Programming curvature using origami tessellations
TLDR
In this article, scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature.Abstract:
Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures-we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.read more
Citations
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References
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Curvature and Creases: A Primer on Paper
TL;DR: In this paper, the authors present fundamental results about how zero-curvature (paper) surfaces behave near creases and apices of cones, which are natural generalizations of the edges and vertices of piecewise-planar surfaces.
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Origami multistability: from single vertices to metasheets.
TL;DR: It is shown how to program the stability features of a single vertex into a periodic fold tessellation, and the resulting metasheets provide a previously unanticipated functionality-tunable and switchable shape and size via multistability.
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Self-organized origami
TL;DR: This paper examines the possible physical origin for the Miura-ori leaf-folding patterns that arise naturally in insect wings, leaves, and other laminae-like organelles, and points out examples where biaxial compression of an elastically supported thin film spontaneously generates these patterns.
Journal Article
Generalization of rigid foldable quadrilateral mesh origami
TL;DR: This work generalizes the geometric condition for enabling rigid motion in general quadrilateral mesh origami without the trivial repeating symmetry and derives the identity of functions from the formula for degree-4 single-vertex origami to ensure the existence of a finite motion.
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The geometry of unfolding tree leaves
TL;DR: In this paper, the leaves of hornbeam (Carpinus betulus) and beech (Fagus sylvaticus) were modelled as plane surfaces, with straight parallel folds, using numerical methods.