Related papers: Folding Polyominoes with Holes into a Cube
We propose a sufficient condition for invertibility of a polynomial mapping function defined on a cube or simplex. This condition is applicable to finite element analysis using curved meshes. The sufficient condition is based on an analysis…
We give short and simple proofs of what seem to be folklore results: * the maximum cardinality of the intersection of a lattice cube with an affine subspace; * the minimum number of affine subspaces needed to cover a lattice cube.
In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given…
In this paper we consider the problem of packing a fixed number of identical circles inside the unit circle container, where the packing is complicated by the presence of fixed size circular prohibited areas. Here the objective is to…
Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type…
Work of Glover and Huneke shows that a cubic graph embeds into the real projective plane if and only if it does not contain one of six topological minors called cubic projective plane obstructions. Here we classify up to equivalence the…
We show that several classes of polyhedra are joined by a sequence of O(1) refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into…
One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair…
Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…
We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.
We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the…
We give a precise definition of folded quivers and folded cluster algebras. We give many examples of including some with finite mutation structure that do not have analogues in the unfolded cases. We relate these examples to the finite…
We prove the perhaps surprising result that given any three polygonal unknots in $\R^3$, then we may form the Borromean rings out of them through rigid motions of $\R^3$ applied to the individual components together with possible scaling of…
One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater [4]. This paper proves the contrasting, but not…
Consider a curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is…
The art and science of folding intricate three-dimensional structures out of paper has occupied artists, designers, engineers, and mathematicians for decades, culminating in the design of deployable structures and mechanical metamaterials.…
All edge-to-edge tilings of the sphere by congruent regular triangles and congruent rhombi are classified as: (1) a $1$-parameter family of protosets each admitting a unique $(2a^3,3a^4)$-tiling like a triangular prism; (2) a $1$-parameter…
We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under…
We give a complete description of all convex polyhedra whose surface can be constructed from several congruent regular pentagons by folding and gluing them edge to edge. Our method of determining the graph structure of the polyhedra from a…
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…