growth will buckle out of plane into a 3D shape, a principle widely exploited in morphogenetic processes such as the formation of wavy leaves [7] and the development of algae caps. [8] In the limit of infinitesimal thickness, Gauss's theorema egregium directly connects the spatial distribution of in-plane growth (or "metric") to the Gaussian curvature K of the equilibrium 3D shape, although there are in general multiple embeddings that satisfy the same metric. [9][10][11] With the help of differential geometry and solid mechanics, researchers have successfully designed inplane growth patterns to achieve targeted 3D shapes in materials including liquid crystal elastomers, [12,13] hydrogels, [14][15][16] and shape memory polymers. [17] In addition to patterning the metric, introducing differential growth through the thickness of a thin sheet is another potent approach to 3D shape morphing. [18,19] This can be achieved by forming discrete bilayers of different materials, [20] or by introducing smooth gradients in deformation through factors including gravity, [21] diffusion, [22] and most commonly, absorption of light. [23][24][25] In contrast to programming Gaussian curvature through in-plane variations, out-of-plane differential growth provides a route to control the mean curvature, an extrinsic variable of a surface.

To uniquely specify a desired 3D shape, it is necessary to prescribe both mean and Gaussian curvatures simultaneously by patterning both in-plane and out-of-plane differential growth, such that the programmed mean curvature can select among the different isometric embeddings possible for a given pattern of Gaussian curvature. However, specifying both in-and out-of-plane differential growth is not trivial, and there have so far been only a few experimental systems providing such concurrent control. In particular, Aharoni et al. [26] and Plucinsky et al. [27] patterned the in-plane nematic director field within liquid crystal elastomer films while also introducing a throughthickness twist using separately patterned anchoring surfaces on the top and bottom of the thin film, while Gladman et al. [28] achieved a similar effect in 3D-printed anisotropically swelling nanocomposite hydrogel bilayers. With isotropically swelling hydrogels, Wang et al. [29] used pairs of physical masks to locally constrain the direction of buckling in a pre-swelling step, thereby leading to control of buckling direction in structures consisting of a series of domes. While effective at controlling both mean and Gaussian curvature, these methods have so far been limited to large (cm-scale) samples, without a clear route to scale them to smaller dimensions.

In recent years, there has been growing interest in shape morphing materials in which initially planar elastic membranes can adopt well-defined 3D shapes in a programmable way. These systems offer promise in a variety of fields including soft robotics, [1,2] microfluidics, [3] and biomimetic materials. [4,5] The ability to start from a flat 2D sheet is particularly interesting because of the attractive scalability of the planar manufacturing steps employed as well as the ease of storing and transporting materials in the flat state. [6] The formation of arbitrary 3D shapes, with simultaneous curvature along both lateral directions, requires differential growth or distortion along the in-plane dimensions of a 2D sheet. Since the stretching energy changes linearly with the sheet thickness, t, while the bending energy changes cubically with t, a sufficiently thin sheet subjected to such differential In recent years, our group has relied on in-plane gradations of crosslinking density within thin films of photocrosslinkable hydrogels to define spatial variations in swelling ratios. [14,15,30,31] This method has so far offered access to dimensions roughly two orders of magnitude smaller (≈100 µm to 1 mm) than the approaches described above, with clear potential for further reductions in size using higher resolution lithography. However, as the method has so far been limited to films that are homogenous through their thickness, it has not been possible to specify the mean curvature using this approach, thereby often yielding samples that fail to adopt the targeted shape. [14] Here, we control the absorption of light through the film thickness to provide an out-of-plane gradient in swelling, thereby adding another degree of freedom. Thus, the preferential buckling direction of each positive K feature defined by a local maximum in swelling can be programmed by controlling whether the film is exposed to UV light from above or below, thereby selecting among different isometries. Compared with the existing methods [26][27][28][29] to simultaneously control both mean and Gaussian curvature described in the previous paragraph, the current approach offers much greater spatial resolution (ultimately limited by the diffraction of light), opening the door to uniquely defined 3D shapes on sub-mm to µm-scales. In addition, the spin-coating and photolithographic steps in our fabrication process are relatively simple compared with the previous methods, which require molding for liquid crystal elastomers or 3D printing for nanocomposite gel bilayers. In light of these advantages, we expect the method described here to offer an appealing alternative especially in the design of biomedical devices, [32] for example, for drug encapsulation, [33] as well as scaffolds for cell culture [34] or forming tissues with complicated geometries. [35]

