We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.

Smoke, fog, jelly, paints, milk and shaving cream are common everyday examples of colloids 1 , a type of soft matter consisting of tiny particles dispersed in chemically distinct host media. Being abundant in nature, colloids also find increasingly important applications in science and technology, ranging from direct probing of kinetics in crystals and glasses 2 to fabrication of third-generation quantum-dot solar cells 3 . Because naturally occurring colloids have a shape that is typically determined by minimization of interfacial tension (for example, during phase separation) or faceted crystal growth 1 , their surfaces tend to have minimum-area spherical or topologically equivalent shapes such as prisms and irregular grains (all continuously deformable-homeomorphic-to spheres). Although toroidal DNA condensates and vesicles with different numbers of handles can exist 4-7 and soft matter defects can be shaped as rings 8 and knots 9 , the role of particle topology in colloidal systems remains unexplored. Here we fabricate and study colloidal particles with different numbers of handles and genus g ranging from 1 to 5. When introduced into a nematic liquid crystal-a fluid made of rod-like molecules that spontaneously align along the so-called "director" 10 -these particles induce threedimensional director fields and topological defects dictated by colloidal topology. Whereas electric fields, photothermal melting and laser tweezing cause transformations between configurations of particle-induced structures, three-dimensional nonlinear optical imaging reveals that topological charge is conserved and that the total charge of particle-induced defects always obeys predictions of the Gauss-Bonnet and Poincaré-Hopf index theorems [11][12][13] . This allows us to establish and experimentally test the procedure for assignment and summation of topological charges in threedimensional director fields. Our findings lay the groundwork for new applications of colloids and liquid crystals that range from topological memory devices 14 , through new types of self-assembly [15][16][17][18][19][20][21][22][23] , to the experimental study of low-dimensional topology 6,7,[11][12][13] .Although a coffee mug and a doughnut look different to most of us, they are topologically equivalent solid tori or handlebodies of genus g = 1, both being different from, say, balls and solid cylinders of genus g = 0, to which they cannot be smoothly morphed without cutting 11,12 . In a similar way, molecules can form topologically distinct structures including rings, knots and other molecular configurations satisfying the constraints imposed by chemical bonds 24 . Although the topology of shapes, fields and defects is important in many phenomena and in theories ranging from the nature of elementary particles to early-Universe cosmology 25,26 , topological aspects of colloidal systems (composed of particles larger than molecules and atoms but much smaller than the objects that we encounter in our everyday life) are rarely explored. Typically dealing with...

The fundamental issues of symmetry related to chirality are discussed and applied to simple situations relevant to liquid crystals. The authors show that any chiral measure of a geometric object is a pseudoscalar (invariant under proper rotations but changing sign under improper rotations) and must involve three-point correlations that only come into play when the molecule has at least four atoms. In general, a molecule is characterized by an infinite set of chiral parameters. The authors illustrate the fact that these parameters can have differing signs and can vanish at different points as a molecule is continuously deformed into its mirror image. From this it is concluded that handedness is not an absolute concept but depends on the property being observed. Within a simplified model of classical interactions, the chiral parameter of the constituent molecules that determines the macroscopic pitch of cholesterics is identified. [S0034-6861(99)00255-X] CONTENTS

We propose a heuristic explanation for the numerous non-close-packed crystal structures observed in various colloidal systems. By developing an analogy between soap froths and the soft coronas of fuzzy colloids, we provide a geometrical interpretation of the free energy of soft spheres. Within this picture, we show that the close-packing rule associated with hard-core interaction and positional entropy of particles is frustrated by a minimum-area principle associated with the soft tail and internal entropy of the soft coronas. We also discuss these ideas in terms of crystal architecture and pair distribution functions and analyze the phase diagram of a model hard-sphere-square-shoulder system within the cellular theory. We find that the A15 lattice, known to be area minimizing, is favored for a reasonable range of model parameters and so it is among the possible equilibrium states for a variety of colloidal systems. We also show that in the case of short-range convex potentials the A15 and other non-close-packed lattices coexist over a broad ranges of densities, which could make their identification difficult.

