Band Unfoldings and Prismatoids: A Counterexample

This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.Comment: 5 pages, 3 figures. v2 replaced Fig.1(b) and Fig.3 to illustrate the angles delta=(1/2)epsilon (rather than delta=epsilon

Ununfoldable Polyhedra with Convex Faces

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.Comment: 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry: Theory and Applications. Major revision with two new authors, solving the open problem about triangular face

Zipper unfolding of domes and prismoids

We study Hamiltonian unfolding—cutting a convex polyhedron along a Hamiltonian path of edges to unfold it without overlap—of two classes of polyhedra. Such unfoldings could be implemented by a single zipper, so they are also known as zipper edge unfoldings. First we consider domes, which are simple convex polyhedra. We find a family of domes whose graphs are Hamiltonian, yet any Hamiltonian unfolding causes overlap, making the domes Hamiltonian-ununfoldable. Second we turn to prismoids, which are another family of simple convex polyhedra. We show that any nested prismoid is Hamiltonian-unfoldable, and that for general prismoids, Hamiltonian unfoldability can be tested in polynomial time.National Science Foundation (U.S.) (Origami Design for Integration of Self-assembling Systems for Engineering Innovation Grant EFRI-1240383)National Science Foundation (U.S.) (Expedition Grant CCF-1138967

Spiral Unfoldings of Convex Polyhedra

The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution

Spiral Unfoldings of Convex Polyhedra

The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution

Unfolding Prismatoids as Convex Patches: Counterexamples and Positive Results

We address the unsolved problem of unfolding prismatoids in a new context, viewing a "topless prismatoid" as a convex patch---a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for "petal unfolding" topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads a positive result for restricted prismatoids. Finally, we suggest suggest studying the unfolding of convex patches in general, and offer some possible lines of investigation.Comment: This paper was prepared for but never submitted to CCCG'12. 12 two-column pages. 27 figure

Frog foams and natural protein surfactants

Foams and surfactants are relatively rare in biology because of their potential to harm cell membranes and other delicate tissues. However, in recent work we have identified and characterized a number of natural surfactant proteins found in the foam nests of tropical frogs and other unusual sources. These proteins, and their associated foams, are relatively stable and bio-compatible, but with intriguing molecular structures that reveal a new class of surfactant activity. Here we review the structures and functional mechanisms of some of these proteins as revealed by experiments involving a range of biophysical and biochemical techniques, with additional mechanistic support coming from more recent site-directed mutagenesis studies

Distinct regulatory mechanisms balance DegP proteolysis to maintain cellular fitness during heat stress

Intracellular proteases combat proteotoxic stress by degrading damaged proteins, but their activity must be carefully controlled to maintain cellular fitness. The activity of Escherichia coli DegP, a highly conserved periplasmic protease, is regulated by substrate-dependent allosteric transformations between inactive and active trimer conformations and by the formation of polyhedral cages that confine the active sites within a proteolytic chamber. Here, we investigate how these distinct control mechanisms contribute to bacterial fitness under heat stress. We found that mutations that increase or decrease the equilibrium population of active DegP trimers reduce high-temperature fitness, that a mutation that blocks cage formation causes a mild fitness decrease, and that combining mutations that stabilize active DegP and block cage formation generates a lethal rogue protease. This lethality is suppressed by an extragenic mutation that prevents covalent attachment of an abundant outer-membrane lipoprotein to peptidoglycan and makes this protein an inhibitor of the rogue protease. Lethality is also suppressed by intragenic mutations that stabilize inactive DegP trimers. In combination, our results suggest that allosteric control of active and inactive conformations is the primary mechanism that regulates DegP proteolysis and fitness, with cage formation providing an additional layer of cellular protection against excessive protease activity.National Institutes of Health (U.S.) (grant AI-16892)Charles A. King Trust (Postdoctoral Fellowship