To fabricate shape-programmed hydrogel sheets, we employ a poly(N,N′-diethylacrylamide) (PDEAm) copolymer containing 7 mol% of pendent benzophenone (BP) UV crosslinkers, and 3 mol% of acrylic acid (AAc) to promote swelling, as well as 0.1 mol% of Rhodamine B methacrylate (RhB) as a fluorescent marker, as shown in Figure 1. When illuminated by UV light, the excited triplet state of a BP acts as a diradical, which then converts into a ketyl radical and generates another aliphatic radical by abstracting an aliphatic hydrogen atom from a neighboring polymer segment. The recombination reaction between these radicals can provide efficient photocrosslinking pathways for BP-containing copolymers. [36][37][38] The crosslinking density of the polymer increases as the exposure dose of UV light increases, and when immersed in an aqueous medium at room temperature, a uniformly crosslinked sheet swells isotropically by an areal factor of Ω, which varies with the crosslinking density. As shown in Figure 1, a range of 2.5-fold in Ω is achieved by varying exposure time from 15 to 600 s. Such variations in swelling ratio can be lithographically patterned through spatial variations in the exposure dose with a resolution ultimately limited by the diffraction limit of light, introducing differential swelling in the photocrosslinkable polymer films. [15] In addition, the PDEAm-based hydrogel can undergo a dramatic reduction in volume upon heating, thanks to its lower critical solution temperature (LCST) phase behavior in water, [39] rendering the swelling-induced deformation of these films reversible due to changes of temperature.

Our group has previously demonstrated the patterning of inplane swelling using poly(N-isopropylacrylamide) (PNIPAm) photocrosslinkable copolymers. [15] Here, we shift to a PDEAm copolymer for several reasons. First, PDEAm develops much smaller gradients in residual stress during film casting, presumably due to its lower glass transition temperature compared to PNIPAm. [40] This allows us to prepare films by spin-coating without introducing substantial through-thickness variations in residual stress, thereby simplifying the process and providing more reproducible film thicknesses compared with the drop-casting method used previously for PNIPAm copolymers. As shown in Figure S1a,b (Supporting Information), a spin-coated PNIPAm gel disk with uniform crosslinking curls after swelling due to the residual stress gradients developed during spin-coating, while a PDEAm sample remains nearly flat. Moreover, the PDEAm copolymer shows a higher rate of photocrosslinking compared to PNIPAm, requiring respective doses for gelation and full crosslinking of 0.2 and 10 J cm -2 (Figure S1c, Supporting Information), which are only ≈30% those of a similar PNIPAm copolymer. [15] Third, intra-and intermolecular hydrogen bonds are known to cause hysteresis during the phase transition of PNIPAm around its LCST, [41] which we suspect leads to slow swelling kinetics of photocrosslinked PNIPAm copolymers. Thus, the lack of hydrogenbond donors within PDEAm-based copolymers provides faster swelling/deswelling kinetics. [42,43] In the limit of vanishing thickness, a flat elastic sheet subjected to a pattern of growth, Ω(r) will adopt a shape that satisfies the target Gaussian curvature, Κ(r) at every point r given by [14,44] ln 2

Relying on the calibration curve of Ω as a function of UV exposure time in Figure 1, we then employ a maskless grayscale photo lithography method [15] to define the swelling at each point r. We first consider three shapes prepared using axisymmetric metrics: a spherical cap, a saddle, and an Enneper's minimal surface with six wrinkles. For the spherical cap, the metric is

where r is the radial position, and c and R are two constants that determine the constant Gaussian curvature K = 4/(cR 2 ).

