This paper describes a procedure for making topologically complex three-dimensional microfluidic channel systems in poly(dimethylsiloxane) (PDMS). This procedure is called the "membrane sandwich" method to suggest the structure of the final system: a thin membrane having channel structures molded on each face (and with connections between the faces) sandwiched between two thicker, flat slabs that provide structural support. Two "masters" are fabricated by rapid prototyping using two-level photolithography and replica molding. They are aligned face to face, under pressure, with PDMS prepolymer between them. The PDMS is cured thermally. The masters have complementary alignment tracks, so registration is straightforward. The resulting, thin PDMS membrane can be transferred and sealed to another membrane or slab of PDMS by a sequence of steps in which the two masters are removed one at a time; these steps take place without distortion of the features. This method can fabricate a membrane containing a channel that crosses over and under itself, but does not intersect itself and, therefore, can be fabricated in the form of any knot. It follows that this method can generate topologically complex microfluidic systems; this capability is demonstrated by the fabrication of a "basketweave" structure. By filling the channels and removing the membrane, complex microstructures can be made. Stacking and sealing more than one membrane allows even more complicated geometries than are possible in one membrane. A square coiled channel that surrounds, but does not connect to, a straight channel illustrates this type of complexity.

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Abstract. Spatial normalization is a key process in cross-sectional studies of brain structure and function using MRI, fMRI, PET and other imaging techniques. A wide range of 3D image deformation algorithms have been developed, all of which involve design choices that are subject to debate. Moreover, most have numerical parameters whose value must be specified by the user. This paper proposes a principled method for evaluating design choices and choosing parameter values. This method can also be used to compare competing spatial normalization algorithms. We demonstrate the method through a performance analysis of a particular nonaffine deformation algorithm, ANIMAL.

Regret minimizing sets are a recent approach to representing a dataset D by a small subset R of size r of representative data points. The set R is chosen such that executing any top-1 query on R rather than D is minimally perceptible to any user. However, such a subset R may not exist, even for modest sizes, r. In this paper, we introduce the relaxation to k-regret minimizing sets, whereby a top-1 query on R returns a result imperceptibly close to the top-k on D.We show that, in general, with or without the relaxation, this problem is NP-hard. For the specific case of two dimensions, we give an efficient dynamic programming, plane sweep algorithm based on geometric duality to find an optimal solution. For arbitrary dimension, we give an empirically effective, greedy, randomized algorithm based on linear programming. With these algorithms, we can find subsets R of much smaller size that better summarize D, using small values of k larger than 1.

This report describes the spontaneous folding of flat elastomeric sheets, patterned with magnetic dipoles, into free-standing, 3D objects that are the topological equivalents of spherical shells. The path of the self-assembly is determined by a competition between mechanical and magnetic interactions. The potential of this strategy for the fabrication of 3D electronic devices is demonstrated by generating a simple electrical circuit surrounding a spherical cavity.folding ͉ microfabrication ͉ 3D structure ͉ soft lithography ͉ soft electronics T he strategies used to form 3D micro-and nanostructures in cells and by humans differ. Proteins, RNAs, and their aggregates, the most complex, 3D molecular structures in nature, form by the spontaneous folding of linear precursors (1). The ubiquity of this strategy reflects the efficiency with which the cell synthesizes linear precursors by sequential formation of covalent bonds. Microelectronic devices, the most complex 3D structures generated by humans, are fabricated by stacking and connecting planar layers (2). This strategy is dictated by the availability of highly developed methods for parallel microfabrication in 2D and the absence of effective, general methods for fabrication in 3D (3).Folding of connected, 2D plates [using robotics (4) or spontaneous folding (5-9)] can yield 3D microelectromechanical systems (MEMS) and microelectronic devices. We (10, 11) and others (4, 12) have explored a number of routes to small 3D shapes based on self-assembly. These strategies are still early in their development.Here, we explore a new strategy for formation of 3D objects that combines the advantages of planar microfabrication with those of 3D self-assembly. Our approach comprises four steps (Fig. 1a): (i) cutting the 3D surface of interest into connected sections that ''almost'' unfold into a plane (unpeeling a sphere as one unpeels an orange is an example); (ii) flattening this surface and projecting it onto a plane; (iii) fabricating the planar projection in the form of an elastomeric membrane patterned with magnetic dipoles; and (iv) allowing this patterned membrane to fold into an ''almost-correct'' 3D shape by self-assembly. This strategy offers the potential to transform easily patterned, functionalized planar sheets into 3D structures and devices. It also raises the problem of designing and generating stable 3D structures by decomposing and projecting these structures into 2D shapes and then balancing the shapes of 2D cuts, the placement of magnetic dipoles, and the mechanical characteristics of the membrane.Converting sheets into 3D objects by folding and creasing is a very old, remarkably interesting, and still incompletely resolved problem in applied mathematics (13-16). The inverse problemmapping the surface of a 3D shape (specifically, the surface of the Earth) onto a flat sheet-has been at the core of cartography since the times of Frisius (1508Frisius ( -1555 and Mercator (1512Mercator ( -1594. Although it is known that a flat, inextensible surface cannot fold i...

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Spatial normalization is a key process in cross-sectional studies of brain structure and function using MRI, fMRI, PET and other imaging techniques. A wide range of 2D surface and 3D image deformation algorithms have been developed, all of which involve design choices that are subject to debate. Moreover, most have numerical parameters whose value must be specified by the user. This paper proposes a principled method for evaluating design choices and choosing parameter values. This method can also be used to compare competing spatial normalization algorithms. We demonstrate the method through a performance analysis of a nonaffine registration algorithm for 3D images and a registration algorithm for 2D cortical surfaces.