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Articles 961 - 990 of 1060

Full-Text Articles in Geometry and Topology

Computational Geometry Column 40, Joseph O'Rourke Dec 2000

Computational Geometry Column 40, Joseph O'Rourke

Computer Science: Faculty Publications

It has recently been established by Below, De Loera, and Richter-Gebert that finding a minimum size (or even just a small) triangulation of a convex polyhedron is NP-complete. Their 3SAT-reduction proof is discussed.

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On Reconfiguring Tree Linkages: Trees Can Lock, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides Sep 2000

On Reconfiguring Tree Linkages: Trees Can Lock, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides

Computer Science: Faculty Publications

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an N-link tree can have 2Ω(N) equivalence classes of configurations.

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Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter Sep 2000

Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter

Mathematical Sciences Technical Reports (MSTR)

The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, T, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.

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Pushpush And Push-1 Are Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke Sep 2000

Pushpush And Push-1 Are Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that two pushing-blocks puzzles are intractable in 2D. One of our constructions improves an earlier result that established intractability in 3D [OS99] for a puzzle inspired by the game PushPush. The second construction answers a question we raised in [DDO00] for a variant we call Push-1. Both puzzles consist of unit square blocks on an integer lattice; all blocks are movable. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover when a …

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Computational Geometry Column 39, Joseph O'Rourke Aug 2000

Computational Geometry Column 39, Joseph O'Rourke

Computer Science: Faculty Publications

The resolution of a decades-old open problem is described: polygonal chains cannot lock in the plane.

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Examples, Counterexamples, And Enumeration Results For Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke Jul 2000

Examples, Counterexamples, And Enumeration Results For Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke

Computer Science: Faculty Publications

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is …

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The Possibility Of Impossible Pyramids, Thomas Q. Sibley Jun 2000

The Possibility Of Impossible Pyramids, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

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Computational Geometry Column 38, Joseph O'Rourke Apr 2000

Computational Geometry Column 38, Joseph O'Rourke

Computer Science: Faculty Publications

Recent results on curve reconstruction are described.

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Lengths Of Geodesics On Klein’S Quartic Curve, Ryan Derby-Talbot Mar 2000

Lengths Of Geodesics On Klein’S Quartic Curve, Ryan Derby-Talbot

Mathematical Sciences Technical Reports (MSTR)

A well-known and much studied Riemann surface is Klein’s quartic curve. This surface is interesting since it is the smallest complex curve with maximal symmetry. In addition to this high degree of symmetry, Klein’s quartic curve can be tiled by triangles,giving rise to a tiling group generated by reflections. Using the tiling group and the universal cover of the tiling group we are able to compile a list of the lengths of the short,simple,closed geodesics on this surface. In particular,w e are able to determine whether the geodesic loops generated by the tiling are the systoles,i.e.,the shortest closed geodesics.

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Rhombic Penrose Tilings Can Be 3-Colored, Thomas Q. Sibley, Stan Wagon Mar 2000

Rhombic Penrose Tilings Can Be 3-Colored, Thomas Q. Sibley, Stan Wagon

Mathematics Faculty Publications

No abstract provided.

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Computational Geometry Column 37, Erik D. Demaine, Joseph O'Rourke Feb 2000

Computational Geometry Column 37, Erik D. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

Open problems from the 15th Annual ACM Symposium on Computational Geometry.

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Pushpush Is Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke Jan 2000

Pushpush Is Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that a particular pushing-blocks puzzle is intractable in 2D, improving an earlier result that established intractability in 3D [OS99]. The puzzle, inspired by the game *PushPush*, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 2D by reduction from SAT.

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A Look At Biseparating Maps From An Algebraic Point Of View, Melvin Henriksen, Frank A. Smith Jan 2000

A Look At Biseparating Maps From An Algebraic Point Of View, Melvin Henriksen, Frank A. Smith

All HMC Faculty Publications and Research

In [ABN], Araujo, Beckenstein, and Narici add the capstone to a series of papers by several groups of authors by showing that if ρ is a biseparating map between two algebras of all real or complex-valued functions on realcompact spaces, then it is a continuous multiple of an isomorphism between these rings. Their proof uses relatively powerful analytic and topological techniques. In what follows, the extent to which such a result can be generalized to a wider class of algebras using algebraic techniques is investigated. We are unable, however to obtain the main result of [ABN] using these techniques.

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Investigation Of The Effectiveness Of Interface Constraints In The Solution Of Hyperbolic Second-Order Differential Equations, Paul Jerome Silva Jan 2000

Investigation Of The Effectiveness Of Interface Constraints In The Solution Of Hyperbolic Second-Order Differential Equations, Paul Jerome Silva

Theses Digitization Project

Solutions to differential equations describing the behavior of physical quantities (e.g., displacement, temperature, electric field strength) often only have finite range of validity over a subdomain. Interest beyond the subdomain often arises. As a result, the problem of making the solution compatible across the connecting subdomain interfaces must be dealt with. Four different compatibility methods are examined here for hyperbolic (time varying) second-order differential equations. These methods are used to match two different solutions, one in each subdomain along the connecting interface. The entire domain that is examined here is a unit square in the Cartesian plane. The four compatibility …

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Constructible Circles On The Unit Sphere, Blaga Slavcheva Pauley Jan 2000

Constructible Circles On The Unit Sphere, Blaga Slavcheva Pauley

Theses Digitization Project

In this paper we show how to give an intrinsic definition of a constructible circle on the sphere. The classical definition of constructible circle in the plane, using straight edge and compass is there by translated in ters of so called Lenart tools. The process by which we achieve our goal involves concepts from the algebra of Hermitian matrices, complex variables, and Sterographic projection. However, the discussion is entirely elementary throughout and hopefully can serve as a guide for teachers in advanced geometry.