Similarly, for the saddle that has a negative constant Gaussian curvature of -4/(cR 2 )

while the metric

corresponds to an Enneper's minimal surfaces with n wrinkles. In all three cases, the experimentally achieved shapes show good qualitative agreement with the programmed shapes. We further measure the Gaussian and mean curvatures of the hydrogel sheets from the confocal microscopy images. Despite some small deviations, both the spherical shell and saddle exhibit relatively uniform Gaussian curvatures. The average Gaussian curvatures excluding edges [15] are 1.1 × 10 -5 and -1.7 × 10 -4 µm -2 , respectively, which are close to the target values. The mean curvature of the Enneper's surface is also close to zero which matches the target profile well. One should note that the measured curvatures are those on the top, bottom, and side surfaces of the nonzero thickness gel sheets, which explains why the regions near the edges of each shape always display significantly different curvatures than the target values. Additionally, for samples where one portion "blocks" another along the optical axis of the confocal microscope, the region more distant from the objective becomes dimmer and its apparent thickness in the reconstructed model decreases, thus inducing deviations in the local surface curvatures.

We next consider a more complicated shape with a nonaxisymmetric metric. Figure 2d,e shows the metric of a corrugated surface described by a height function H(x,y) = H 0 [cos(2πx/L) + cos(πx/L + (√3πy)/L)], where we choose L = 5H 0 . Similar to the report by Kim et al., [14] two different shapes are observed experimentally, as shown in Figure 2d,e. In Figure 2d, the gel sheet adopts the desired corrugated shape. In this situation, the three local maxima in the metric indicated by the blue boxes buckle in alternating directions, that is, upwarddownward-upward (or downward-upward-downward), as targeted. The curvature maps show that the regions around each local swelling maximum exhibit positive Gaussian curvature, among which the two regions on the sides have negative mean curvature while the central region has positive mean curvature However, in some cases, these local swelling maxima may also buckle toward the same direction and form the shape shown in Figure 2e-in this case all three regions have negative mean curvature, while the Gaussian curvature map is still very similar to that in Figure 2d. This represents one simple example of the general principle that the prescribed Gaussian curvature does not uniquely determine the shape of the gel sheet, since there are multiple nearly isometric shapes that have different distributions of mean curvature and that are mechanically stable.

In contrast to Gaussian curvature, mean curvature is an extrinsic property of a surface, implying that out-of-plane differential growth is essential to its prescription. Here, we introduce control over the sign of mean curvature preferred by the gel sheets simply by introducing a gradient of crosslinking density through the thickness due to absorption of light as it travels through the films. According to the Beer-Lambert law, the light intensity I will decay exponentially from its initial value I 0 upon traveling through a path length l, according to I/I 0 = e -εcl , where ε is the absorption coefficient of the light absorbing species and c is its concentration. Equivalently, a characteristic penetration depth can be defined as (εc) -1 . Thus, a substantial gradient in light intensity can be achieved only when the penetration depth is comparable to, or smaller than, the film thickness. Based on the absorbance spectrum of the PDEAm copolymer (Figure S2, Supporting Information), with an absorption coefficient of 0.17 L g -1 cm -1 at 365 nm (the wavelength used for UV crosslinking), the penetration depth is around 25 µm, while the film thickness is only 5 µm. To increase UV absorption and make the penetration depth comparable to the film thickness, we therefore dope 2-(2H-benzotriazol-2-yl)-4,6-di-tert-pentylphenol (Tinuvin 328) into the PDEAm copolymer solutions prior to spin-coating. Based on its absorption coefficient at 365 nm of 30 L g -1 cm -1 , 0.5 and 1 wt% of the absorber, relative to copolymer, reduce the penetration depth to 13 and 9 µm, respectively. We note that similar methods have been used previously to define through-thickness crosslinking gradients, for example, in the context of wrinkled surfaces. [45] To explore the effects of UV absorber, we expose gel disks to increasing doses of light with in-plane homogeneity. Due to the significant absorption of light, the surface of the polymer film closer to the light source will have a higher crosslinking density, and accordingly a lower swelling ratio compared with the bottom. As a result, the samples exhibit dose-dependent mean curvature, as shown in Figure 3b. The mean curvature decreases as the exposure dose increases, which is expected since the swelling ratio saturates to a constant value for long exposure times as the benzophenone crosslinkers are completely converted. By an exposure time of ≈600 s, the disk is fully crosslinked through its entire thickness, so it remains flat after swelling in water. For comparison, a PDEAm gel disk without UV absorber still remains nearly flat even when the exposure time is relatively short, as shown in Figure 3c. By depositing the polymer film on a UV transparent quartz substrate, it is possible to irradiate the sample from both directions, such that the sign of mean curvature can be simply controlled by the incident direction of UV light (Figure 3a). For example, Figure 3d shows a 3D shape obtained by prescribing both positive and negative mean curvatures using this doublesided lithography method. Specifically, a wavy stripe is achieved by patterning three rectangular regions from the "top" (the air/ polymer interface side) and the two intervening rectangular regions from the "bottom" (the quartz/polymer interface side).

