A unified geometric design framework for kirigami structures

Qinghai Jiang; Gary P. T. Choi

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A unified geometric design framework for kirigami structures

Qinghai Jiang, Gary P. T. Choi

arXiv:2608.30032v1Aug 30, 2026cond-mat.softcond-mat.mtrl-scics.CGphysics.app-ph

TL;DR

Existing kirigami design methods apply to limited classes of structures, motivating a unified approach for broader shape morphing and additional geometric properties. The paper formulates design as length-based constrained optimization over multiple states, demonstrates diverse 2D and 3D designs, and identifies theoretical design limitations including curvature-related angle defects.

Problem
Existing kirigami design methods are difficult to apply and generalize to other scenarios, limiting the range of structures they can design.
Method
The framework optimizes vertex coordinates across contracted and deployed states using edge-length constraints, angle constraints, and optional geometric-property constraints.
Results
The framework produces diverse shape-morphing designs and supports compact reconfigurability and rigid deployability across 2D and 3D settings.
Takeaways & Limitations
The theoretical analysis provides design rules and limitations involving inertia transposition, aspect ratio, and angle defects.
Takeaways & Limitations
A compact rectangular kirigami pattern cannot reconfigure into an arbitrary positively curved surface while keeping all vertices exactly on the target geometry.

Abstract

from arXiv · show

In recent years, kirigami metamaterials have been widely studied and applied in science and engineering. While various two- and three-dimensional kirigami design methods have been developed, most of them are only applicable to a limited class of kirigami structures. In this work, we develop a unified framework for kirigami design that encompasses a wide range of 2D-to-2D, 2D-to-3D, and 3D-to-3D shape-morphing effects, as well as additional geometric and physical properties such as compact reconfigurability and rigid deployability. In particular, by reformulating the design task as a length-based constrained optimization problem and solving it simultaneously for multiple target states of the kirigami structure, our unified design framework enables greater design flexibility and stronger theoretical support. Experimental results with a wide range of shape-morphing effects are presented to demonstrate the effectiveness of our framework. We further present a rigorous theoretical analysis of several key aspects of kirigami design, covering inertia transposition, aspect-ratio law, and angle defects, thereby elucidating important design rules and limitations. Altogether, our work paves a new way for the design of shape-morphing mechanical metamaterials.

I. INTRODUCTION

The paper addresses the limited generality of existing kirigami design methods by introducing a unified framework for 2D and 3D shape morphing. It formulates design as length-based constrained optimization over multiple configurations while supporting additional properties such as compact reconfigurability and rigid deployability.

Motivation: Existing kirigami design methods are difficult to apply and generalize beyond limited classes of structures.
Unified framework: The framework designs 2D and 3D kirigami structures with prescribed target shapes across multiple states.
Optimization formulation: The design variables are vertex coordinates in contracted and deployed configurations, with edge-length equalities enforcing tile correspondence and isometry.
Optimization formulation: Angle inequalities supplement edge-length constraints to avoid undesirable deployed configurations such as tile overlap.
Optimization procedure: The framework uses an interior-point method to solve an optimization problem whose constraints encode geometric and physical design requirements.
Additional properties: Compact reconfigurability is enabled by dual geometric constraints that enforce equal edge lengths in negative spaces, while redundant constraints must be removed as the problem grows.

C. Rigid deployability

The framework supports rigid deployment and flexible target-shape matching by combining geometric constraints with objectives defined on vertex coordinates. Its extensions cover 2D-to-3D and 3D-to-3D morphing, but rigid deployment requires more than straight slits alone.

Rigid deployability: Rigid deployability requires equal edge lengths around each negative space and straight slits in the contracted state.
Rigid deployability: The framework imposes slit straightness through angle inequalities, which are preferred to hard collinearity equalities in the interior-point formulation.
Rigid deployability: A straight slit is necessary but insufficient for global rigid deployment unless the rhombus negative-space constraint is also enforced.
Shape matching: Target-shape matching can be encoded either through equality constraints or through an objective based on distances to the target shape.
3D extensions: The same coordinate-based formulation extends naturally to 3D, including 2D-to-3D designs with surface matching and compact reconfigurability.
3D extensions: 3D-to-3D designs can exploit target symmetry by assembling compatible 2D-to-3D patches with additional boundary constraints.

F. Implementation

The framework is implemented as a constrained optimization system and demonstrates shape morphing across 2D-to-2D, 2D-to-3D, and 3D-to-3D structures. Its examples also expose theoretical flexibility and limitations, including compact reconfiguration and inertia transposition.