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Finite Type Link Concordance Invariants, Blake Mellor Jan 2000

Finite Type Link Concordance Invariants, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components.

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Finite Type Link Homotopy Invariants Ii: Milnor's Invariants, Blake Mellor Jan 2000

Finite Type Link Homotopy Invariants Ii: Milnor's Invariants, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

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The Intersection Graph Conjecture For Loop Diagrams, Blake Mellor Jan 2000

The Intersection Graph Conjecture For Loop Diagrams, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loop. We also compute the size of the subalgebra generated by these "loop diagrams."

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Rental Harmony: Sperner's Lemma In Fair Division, Francis E. Su Dec 1999

Rental Harmony: Sperner's Lemma In Fair Division, Francis E. Su

All HMC Faculty Publications and Research

No abstract provided in this article.

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Computational Geometry Column 36, Joseph O'Rourke Dec 1999

Computational Geometry Column 36, Joseph O'Rourke

Computer Science: Faculty Publications

Two results in "computational origami" are illustrated.

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Pushpush Is Np-Hard In 3d, Joseph O'Rourke, The Smith Problem Solving Group Nov 1999

Pushpush Is Np-Hard In 3d, Joseph O'Rourke, The Smith Problem Solving Group

Computer Science: Faculty Publications

We prove that a particular pushing-blocks puzzle is intractable in 3D. The puzzle, inspired by the game PushPush, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 3D by reduction from SAT. The corresponding problem in 2D remains open.

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Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton Sep 1999

Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines …

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Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith Aug 1999

Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith

Mathematical Sciences Technical Reports (MSTR)

We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.

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Art And Geometry: Proportion And Similarity, Catherine A. Gorini Jul 1999

Art And Geometry: Proportion And Similarity, Catherine A. Gorini

Humanistic Mathematics Network Journal

No abstract provided.

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Zero-Parity Stabbing Information, Joseph O'Rourke, Irena Pashchenko Jun 1999

Zero-Parity Stabbing Information, Joseph O'Rourke, Irena Pashchenko

Computer Science: Faculty Publications

Everett et al. [EHN96, EHN97] introduced several varieties of stabbing information for the lines determined by pairs of vertices of a simple polygon P, and established their relationships to vertex visibility and other combinatorial data. In the same spirit, we define the “zero-parity (ZP) stabbing information” to be a natural weakening of their “weak stabbing information,” retaining only the distinction among {zero, odd, even > 0} in the number of polygon edges stabbed. Whereas the weak stabbing information’s relation to visibility remains an open problem, we completely settle the analogous questions for zero parity information, with three results: (1) ZP information …

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Tilings Which Split A Mirror, Jim Belk Jun 1999

Tilings Which Split A Mirror, Jim Belk

Mathematical Sciences Technical Reports (MSTR)

We consider the mirror of a reflection which consists of its subset of fixed points. We investigate a number of conditions on the tiling that guarantee that the surface splits at a mirror.

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Ossa's Theorem And Adams Covers, Robert R. Bruner Mar 1999

Ossa's Theorem And Adams Covers, Robert R. Bruner

Mathematics Faculty Research Publications

We show that Ossa’s theorem splitting ku ∧ BV for elementary abelian groups V follows from general facts about ku ∧ BZ/2 and Adams covers. For completeness, we also provide the analogous results for ko ∧ BV .

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Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton Jan 1999

Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

We have all seen many of the beautiful patterns obtained by tiling the hyperbolic plane H by repeated reflection in the sides of a "kaleidoscopic" polygon. Though there are such patterns on the sphere and the euclidean plane, these positively curved and fiat geometries lack the richness we see in the hyperbolic plane. Many of these patterns have been popularized by the beautiful art of M.C. Escher. For a list of references and a more complete discussion on the construction of artistic tilings see [6].

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Knot Theory And Wild Knots, Cherie Annette Reardon Jan 1999

Knot Theory And Wild Knots, Cherie Annette Reardon

Theses Digitization Project

No abstract provided.

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The Topological Snake Lemma And Corona Algebras, Claude Schochet Jan 1999

The Topological Snake Lemma And Corona Algebras, Claude Schochet

Mathematics Faculty Research Publications

We establish versions of the Snake Lemma from homological algebra in the context of topological groups, Banach spaces, and operator algebras. We apply this tool to demonstrate that if ƒ : B → B′ is a quasi-unital C*-map of separable C*-algebras, so that it induces a map of Corona algebras ƒ̄ : QB → QB′, and if ƒ is mono, then the induced map ƒ̄ is also mono.

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