Next, we focus on the simultaneous control of both mean and Gaussian curvature, using the corrugated surface introduced in Figure 2d,e as an example of how nearly isometric shapes can be distinguished by providing a preferential buckling direction. As previously, Gaussian curvature is prescribed by patterning the in-plane differential swelling through spatial variations in exposure time. Meanwhile, 0.5 wt% of UV absorber is doped to provide a preferential buckling direction by introducing a slight variation in swelling through the film thickness to break up-down symmetry without substantially altering the target inplane metric (Figure S1c, Supporting Information). When one unit in the sinusoidal surface is exposed from the top of the film, the swelling ratio through the thickness increases from the top to the bottom, thus this unit is programmed to buckle downward, and vice versa (Figure 4a).

For simplicity, we start with a metric that has only one local maximum in swelling (Figure 4b), corresponding to a single "unit cell" of the corrugated surface metric shown in Figure 2d. A small rectangular tab is added at one corner to break the reflection symmetry, so that we can distinguish the buckling direction even when the gel sheet is flipped over. We define the buckling direction as up or down relative to the original orientation of the sheet during patterning, as illustrated in Figure 4a. When undoped and doped PDEAm films are patterned from the top side with the same metric, they deform into the shapes shown in Figure 4d,e absorber, the intensity gradient through the thickness is not sufficient to break up/down symmetry. Although it is difficult to judge the buckling direction from the 2D fluorescence images shown in Figure 4d, each sample has been oriented with the apex of the positive-K feature pointing away from the microscope objective, and thus the position of the small tab at the corner of the sample uniquely reveals whether the sample has buckled downward (preserving its as-fabricated position at the "top-left" corner) or upward (switching the location of the tab to the "top-right" corner). As summarized in Figure 4c, among four such samples, two sheets buckle upward while the other two buckle downward. With the addition of UV absorber, on the other hand, all seven gel sheets fabricated are found to buckle downwards, as programmed, validating that our method can reliably control the buckling direction of a single-unit structure.

We next test a metric corresponding to three such units connected in a line. Instead of exposing all of them simultaneously from the same direction, the first and third units are irradiated from the top of the film and the second unit from the bottom. Ideally, a corrugated surface should form, with the three units buckling in alternating directions due to the preferences programmed through the irradiation direction. However, all three units choose the same direction, leading to the shape shown in Figure 4f. Interestingly, the middle unit clearly shows a boundary layer [46] in which the curvature inverts near the edges, consistent with the fact that this unit is inverted compared to its preferred curvature. This finding suggests that a coupling exists between these units and even though we introduce a bias for each unit individually, the coupling effect is strong enough to overcome the preference of an individual unit. To test this hypothesis, softened boundaries between each unit are incorporated into the pattern of swelling by introducing narrow regions with a high areal swelling ratio of 2.75, and thus low modulus, to decouple the interaction between the buckled units (see Figure S3, Supporting Information, for additional details). Notably, this also introduces a local change to the metric, which could distort the shape, though since the softening is confined to a narrow region of width comparable to the film thickness, we do not expect this to be a major effect. With the possibility for each repeating unit to buckle either up or down, the total number of possible configurations is 8; however, symmetry reduces the number of distinct shapes to three. By incorporating softened regions, all three shapes are successfully programmed as shown in Figure 4g, establishing that the buckling direction of each unit can be controlled individually by the irradiation direction. Notably, since the shape transformations result from the LCST phase behavior of PDEAm in water, which is a highly reversible process, heating and cooling of patterned gel sheets causes them to flatten and then regain their swelled shapes, respectively, as shown in Figure S4 (Supporting Information).