Programmable shape-shifting materials can take different physical forms to achieve multifunctionality in a dynamic and controllable manner. Although morphing a shape from 2D to 3D via programmed inhomogeneous local deformations has been demonstrated in various ways, the inverse problem-finding how to program a sheet in order for it to take an arbitrary desired 3D shape-is much harder yet critical to realize specific functions. Here, we address this inverse problem in thin liquid crystal elastomer (LCE) sheets, where the shape is preprogrammed by precise and local control of the molecular orientation of the liquid crystal monomers. We show how blueprints for arbitrary surface geometries can be generated using approximate numerical methods and how local extrinsic curvatures can be generated to assist in properly converting these geometries into shapes. Backed by faithfully alignable and rapidly lockable LCE chemistry, we precisely embed our designs in LCE sheets using advanced top-down microfabrication techniques. We thus successfully produce flat sheets that, upon thermal activation, take an arbitrary desired shape, such as a face. The general design principles presented here for creating an arbitrary 3D shape will allow for exploration of unmet needs in flexible electronics, metamaterials, aerospace and medical devices, and more.

We analyze the energetics of sphere-like micellar phases in diblock copolymers in terms of wellstudied, geometric quantities for their lattices. We argue that the A15 lattice with P m3n symmetry should be favored as the blocks become more symmetric and corroborate this through a self-consistent field theory. Because phases with columnar or bicontinuous topologies intervene, the A15 phase, though metastable, is not an equilibrium phase of symmetric diblocks. We investigate the phase diagram of branched diblocks and find that the A15 phase is stable.The ability to control the self-assembly of complex lattices by manipulating molecular architecture remains an essential aspect in the creation of new, functional materials. With only a few tunable parameters, diblock copolymer melts exhibit a wide variety of equilibrium phases which can be understood via the mean-field Gaussian chain model of "AB" diblock copolymers composed of immiscible A and B blocks [1,2]. Indeed, in a system where the A and B-blocks are otherwise identical, there are only two thermodynamic variables, φ, the volume fraction of A type monomers, and χN , where χ is the Flory-Huggins parameter characterizing the repulsive interactions between the A-and B-type monomers and N is the degree of polymerization [3]. In this letter, we present a model which predicts that the the A15 (shown in Fig. 1a) lattice of diblocks is stable relative to other sphere-like phases for sufficiently large φ or, in other words, sufficiently symmetric diblocks. We corroborate this prediction by recalculating the phase diagram for symmetric diblocks (Fig. 1b) via a self-consistent field theory (SCFT) for diblock copolymer melts [4].The "classical" diblock phases are well-understood: near the order-disorder transition (ODT), Leibler developed a Landau-like theory in the weak-segregation regime to establish the stability of a body-centered cubic (BCC) phase, a hexagonal phase of columns and a lamellar phase [5]. Moreover, Semenov's picture of spherical micelles interacting through a disordered copolymer background when φ ≪ 1 accounts for the appearance of the facecentered cubic (FCC) lattice near the ODT in the meanfield phase diagram [6]. The more exotic gyroid phase was discovered [7,8,9] and was explained successfully by Matsen and Schick via SCFT [4]. In our study of the A15 lattice, we find that the hexagonal and gyroid phases intervene and thus there should be no stable A15 lattice for simple diblocks. However, sphere-like topologies are favored by branched diblock copolymers [10,11,12] and dendritic polymers [13,14]; with this in mind we predict that sphere-like phases are stabilized and that the A15 phase is a ground state for this class of structures. By implementing, to our knowledge, the first full SCFT treatment of branched molecules (shown in Fig. 1c) we have verified our theory.In the dilute regime, Semenov's picture treats each micelle as an undistorted sphere so that the outer block extends to a spherical unit cell of radius R S . This unit-cell approximation pr...

The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress has been made in controlling and measuring colloidal inclusions in liquid crystalline phases. The topological structure of these systems is quite rich but, at the same time, subtle. Motivated by experiment and the power of topological reasoning, we review and expound upon the classification of defects in uniaxial nematic liquid crystals. Particular attention is paid to the ambiguities that arise in these systems, which have no counterpart in the much-storied XY model or the Heisenberg ferromagnet.

The helix is a ubiquitous motif for biopolymers. We propose a heuristic, entropically based model that predicts helix formation in a system of hard spheres and semiflexible tubes. We find that the entropy of the spheres is maximized when short stretches of the tube form a helix with a geometry close to that found in natural helices. Our model could be directly tested with wormlike micelles as the tubes, and the effect could be used to self-assemble supramolecular helices.