24 research outputs found
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
Vertex-Unfoldings of Simplicial Polyhedra
We present two algorithms for unfolding the surface of any polyhedron, all of
whose faces are triangles, to a nonoverlapping, connected planar layout. The
surface is cut only along polyhedron edges. The layout is connected, but it may
have a disconnected interior: the triangles are connected at vertices, but not
necessarily joined along edges.Comment: 10 pages; 7 figures; 8 reference
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
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
Compactness Determines the Success of Cube and Octahedron Self-Assembly
Nature utilizes self-assembly to fabricate structures on length scales ranging from the atomic to the macro scale. Self-assembly has emerged as a paradigm in engineering that enables the highly parallel fabrication of complex, and often three-dimensional, structures from basic building blocks. Although there have been several demonstrations of this self-assembly fabrication process, rules that govern a priori design, yield and defect tolerance remain unknown. In this paper, we have designed the first model experimental system for systematically analyzing the influence of geometry on the self-assembly of 200 and 500 µm cubes and octahedra from tethered, multi-component, two-dimensional (2D) nets. We examined the self-assembly of all eleven 2D nets that can fold into cubes and octahedra, and we observed striking correlations between the compactness of the nets and the success of the assembly. Two measures of compactness were used for the nets: the number of vertex or topological connections and the radius of gyration. The success of the self-assembly process was determined by measuring the yield and classifying the defects. Our observation of increased self-assembly success with decreased radius of gyration and increased topological connectivity resembles theoretical models that describe the role of compactness in protein folding. Because of the differences in size and scale between our system and the protein folding system, we postulate that this hypothesis may be more universal to self-assembling systems in general. Apart from being intellectually intriguing, the findings could enable the assembly of more complicated polyhedral structures (e.g. dodecahedra) by allowing a priori selection of a net that might self-assemble with high yields