20 Result(s)
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Chapter and Conference Paper
Negative Instance for the Edge Patrolling Beacon Problem
Can an infinite-strength magnetic beacon always “catch” an iron ball, when the beacon is a point required to be remain nonstrictly outside a polygon, and the ball is a point always moving instantaneously and m...
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Chapter and Conference Paper
Packing Cube Nets into Rectangles with O(1) Holes
We show that the 11 hexomino nets of the unit cube (using arbitrarily many copies of each) can pack disjointly into an \(m \times n\) ...
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Chapter and Conference Paper
Toward Unfolding Doubly Covered n-Stars
We present nonoverlap** general unfoldings of two infinite families of nonconvex polyhedra, or more specifically, zero-volume polyhedra formed by double-covering an n-pointed star polygon whose triangular point...
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Chapter and Conference Paper
Universal Hinge Patterns for Folding Strips Efficiently into Any Grid Polyhedron
We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid—for example, the surface of any polycube. The folding is e...
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Chapter and Conference Paper
Dissection with the Fewest Pieces is Hard, Even to Approximate
We prove that it is NP-hard to dissect one simple orthogonal polygon into another using a given number of pieces, as is approximating the fewest pieces to within a factor of
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Chapter and Conference Paper
Continuous Flattening of Orthogonal Polyhedra
Can we flatten the surface of any 3-dimensional polyhedron P without cutting or stretching? Such continuous flat folding motions are known when P is convex, but the question remains open for nonconvex polyhedra. ...
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Chapter and Conference Paper
Polynomial-Time Algorithm for Sliding Tokens on Trees
Suppose that we are given two independent sets I \(_{b}\) and I ...
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Chapter and Conference Paper
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles (from among {0,180°, 360°}) be folded flat to lie in an infinitesimally thick line, without crossings? This problem generalizes the cla...
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Chapter
Variations on Instant Insanity
In one of the first papers about the complexity of puzzles, Robertson and Munro [14] proved that a generalized form of the then-popular Instant Insanity puzzle is NP-complete. Here we study several variations ...
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Chapter
Meshes Preserving Minimum Feature Size
The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to ...
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Chapter and Conference Paper
Folding Equilateral Plane Graphs
We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical con...
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Chapter and Conference Paper
Common Unfoldings of Polyominoes and Polycubes
This paper studies common unfoldings of various classes of polycubes, as well as a new type of unfolding of polyominoes. Previously, Knuth and Miller found a common unfolding of all tree-like tetracubes. By co...
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Chapter and Conference Paper
Algorithms for Solving Rubik’s Cubes
The Rubik’s Cube is perhaps the world’s most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik’s Cube also has a rich u...
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Chapter and Conference Paper
Making Polygons by Simple Folds and One Straight Cut
We give an efficient algorithmic characterization of simple polygons whose edges can be aligned onto a common line, with nothing else on that line, by a sequence of all-layers simple folds. In particular, such...
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Chapter and Conference Paper
Minimal Locked Trees
Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomial-time characterization. This paper examines the properties needed for planar trees to l...
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Chapter and Conference Paper
Deflating the Pentagon
In this paper we consider deflations (inverse pocket flips) of n-gons for small n. We show that every pentagon can be deflated after finitely many deflations, and that any infinite deflation sequence of a pentago...
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Chapter and Conference Paper
Hinged Dissection of Polypolyhedra
This paper presents a general family of 3D hinged dissections for polypolyhedra, i.e., connected 3D solids formed by joining several rigid copies of the same polyhedron along identical faces. (Such joinings are p...
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Chapter and Conference Paper
When Can You Fold a Map?
We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are se...
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Chapter and Conference Paper
Balanced k-Colorings
While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k ≥ 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The...
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Chapter and Conference Paper
Planar Drawings of Origami Polyhedra
This work studies the structure of origami bases via graph drawings of origami polyhedra. In particular, we propose a new class of polyhedra, called extreme-base polyhedra, that capture the essence of “extreme...
