In this paper we discuss the transformation of a sheet of material into a wide range of desired shapes and patterns by introducing a set of simple cuts in a multilevel hierarchy with different motifs. Each choice of hierarchical cut motif and cut level allows the material to expand into a unique structure with a unique set of properties. We can reverse-engineer the desired expanded geometries to find the requisite cut pattern to produce it without changing the physical properties of the initial material. The concept was experimentally realized and applied to create an electrode that expands to >800% the original area with only very minor stretching of the underlying material. The generality of our approach greatly expands the design space for materials so that they can be tuned for diverse applications.he physical properties of materials are largely determined by structure: atomic/molecular structure, phase distribution, internal defects, nano/microstructure, sample geometry, and electronic structure. Among these, engineering the geometry of the sample can provide a direct, intuitive, and often materialindependent approach to achieve a predetermined set of properties. Metamaterials are fabricated based on geometric concepts (1-16). In two dimensions, periodic geometries have been adopted to tune the mechanical properties of membranes (3-8, 10, 12-14). From simple shapes such as circles (3), triangles (6,7,12,13), and quadralaterals (4, 5, 14) to more complex shapes (8, 10), a broad range of mechanical behavior has been observed, including pattern transformation, negative Poisson's ratio (auxetic), elastic response, and isostaticity. Origami and kirigami, the arts of paper folding and paper cutting, create beautiful patterns and shapes that have attracted the attention of scientists to two-dimensional materials (e.g., graphene, polymer films, and so on) (11,(17)(18)(19). However, application of conventional origami and kirigami approaches to achieve desired material response requires complex cutting and/or folding patterns that are often incompatible with engineering materials. In this paper we propose an advanced approach to the design of two-dimensional structures that can achieve a wide range of desirable programmed shapes and mechanical properties.This study starts from the question, Can we design twodimensional structures that can be formed by simply cutting a sheet, that can morph into a specific shape? In nature, many biological and natural system (20) can be found that use hierarchical structure to produce different properties and/or shapes. One such example is a stem cell. An embryonic, pluripotent stem cell can differentiate into any type of cell in the body (21). By recursively dividing, the stem cell can transform into particular cell types or remain unspecialized with the potential to transform. For a material, one aspect of recursive hierarchical geometry was recently discussed for applications in flexible electronics (22). Here, by analogy to the stem cell, we demonstrate that starting from a simpl...

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Biomimicking ceramics have been developed to induce efficient recovery of damaged hard tissues. Among them, calcium phosphate-based bioceramics have been the most widely used because of their similar composition with human hard tissue and excellent biocompatibilities. However, the incomplete understanding of entire inorganic phases in natural bone has limited the recreation of complete bone compositions. In this work, broad biomedical evaluation of whitlockite (WH: Ca18Mg2(HPO4)2(PO4)12), which is the secondary inorganic phase in bone, is conducted to better understand human hard tissue and to seek potential application as a biomaterial. Based on the recently developed gram-scale method for synthesizing WH nanoparticles, the properties of WH as a material for cellular scaffolding and bone implants are assessed and compared to those of hydroxyapatite (HAP: Ca10(PO4)6(OH)2) and β-tricalcium phosphate (β-TCP: β-Ca3(PO4)2). WH-reinforced composite scaffolds facilitate bone-specific differentiation compared to HAP-reinforced composite scaffolds. Additionally, WH implants induce similar or better bone regeneration in calvarial defects in a rat model compared to HAP and β-TCP implants, with intermediate resorbability. New findings of the properties of WH that distinguish it from HAP and β-TCP are significant in understanding human hard tissue, mimicking bone tissue at the nanoscale and designing functional bioceramics.

In two-dimensional interfacial assemblies, there is an interplay between molecular ordering and interface geometry, which determines the final morphology and order of entire systems. Here we present the interfacial phenomenon of spontaneous facet formation in a water droplet driven by designed peptide assembly. The identified peptides can flatten the rounded top of a hemispherical droplet into a plane by forming a macroscopic two-dimensional crystal structure. Such ordering is driven by the folding geometry of the peptide, interactions of tyrosine and crosslinked stabilization by cysteine. We discover the key sequence motifs and folding structures and study their sequence-specific assembly. The well-ordered, densely packed, redox-active tyrosine units in the YYACAYY (H-Tyr-Tyr-Ala-Cys-Ala-Tyr-Tyr-OH) film can trigger or enhance chemical/electrochemical reactions, and can potentially serve as a platform to fabricate a molecularly tunable, self-repairable, flat peptide or hybrid film.

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