Implementation: The framework is implemented in MATLAB using an interior-point method with IPOPT to solve its constrained optimization problem.The supplementary material covers objective and constraint formulations, Jacobians, KKT degeneracy, error analysis, and practical solution setups.
2D-to-2D structures: The framework designs 2D-to-2D kirigami transformations, including circle-to-rainbow, rainbow-to-square, and rigidly deployable compact reconfigurable patterns.A TPU 3D-printed square-to-circle model successfully underwent the desired transformation.
2D-to-3D structures: The framework produces 2D-to-3D structures that approximate complex targets such as human faces, Max Planck surfaces, and half-vase geometries under different contracted and deployed conditions.The optimized face design uses elongated tiles near the nose bridge, while initial cutting angles control alternative 2D/3D state combinations.
2D-to-3D structures: 2D-to-3D design has theoretical limits: a compact 2D rectangle cannot morph into a compact 3D vase with every vertex lying on the target geometry.The restriction follows from the target shape and desired kirigami properties rather than from the optimization implementation.
3D-to-3D structures: The framework supports 3D-to-3D transformations such as cube-to-sphere, square-ring-to-torus, and spherical-cap-to-saddle morphing.Symmetry, partitioning, and local 2D-to-3D reductions simplify some designs, while direct optimization over 3D vertex coordinates handles other cases.
Theoretical analysis: For compact reconfigurable patterns, the moment aspect ratio of the two compact states is inverted, with the law characterized through inertia transposition.The analysis establishes the result for general compact states and connects it to centroid-variance balance under a vanishing-diameter hypothesis.

B. The aspect-ratio law for rectangular kirigami patterns

For compact rectangular kirigami patterns, the target shape’s inertia ratio determines the rectangle’s aspect ratio through a reciprocal moment law. Numerical designs across seven target shapes support the predicted relation and yield a practical initialization rule.

Aspect-ratio law: The rectangle’s moment aspect ratio is HC/WC, and the reciprocal law gives rin(RM) → 1/rin(Ω).This follows by combining the rectangle’s central moments with the general inertia-transposition theorem.
Numerical verification: For seven target shapes and resolutions M × M from 8 to 24, the compact aspect ratio converges rapidly and agrees closely with the target inertia ratio.The numerical limits are highly consistent with the theoretical prediction.
Design rule: With area preservation WCHC = A, the target inertia ratio fixes the compact rectangle’s dimensions.The law determines the rectangle from the target without reference to individual tiles.
Design rule: A tall target with large rin requires a wide compact rectangular initialization, whereas a low-rin target requires the opposite.The prescribed relation reverses the target’s inertia ratio when selecting the compact rectangle.
Scope: The inertia-transposition analysis also applies when the target is prescribed in the deployed state with open voids.Moments are computed over the union of tiles; voids contribute nothing directly and affect the result through surrounding tile centroids.

C. The discrete transposition of the ideal mechanism

The ideal kirigami mechanism transposes compact-state geometry: row and column spacings exchange roles, producing exact discrete identities and a limiting aspect-ratio relation. The result depends on an exact rectangular initial state, checkerboard ±π/2 tile rotations, and controlled fluctuation terms for nonsquare grids.

Figure 6: The figure relates compact aspect ratio to pattern resolution and target inertia, while illustrating centroid-link projections used in the discrete-transposition proof.The fitted continuum limit c∞ is compared with rin, and the rectangle-to-egg example visualizes row and column links.
Discrete formulation: The lattice formulation uses vertex-mean tile centers and measures mean vertical and horizontal projections of row and column links.These centers depend only on tile vertices, providing a discrete geometry independent of tile shape.
Theorem assumptions: The discrete transposition theorem assumes an exact M × N rectangle and checkerboard tile rotations of exactly ±π/2.Under these conditions, every tile row telescopes to WC and every column to HC.
Spacing identities: The reconfigured lattice’s dominant spacings equal the initial compact dimensions divided by the grid counts.The row and column spacing identities are derived from centroid-link projections and layer cancellations.
Square grids: For square resolution M = N, the unweighted variance ratio VX/VY approaches (HC/WC)^2, restoring a discrete form of transposition.The row-mean levels form an arithmetic progression, enabling the exact variance calculation.
Nonsquare grids: For M ≠ N, fluctuation corrections can remain order one because the Ex prefactor is amplified by (M/N)^4.In the 20 × 6 wavy pattern, the fluctuation ratio is approximately 0.94 and explains the apparent gap between the nominal aspect-ratio estimate and rin.

D. Angle defects and limitations on target shape design

Angle defects impose a fundamental boundary-shape limitation when kirigami structures target general curved surfaces. In particular, positively curved compact targets cannot generally retain rectangular planar boundaries.

