Growth rates are uniformized by introducing anisotropy. Optimized constant growth patterns for set of synthetic surfaces. Initial configuration for all surfaces is the flat unit disk. Circular texture on the surfaces represents the deformation of material patches under the flow. (a) A conformal growth pattern linking the unit disk and the hemispherical surface. Growth occurs primarily at the apex of the dome. (b) The optimized constant growth rates for the same final shape. Growth rates are heavily uniformized relative to the conformal growth pattern. (c) The absolute value of the Beltrami coefficient for the optimized constant growth pattern. (d) A conformal growth pattern linking the unit disk and an elliptic paraboloid. Growth occurs primarily at the poles. (e) The optimized constant growth rates for the same final shape. Growth rates are uniformized, although not as completely as the case of the spherical surface. (f) The absolute value of the Beltrami coefficient for the optimized constant growth pattern. (g) The absolute value of the Beltrami coefficient for the optimized constant growth pattern transforming a flat disk into a hemispherical surface and the corresponding nematic field illustrating the orientation along which material patches extend. The texture contains a +1 topological defect with phase ψ = 0. (h) A synthetic Beltrami coefficient with the same total integrated anisotropy constructed by placing two +1 defects with phase ψ = π/2 at opposite poles of the disk boundary. (i) A new surface generated using the exact same growth rates as the surface in (b), but with the anisotropy texture shown in (h).

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... begin by calculating optimal growth patterns for a set of simple synthetic target shapes grown from an initially flat disk configuration (Fig. 3). In particular, we solve the optimal growth problem for a hemisphere (Fig. 3(a-c)) and an elliptic paraboloid ( Fig. 3(d-f)) using a conformal parameterization as our initial guess. For all systems studied, growth rates in the optimal configuration are uniformized at the cost of introducing anisotropy. A(T)/A(0)-1 These results ...

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... begin by calculating optimal growth patterns for a set of simple synthetic target shapes grown from an initially flat disk configuration (Fig. 3). In particular, we solve the optimal growth problem for a hemisphere (Fig. 3(a-c)) and an elliptic paraboloid ( Fig. 3(d-f)) using a conformal parameterization as our initial guess. For all systems studied, growth rates in the optimal configuration are uniformized at the cost of introducing anisotropy. A(T)/A(0)-1 These results suggest that growth rate uniformization may be a generic mechanism underlying the ...

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... begin by calculating optimal growth patterns for a set of simple synthetic target shapes grown from an initially flat disk configuration (Fig. 3). In particular, we solve the optimal growth problem for a hemisphere (Fig. 3(a-c)) and an elliptic paraboloid ( Fig. 3(d-f)) using a conformal parameterization as our initial guess. For all systems studied, growth rates in the optimal configuration are uniformized at the cost of introducing anisotropy. A(T)/A(0)-1 These results suggest that growth rate uniformization may be a generic mechanism underlying the presence of anisotropy in observed developmental ...

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... at the cost of introducing anisotropy. A(T)/A(0)-1 These results suggest that growth rate uniformization may be a generic mechanism underlying the presence of anisotropy in observed developmental growth patterns. Our formalism enables us to decode the contributions of areal expansion and anisotropic deformation towards determining 3D shape. Fig. 3(c) shows the cumulative anisotropy of the optimal constant growth pattern transforming a flat disk into a hemispherical shell. Displaying the local orientation along which tissue parcels extend due to anisotropic deformation results in a nematic texture supporting topological defects (Fig. 3(g)). Such textures have recently been studied ...

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... anisotropic deformation towards determining 3D shape. Fig. 3(c) shows the cumulative anisotropy of the optimal constant growth pattern transforming a flat disk into a hemispherical shell. Displaying the local orientation along which tissue parcels extend due to anisotropic deformation results in a nematic texture supporting topological defects (Fig. 3(g)). Such textures have recently been studied in the context of epithelial morphogenesis coupled to active nematic biomolecular components [43], where defects in nematic textures have been identified as organizing centers of curvature and mass accretion leading to shape change [44][45][46]. The optimal growth texture is characterized by a ...

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... in the context of epithelial morphogenesis coupled to active nematic biomolecular components [43], where defects in nematic textures have been identified as organizing centers of curvature and mass accretion leading to shape change [44][45][46]. The optimal growth texture is characterized by a +1 defect with phase ψ = 0 at the pole of the sphere. Fig. 3(h) displays a modified texture with the same total integrated anisotropy, i.e. D |µ( x)| 2 d 2 x, but with an alternative texture constructed by placing two +1 defects with phase ψ = π/2 at opposite poles on the disk boundary. Simulating a growth process using this modified anisotropy texture, but keeping the areal growth rates exactly ...

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... a modified texture with the same total integrated anisotropy, i.e. D |µ( x)| 2 d 2 x, but with an alternative texture constructed by placing two +1 defects with phase ψ = π/2 at opposite poles on the disk boundary. Simulating a growth process using this modified anisotropy texture, but keeping the areal growth rates exactly the same as in Fig. 3(b), produces an entirely new final shape (Fig. 3(i)). This proof-of-principle demonstrates how our methods can be used as a platform for growth pattern design in terms of nematic ...

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... anisotropy, i.e. D |µ( x)| 2 d 2 x, but with an alternative texture constructed by placing two +1 defects with phase ψ = π/2 at opposite poles on the disk boundary. Simulating a growth process using this modified anisotropy texture, but keeping the areal growth rates exactly the same as in Fig. 3(b), produces an entirely new final shape (Fig. 3(i)). This proof-of-principle demonstrates how our methods can be used as a platform for growth pattern design in terms of nematic ...

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... γ ≈ ˙ µ. From Eq. (9), we see that this approximation holds for |µ| 2 << 1 and e iψ = (∂ z w)/(∂ z w) ≈ 1. Writing z = r e iθ , this latter condition is exactly true, for instance, when w(z, t) = ρ(r, t) e iθ , which applies to a broad variety of relevant growth patterns, including the growth pattern generating the hemispherical cap shown in Fig. 3(a-c). In this new approximation scheme, sincësincë µ ≈ ˙ γ = 0, the Beltrami coefficient as a function of time is simply µ( x, t) ≈ t ˙ µ( x) ≈ t γ(x), where we have exploited the fact that we are always free to choose a conformal parameterization for our initial time point. The modified constant growth functional is given ...