To better understand the origin of the coupling between the buckled units, we next step back to a single-unit gel sheet without any UV absorber. Confocal microscopy measurements reveal that for a single unit that has a large in-plane size (500 µm × 500 µm, ≈2.5 times larger than the units in the three-unit corrugated samples), the actual 3D shape of the gel sheet after swelling (Figure 5b) is very similar to the target (Figure 5a), which has flat boundaries. However, when the lateral dimension of the unit is reduced but the thickness is kept the same, the shape becomes distorted from the target, presumably because the bending energy required to adopt the target shape becomes progressively larger as the effective thickness of the sheet is increased. As shown in Figure 5c,d, this distortion causes the boundaries of smaller gel sheets to curve. We measure the bending content [9] distribution (defined as 4H 2 -2(1-ν)K, where ν is the Poisson ratio, taken here to be 0.3) of the full-size gel sheet (Figure 5b) as shown in Figure 5f and the half-size gel sheet (Figure 5c) as shown in Figure 5g, using a method adapted from Na et al. [15] The bending content of the target surface is shown in Figure 5e for comparison. As shown in Figure 5e-g and Table S1 (Supporting Information), when the effective thickness of the gel sheet increases, the normalized bending content decreases from 2.52 × 10 -5 µm -2 for the full-size sheet to 2.06 × 10 -5 µm -2 for the half-size gel sheet, whereas that of the target shape is 2.96 × 10 -5 µm -2 , further supporting that the distortion from the target shape serves to lower the bending energy. Notably, this conclusion is not sensitive to the choice of Poisson's ratio and still valid when the material is taken to be incompressible, that is, ν = 0.5. Further evidence for this picture is provided by simulations. We compare the energies of a single buckled unit where the boundaries are allowed to freely curve and one where the boundary is fixed to be planar at different normalized thicknesses. As shown in Figure 5j, a sufficiently thin sheet prefers to maintain a flatboundary while thicker sheets prefer the curved-boundary state.

The curvature of the boundaries for an individual unit, as driven by bending energy, leads to an incompatibility when neighboring units buckle in opposite directions, as shown in Figure 5h. To fill the "gap" that would be formed at the boundary, both units would need to distort substantially, causing an increase in elastic energy relative to the case of two with the same buckling direction, which can fit together with much less distortion. Thus, the incorporation of a flexible region at the boundary can help lower the extra energy due to the incompatibility of boundary lines and decouple the Adv. Funct. Mater. 2019, 1905273 and d) a gel sheet one quarter the size of (b); e) normalized bending content distribution of the target unit in (a), f) the full-size gel sheet in (b), and g) the half-size gel sheet in (c). h) A gap exists when two units buckled in opposite directions meet, while there is no such gap when the two units buckle in the same direction. i) A 3D-printed elastic model with three curved units, and the shapes when one of the units buckles in the opposite direction to the others. j) Simulated normalized energy versus normalized thickness for singleunit surfaces with flat (red) versus curved boundaries (blue). Inset images correspond to thicknesses of 0.03. k) Simulated normalized energy versus normalized thicknesses for a double-unit surface with two units buckling in the same (red) or opposite (blue) directions. The color in each inset image (corresponding to thicknesses of 0.03) indicates the difference of Gaussian curvature from the target shape.

buckling direction of adjacent units. [29] This also explains why smaller units are coupled with each other more strongly, since the edges of a smaller unit are more curved. To support this hypothesis, an elastomer model with three curved units shown in Figure 5i is made by 3D printing. When one of the three units is mechanically pressed to buckle in the opposite direction, its neighboring unit(s) folds toward this unit to fill the gap between incompatible curved boundaries. A sharp bend is observed at the boundaries between neighboring units buckled in opposite directions, suggesting a high-energy state. Thus, a curved unit is more likely to buckle to the same direction as its neighbor(s), which results in the coupling effect in buckling direction of adjacent units. We also simulate two-unit shapes as shown in Figure 5k. For films with normalized thickness below ≈0.003, the energy is lower in the up-down state, but thicker films prefer the up-up state. Notably, the normalized thickness in our experiment is roughly 0.03, an order of magnitude thicker than the cross-over predicted by simulations, consistent with the observation of significant coupling between neighboring units in experiments.

