up to rotations.
- SQUARE CUT DESIGN: Stitching these units together programs stepped target surfaces by projecting suitable nearest odd or even integer heights onto the square lattice.
- SQUARE CUT DESIGN: The allowed folded-state structures are enumerated as six reduced square-lattice building blocks, with some related by rotations.