Angle defects encode Gaussian curvature in piecewise-flat kirigami surfaces, creating a mismatch with planar compact states.Interior angle defects are positive on positively curved targets, whereas planar compact states have zero interior deficit.
For a sphere patch, positive interior defects prevent both boundary-angle pairs from remaining close to π as on a rectangle.The constraint follows by balancing angle sums in a 2N × 2 quad strip with N interior vertices on the sphere patch.
A compact rectangular kirigami pattern cannot reconfigure into an arbitrary positively curved surface with every vertex exactly on the target geometry.The half-vase example illustrates this theoretical limitation.
A compact-cube-to-compact-sphere transformation is likewise impossible because it would require six square-to-spherical-cap compact designs.Each cube side would need a compact reconfigurable design that violates the angle-defect limitation.

V. DISCUSSION

Prior kirigami design approaches have been limited in their generalizability across structures and shape-morphing problems.

Many prior kirigami design approaches are case-specific and have limited generalizability.

Appendix A: Constrained optimization formulation

The appendix formulates kirigami design as a sparse constrained optimization problem over compact and deployed vertex coordinates. Its constraints enforce isometry, valid tile geometry, optional rigid deployability, and planarity while omitting a redundant diagonal constraint.

1. Residual and Jacobian: Vertex coordinates in both initial and deployed configurations form the optimization variables, with sparse Jacobians assembled for an interior-point solver.Each constraint involves only a small fixed number of vertices.
1. Residual and Jacobian: Edge-length equality constraints match corresponding compact and deployed edges, with Jacobian rows derived from endpoint differences.The listed nonzero derivatives use the coordinate differences along each corresponding edge.
1. Residual and Jacobian: Smoothed length differences replace squared-length residuals to keep the gradient bounded.The Jacobian entries are divided by the corresponding smoothed length.
1. Residual and Jacobian: Signed-area inequalities enforce positive cutting orientation and interior angles strictly between 0 and π.The orientation condition uses rarea ≥ ρ > 0, while the angle bounds are applied to all four corner triplets.
1. Residual and Jacobian: Slit collinearity is imposed for rigid deployability, while 3D tile planarity is enforced by setting the tetrahedral signed volume to zero.Together with five edge-length equalities, the planarity residual guarantees each tile is isometric and planar.
2. Possible degeneracy in KKT: The formulation explicitly omits the redundant diagonal constraint and relies on angle inequalities to select the correct isometric branch.The dependency is nonlinear in the variables and therefore escapes standard constant-coefficient presolvers.
2. Possible degeneracy in KKT: Including all six quadrilateral edge constraints would cause severe numerical instability because the sixth Jacobian row is dependent on the first five.This dependency holds around any feasible point.

3. Error analysis for the optimization scheme

The error analysis bounds discrepancies caused by omitting the second diagonal constraint under well-scaled, nondegenerate conditions. Near-collapsed or highly distorted quadrilaterals weaken these guarantees.

The omitted diagonal error is analyzed because five enforced constraints are solved only to tolerance ε.The analysis asks how this tolerance propagates to the omitted diagonal and actual edge lengths.
Squared-length residuals generally control actual length error, but near-zero edges can degrade the bound to O(ε^1/2).Angle inequalities prevent zero-area collapse when the problem is not over-constrained, but thin quads weaken the guarantee.
Extreme area changes or poor initial guesses can produce thin quadrilaterals and weaken the length-error bound.Rescaling target geometries to O(1) edge lengths mitigates this risk in practice.
The tessellation construction relates compact dimensions to tile aspect ratio and generates slit-offset variants through linear parameterization.At θ = 0, h/w = (N/M)r; with deployment angle ϕ, the standard tessellation uses θ = π − ϕ.
For uniform quadrilaterals, the local amplification factor Cq is bounded at the target geometry.
The omitted-diagonal residual satisfies |e6| ≤ Cq ε/(d2 + ˜d2) under the stated nondegeneracy conditions.The bound follows from applying the mean-value theorem to the diagonal expression F.
For ε = 10^-6 to 10^-8 with lc + ld = O(1) and Cq = O(1), both actual edge-length error and omitted-diagonal discrepancy are O(ε).The guidance assumes a well-scaled problem and recommends monitoring both discrepancies as diagnostics.

4. Basic tessellation settings

The framework uses a rotating-rectangle tessellation whose geometry, initialization, boundary correspondence, and resolution are chosen to support target-shape optimization. These choices jointly affect deformation quality and computational efficiency.

Tessellation geometry: The standard tessellation is an M × N array of congruent rectangular tiles with width b, height a, and aspect ratio r := a/b.Deployment rotates each tile about its center by θ, with adjacent tiles rotating oppositely to open four-sided negative spaces.
Initialization: The framework optimizes vertex coordinates in both contracted and deployed states, requiring initial guesses for each state.Initial guesses may use standard configurations, rescaled patterns, conformal or quasi-conformal maps, or other locally injective mappings.
Initialization: For an egg-shaped target, an initial rectangle with a larger cutting angle generally produces more uniform deformation between compact and deployed patterns.Two initialization choices are described: a taller rectangle with cutting angle below π/2, or a wider rectangle with cutting angle above π/2.
Resolution selection: For a wavy target, changing resolution from 20 × 20 to 20 × 6 improves tile regularity because the finite-grid law prescribes a much larger compact width-to-height ratio.The predicted ratios are WC/HC ≈0.675 for 20 × 20 and WC/HC ≈7.07 for 20 × 6.
Boundary correspondence: Splitting a target boundary into pieces can improve symmetry and efficiency, but may exclude potentially better solutions.Equal, uneven, or single-curve splitting strategies can yield substantially different optimized designs.