Finally, we explore the possibility to pattern buckling direction in 2D arrays (3 × 2 and 3 × 3) of such units. The number of possible configurations of each is large, and here we study only a few representative cases. As shown in Figure 6, for "checkerboard" patterns where each repeating unit is programmed to buckle in the opposite direction of its neighbors (the first configuration in each figure), the gel sheets do adopt the desired configurations. However, not all configurations are achievable even with the softened regions at unit boundaries. When all three units in a row are designed to buckle in the same direction but the neighboring rows are assigned to the opposite direction, all samples fail to adopt the desired shapes and instead show buckling of all features in the same direction. This is not difficult to understand using a similar argument as above. When all three units in a row have the same buckling direction, the overall shape will also bend toward the same direction to relax the bending energy near the boundaries of adjacent units, thus curving the border, just like the shape with a down-down-down configuration shown in Figure 4g. Thus, stitching two rows that bend in opposite directions together would require a great deal of distortion. This is also supported by the simulation results in Figure S5 (Supporting Information), which demonstrate that such a target shape has much higher energy compared with other buckling states. We also note that if a single unit is programmed to buckle in the opposite direction from all of its neighbors, it follows the programmed direction when this unit sits at the corner, but fails when it sits in the middle of a row at the edge of the sheet (the two configurations on the right in Figure 6a), consistent with the expectation that a unit with more neighbors will have stronger overall coupling, and therefore pay a larger penalty to adopt the inverse curvature.

As we have discussed, the curved boundary arising from the bending energy of hydrogel sheets limits the shapes that can be achieved. Presumably, if the mean curvature could be precisely patterned at each point to be compatible with the metric, rather than simply biasing the buckling direction unit-by-unit as done here, it would be possible to select the target shapes even without weakening the connection between neighboring features. However, the method described here provides a very simple route to bias the buckling direction of positive-K features in sheets with programmed metrics, and thus provides a useful tool in shape-programmable materials.

To summarize, a photocrosslinkable polymer of which the crosslinking density can be patterned by varying the dose of UV light received locally is employed to photolithographically pattern the 3D shapes adopted by hydrogel sheets upon swelling. Most importantly, we establish a simple double-sided exposure strategy in which the introduction of controlled amounts of a UV absorber allow for the definition of through-thickness gradients while simultaneously allowing for prescription of inplane variations in swelling. This allows us to strongly bias the direction of mean curvature adopted by sheets with nonzero Adv. Funct. Mater. 2019, 1905273 Gaussian curvature with high spatial resolution and a simple fabrication process, substantially expanding the range of 3D shapes that can be achieved compared with methods to control either Gaussian or mean curvature alone.

Materials: The PDEAm copolymer was synthesized by free radical polymerization, with a feeding ratio of 90.9 mol% of N, N-diethylacrylamide (DEAm, TCI), 7 mol% of acrylamidobenzophenone (AmBP), 2 mol% of acrylic acid, and 0.1 mol% of RhBMA (Rhodamine B methacrylate) as a fluorescent marker. Monomers were dissolved in 1,4-dioxane and polymerized at 80 °C overnight with azobisisobutyronitrile (AIBN) as the initiator after a nitrogen purge for 1 h. The copolymer was then purified by precipitating into cold hexanes and finally dried in a vacuum oven at room temperature for 24 h. Here, DEAm was purified by passing through an alumina column before use. AmBP was prepared as the previous report. [14] AIBN was purified by recrystallization. 2-(2H-benzotriazol-2-yl)-4,6-di-tert-pentylphenol (Tinuvin 328) was purchased from Sigma.

Sample Preparation: Spin-coating was used to prepare PDEAm films with a uniform thickness. First, 0.2 mm thick quartz slides (Technical Glass Products) covered with a sacrificial layer of Ca 2+ crosslinked polyacrylic acid were prepared as described previously. [30] A solution of the PDEAm copolymer in 1-propanol (20 wt%) was subsequently spin-coated onto the substrate at a speed of 1000 rpm for 90 s. Such spin-coating was conducted twice to obtain the film thickness of 4.5-5 µm (measured with a Dektak 150 stylus profilometer, Veeco Instruments). The polymer films were dried overnight with nitrogen flow before use.

