Continuous Blooming of Convex Polyhedra

We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.Comment: 13 pages, 6 figure

Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings

This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S from the tangent space at v. We characterize the `cut locus' (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R^2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo

Folding Polyominoes into (Poly)Cubes

We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing faces of $Q$ to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of $P$ or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of $180^\circ$), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of $P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.Comment: 30 pages, 19 figures, full version of extended abstract that appeared in CCCG 2015. (Change over previous version: Fixed a missing reference.

Locked and unlocked smooth embeddings of surfaces

We study the continuous motion of smooth isometric embeddings of a planar surface in three-dimensional Euclidean space, and two related discrete analogues of these embeddings, polygonal embeddings and flat foldings without interior vertices, under continuous changes of the embedding or folding. We show that every star-shaped or spiral-shaped domain is unlocked: a continuous motion unfolds it to a flat embedding. However, disks with two holes can have locked embeddings that are topologically equivalent to a flat embedding but cannot reach a flat embedding by continuous motion.Comment: 8 pages, 8 figures. To appear in 34th Canadian Conference on Computational Geometr

On Folding: Towards a New Field of Interdisciplinary Research

It is only recently, with the increasing interest in origami and folding in natural sciences and the humanities, that the fold as a new conception in a whole range of disciplines has begun to be conceived in a broader way. Folding as a material and structural process offers a new methodology to think about the close relationship of matter, form and code. It henceforth crosses out old dichotomies, such as the organic and the inorganic or nature and technology, and blurs the boundaries between experimental, conceptual and historical approaches. This anthology aims to unfold this new interdisciplinary field and its disciplinary impact, ranging from materials science, biology, architecture, and mathematics to literature and philosophy

Amorphous mirror coatings for ultra-high precision interferometry

The dominant noise source in aLIGO is Brownian thermal noise, due to mechanical losses in the atomic structure of the amorphous titania doped tantala end test-mass mirror coatings. This thesis investigates the structural source of these losses. The effect of titania doping and thermal annealing upon the atomic structure of amorphous tantalum pentoxide coating preparations are studied using advanced electron diffraction techniques. Significant differences between the coating atomic structures have been identified for the first time in detail. The tantala based coatings studied have been demonstrated as better described by a heterogeneous phase separated model, rather than the continuous random network model for covalently bonded amorphous metal-oxides. The short-range ordering (SRO) of the coating atomic structures was investigated using pair-distribution function analyses, with an upper limit found to be ~4 Å. Correlations spanned ~9 Å, and have been related to model structures; between 4 - 5 Å, correlations were identified as signatures for 3D structural ordering. Fluctuation Electron Microscopy (FEM) was employed to investigate the MRO of the coating atomic structures. A novel approach to FEM was developed by the author during this PhD, in which the structural variance was computed using normalised cross-correlation coefficients. This made absolute intensity irrelevant, with the shape and the spatial distribution of the diffracted intensity taking precedence. The method is insensitive to poor SNR, illumination conditions, slight differences in experimental facility, and slight thickness variations in the samples. Virtual Dark-Field (VDF) imaging was adapted to amorphous structures in novel ways for the first time in this thesis. Simultaneous representation of the FEM data in real and reciprocal space, spatially resolved the structures responsible for the FEM signal. Correlation analyses were performed between VDF images of the structural ordering that relate to specific atom-pair correlations, including the use of novel annular variance images. The images and correlations clearly highlight the heterogeneous ordering and phase separation within the structures. Mechanisms responsible for the coating mechanical losses have been proposed, relating to the MRO, tensile-stress, as well as its reduction by titanium doping

On Computer Stereo Vision with Wire Frame Models

Coordinated Science Laboratory changed its name from Control Systems LaboratoryShould have been numbered UILU-ENG 77-2252, and that number may have been distributed on some copies.Joint Services Electronics Program / DAAB-07-72-C-0259Ope