Appendix B: Additional results

Additional experiments demonstrate that the framework supports diverse 2D, 2D-to-3D, and 3D-to-3D morphing effects, including compact reconfigurability and rigid deployability. The examples also show flexibility across resolutions and target shapes.

2D designs: The framework produces 2D shape changes including circle-to-rainbow and rainbow-to-rectangle morphing, as well as compact reconfigurable rainbow and face targets.These examples use independently chosen initial guesses for contracted and deployed states.
Rigid deployability: The framework designs rigidly deployable structures with non-rhombus parallelogram negative spaces whose contracted slits are straight lines.Each negative space has four distinct points rather than the three points associated with rhombus negative spaces.
2D-to-3D and assembly: 2D-to-3D designs morph compact planar shapes into spherical caps and partial tori that can be assembled into larger surfaces.The construction uses surface and boundary matching, duplication, and reflection procedures.
3D-to-3D reconfiguration: A spherical-cap compact state can be reconfigured into saddle, catenoid, vase, or partial-torus targets in 3D-to-3D designs.The examples use the same target shape for the first compact state but different second compact-state targets.
Extreme changes: The experiments demonstrate flexibility in achieving extreme shape and size changes across square-to-circle, rectangle-to-circle, and varied inertia-ratio targets.Free-to-circle, free-to-egg, and free-to-rainbow patterns are also analyzed through centroid-variance quantities and a cross-state residual.
Resolution variation: 96, 216, 384, and 864 facets all produce cube-to-sphere deployable structures satisfying the desired shape-morphing effect.The four resolutions are 16 × 6, 36 × 6, 64 × 6, and 144 × 6 facets, respectively.

2. The balanced-fluctuation condition for the rectangular kirigami patterns

The balanced-fluctuation analysis decomposes rectangular-pattern behavior into kinematic and mismatch contributions. Numerical examples examine centroid variances, transposition relations, and residuals across compact reconfigurable designs.

Geometric quantities: The analysis separates vertex-average levels, area centroids, row mismatches, and variance contributions rather than treating them as interchangeable quantities.The distinction is needed because area-centroid substitution can alter the arithmetic-progression relation.
Transposition relation: The discrete transposition theorem gives an arithmetic progression for vertex-average row levels, with Yj+1 − Yj = WC/M.This progression underlies the rectangular-pattern variance decomposition.
Numerical evaluation: Free-to-circle, free-to-egg, and free-to-rainbow patterns are evaluated using area-weighted centroid variances and the balanced-fluctuation residual.Both compact states are rescaled to unit total area, making the reported quantities scale-invariant.
Numerical evaluation: The balanced-fluctuation residual is of order 10^-4 to 10^-3 under the stated resolutions, while transposition-ratio products equal 1 when the two ratios coincide.The residual measures the H3 imbalance between the two compact states.

C M 2 Varp(J) + 2WC

The finite-grid correction decomposes the rectangular-pattern residual into mass-profile, cross, row-mismatch, and within-row terms. A 20 × 6 example shows how aspect-ratio and fluctuation corrections explain the apparent discrepancy between finite-grid predictions and target inertia.

Residual decomposition: The residual decomposition contains mass-profile, cross, row-mismatch, and within-row terms, each representing a distinct contribution to Ey/W^2_C.The row-mismatch term is small but necessary for exact closure of the identity.
Residual closure: The row-mismatch term is approximately 10^-5 in the reported experiments, while direct and decomposed values agree to numerical precision when all terms are retained.Replacing vertex averages with area centroids would leave a closure residual of approximately 10^-5.
Finite-grid example: For the 20 × 6 wavy pattern, WC/HC = 7.0717, (WC/HC)(N/M)^2 = 0.6365, rin = 0.675, and Ey/VY = +0.109.The example also reports Ex/VX = −0.0075 and a reconfigured-moment inertia ratio of 0.6650.
Finite-grid correction: The finite-grid geometric factor and fluctuation correction account for the apparent gap between (WC/HC)(N/M)^2 and rin.The fluctuation correction is 0.9461, and the target-versus-reconfigured inertia difference reflects finite-grid approximation.
Limitation: When M ≫ N, the balanced-fluctuation assumption is not met and the closure residual grows to approximately 10^-3.This marks a limitation of applying the balanced-fluctuation approximation to strongly non-square grids.