Photopatterning of the crosslinkable polymer films was conducted with a maskless photolithography setup introduced previously. [15] Specifically, collimated UV light (365 nm, pE-100, CoolLED) was reflected by a digital micromirror array device (DMD, DLP Discovery 4100, 0.7 XGA, Texas Instruments) in which each mirror was controlled using MATLAB. Then the generated light pattern was projected onto samples through an inverted microscope (Nikon ECLIPSE Ti) to photocrosslink desired regions on the polymer films. The UV intensity was 18 mW cm -2 , as measured with a power meter (Thorlabs, PM100USB). For doublesided lithography, an alignment step was conducted to ensure that the top and bottom patterns were correctly registered. An intense exposure of ≈20 J cm -2 was employed to bleach the RhB fluorophore to pattern an alignment mark (Figure 3d), following the normal exposure of the top pattern. Such a bleached mark could still be clearly seen with fluorescence filters inserted after flipping the sample over for bottom exposure. Meanwhile, the same alignment mark pattern was projected onto the sample by DMD with visible light. Thus, the original sample location before flipping was restored by translating the sample until the bleached region overlapped with the pattern of projected light.

After UV irradiation, samples were immersed into a marginal solvent (a 3:1 toluene/hexanes mixture, v/v) to dissolve uncrosslinked regions without excessively swelling crosslinked regions, followed by blowing dry with nitrogen. Finally, the PAA sacrificial layer was dissolved by immersion in phosphate buffered saline (PBS, pH 7.2, 1 × 10 -3 m NaCl) solution, allowing the patterned samples to release from the substrate and swell freely. In programming corrugated surfaces, the choice of buckling direction can be sensitive to the initial process of swelling, since once a given feature buckles in a certain direction, reversing its direction generally requires an energy barrier to be overcome. Since the process of detachment from the substrate might influence the buckling direction of the gel sheet, the PAA sacrificial layer was first removed in a high-temperature (50-55 °C) PBS solution, where the swelling of the polymer sheet was minimal. After release from the substrate, freestanding polymer sheets were allowed to swell slowly and deform into the desired shapes by decreasing the bath temperature at a modest rate of 2 °C min -1 . 3D-printed parts were designed in Mathematica and Rhinoceros, and then printed using a Formlabs Form2 printer with Formlabs Flexible resin.

Metric Calculation and Simulation of Buckled Sheets: Given a surface with metric g ¯, a parameterization X was constructed in some domain in the Euclidean plane with metric g, such that g ¯ = Ω(r)g, where Ω(r) = e u(r) . As a result of Gauss' theorema egregium, 2K(r)e u(r) = ∇ 2 u(r). Then the target surface was triangulated, real-valued numbers, u i , were assigned to the vertices of the triangulation to represent a discrete approximation of the function u(r). [47] u i was found by imposing the law of cosines and the fact that the sum of interior angles around each vertex on the flat domain must be zero. For the original corrugated surfaces shown in Figure 2d,e and their derivatives, the metric from Kim et al. [14] was used. To simulate the sheets in different configurations and compare their elastic energies, we used the method described in Duque et al. [47] Here, uniform thickness and elastic modulus were assumed throughout each gel sheet for simplicity, and any preferred curvature was not introduced. Simulations were performed by triangulating the initially flat swelling pattern. A preferred length, l ij , is associated to an edge connecting pair of vertices i and j. If X i and X j are the positions of vertices i and j, the contribution to the elastic energy coming from this edge is then given by

Summing over all the edges gives the total stretching energy. To study bending effects, the indices I and J of two triangular faces sharing a common edge in the triangulation can be considered and sum over the terms of the type

where t is the normalized elastic thickness and n I is the unit normal to the triangular face with index I. Measurement and Instruments: Swollen gel samples were measured by a Zeiss Axiotech Vario upright microscope or a Zeiss Axivert 200 inverted microscope. The temperature of the solution was controlled using a temperature stage (INSTEC HCS621V). Swelling ratios of gels with different exposure doses were measured by comparing the area of uniformly crosslinked gel disks in the fully swollen state after soaking in the buffer solution for 24 h to the area in the dry, as-crosslinked state. Some samples were also measured by a laser scanning confocal fluorescence microscope (Nikon, A1R) to obtain the 3D shape. When quantifying the mean and Gaussian curvatures of a sample, 3D-stacked confocal images were first extracted using image processing software, Fiji, [48] as an image sequence with the nominal spacing expanded by a factor of 1.33 to correct for the refractive index mismatch between water and air, and then converted to black-and-white images with a proper threshold. Subsequently, an isosurface mesh of the sample was extracted from the processed image sequence in MATLAB and smoothed to minimize surface imperfections, from which the mean and Gaussian curvatures were finally calculated with a user-developed MATLAB function. [49]