DOCSLIB.ORG

CUBE-NETS AS a STUDENT MATH RESEARCH PROJECT Reginald Luke, Ph.D

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

CUBE-NETS AS A STUDENT MATH RESEARCH PROJECT Reginald Luke, Ph.D. with students, Mara Olivares and Sindhu Murthy Introduction and Background qRetired from community college (Middlesex CC) teaching, mainly, precalculus, calculus, and linear algebra, and adjuncting at Rutgers University in graduate statistics. qReflecting on student math research work as a few enterprising students had approached me to engage in the doing research beyond typical homework. qFirst, Mara Olivares, who got involved with cutting cubes and counting paths. She was interested in a NASA scholarship and later transferred to Montclair U majoring in math education. qLater, Sindhu Murthy, who focused on trees and symmetry, and presented her math results on a Science Day at MCC. She graduated MCC as valedictorian and transferred to Rutgers U. The Cube-Nets Puzzle from NCTM Illuminations website page qInteractive geometric puzzle for elementary and middle-school students (from https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Cube- Nets/ qParenthetically, the site states that there are 11 different (non-isometric)Cube- Net configurations, which we labelled C1 to C11. qThe question I posed to my students was: Why are there 11 such configurations? qThe first student researcher, Mara, began cutting edges of a cube and tracing paths which ultimately allowed an unfolding of the cube into a planar polyhedron, 6 contiguous squares, the Cube-Nets designated C1 to C11. The 11 Cube-Net Configurations, C1 to C11 Trees and Minimum Spanning Trees qWe realized that the cube cutting paths were spanning trees that covered all 8 vertices of the cube, since each corner of the cube had to be unzipped to become planar. No loops were allowed since the cube-nets were connected polyhedral. The spanning tree had to have nodes of degree 3 or less. qWe found 6 different minimum spanning trees, none with more than two deg 3 nodes and none with an even number of edges between deg 3 nodes. Surprisingly, trees with 8 vertices and 7 edges that did not make the cut include (right figure): Counting Minimum Spanning Trees Kirchoff’s Matrix-Tree Theorem (1847) Hypercube Version (Stanley, 1999) qLet G(V, E) be a graph and the qFor any hypercube, Qn (Q3 being the Laplacian matrix L = D – A, where D is 3-d cube), the determinant value from the degree matrix (with values for Kirchoff’s Theorem works out to be: each vertex along the main diagonal) and L the adjacency matrix with n 2 – n - 1 $ C(n,k) values 1 or 0, depending on whether 2 ∏!"# � there is an edge or not between the two vertices being considered (note qKirchoff is usually known for his the zeroes along the diagonal). Then connection to electrical circuits the number of spanning trees of G is and chemistry/spectroscopy. the determinant of any cofactor of L. A Simpler Case: The Tetrahedron (Smallest Platonic Solid) � −� −� −� L = D - A = −� � −� −� −� −� � −� −� −� −� � � � � � � � � � � −� −� D = � � � � A = � � � � One cofactor of L = −� � −� � � � � � � � � −� −� � � � � � � � � � with determinant value of 16. By Kirchoff’s Matrix-Tree Theorem there are 16 different ways to cut open a tetrahedron and unfold it into a planar figure, a Tetra-Net. How many are there of these? A Simpler Case: The Tetrahedron (Only two spanning sets leading to two Tetra-Nets) q Linear spanning set: How many ways qOne deg 3 vertex: There are 4 nodes to to linearly cut open a tetrahedron? position this branched spanning set. q There are 4 starting vertices possible, THUS, we have a total of 12 + 4 = 16 leading to 3 ending vertices, but each different spanning sets that can cut path is reversible. So there are 4 . 3/2 = open a tetrahedron into two possible 12 ways to linearly cut it open. configurations. ◦ The Cube-Net Case qLessons from the Tetra-Net case: The spanning tree cuttings will create the perimeter edges of the Tetra-Net (darker shaded), and leave the interior edges between contiguous triangles untouched. This will help the visualization in the case of Cube-Nets, to determine which square faces lie adjacent. qThe use of Kirchoff’s Theorem applied to the Cube leads to a 8 x 8 Laplacian matrix and requires the calculation of the determinant of a 7 x 7 cofactor. Think n 2 – n - 1 $ C(n,k) about that! We use the alternate Hypercube Formula: 2 ∏!"# � = 24 (13 )(23)(31) = 384 different Cube spanning trees. qThis result already appears in Tulley (2012) and uses a more sophisticated technique. For us, the question was how the 6 cube-spanning sets generating 384 different possibilities led to the 11 cube-net configurations, using only combinatorics and geometric visualization, a more simplistic approach, which we now continue. Cube- Nets: The Linear Spanning Tree, Case 1: End Nodes of Tree Cross Cube Diagonal qWe attempt to fit a linear spanning tree qThe tree diagram of possible spanning trees is shown between pairs of vertices lying across the below with the topmost branch for the diagram on left. cube’s main diagonal, as below, V1 - V8. Note that by our vertex labeling the parity of vertex Three other possible pairings are V2 – V5, numbering is alternatively odd and even. Thus, our V3 – V6 & V4 – V7. linear spanning set of 7 edges and starting at V1 cannot ever end at V3, V5 or V7. qThere are 6 different linear spanning trees (two junctions of 3 and 2 branches) possible for the specific vertex pair (V1 - V8). Since there are 4 such cross diagonal vertex pairs, we get a total 24 spanning trees for this Linear 1 case. Linear Tree 1, Cross Vertices, Cube Unfolded to C4 qSimilar to this linear V1-V8 example, all of the 24 linear L1 spanning trees, cuts open the cube to become Cube-Net C4. The cube faces are labeled to assist in the unfolding visualization. Cube- Nets: The Linear Case 2, Adjacent Vertices qAgain envision a 7-edges linear spanning tree starting at V1 and ending at an adjacent even-numbered vertex, V2, V4 & V6. We only show the graphics for V1 to V2 going first through V4. The results for traversing V6 first is similar from symmetry qThere are 4 odd and 4 even-numbered vertices, thus 16 vertex pairs. We already covered the 4 across-main diagonal one, leaving 12 for adjacent vertex pairs. Double this amount for whether we go through V4 or V6 first, so there should be 24 Cube-nets for each of the cube cuttings below. But what are they? the same? Cube Unfolding Using a Linear Spanning Tree Between Adjacent Vertices The Linear Spanning Tree L2, that connects adjacent vertices, surprisingly unfolds into two different Cube-Nets, C1 and C10. There are 24 of each configuration type. Spanning Trees with One 3-deg Node Fitting qSince this tree is not symmetric, place the degree 3 node at V1 (WLOG for any of 8 possible vertices) and fit the branches with 1, 2 and 4 edges emanating from V1 for 3! = 6 variations. Thus, depending on how the rest of the edges end up, we already have a multiplier of 48 for this spanning tree. In what follows, we find out there are 3 different ways to complete the 1, 2 & 4 edges branching emanating from V1 (WLOG) designated Types 1, 2 & 3. Spanning Trees with One 3-deg Node, Types 2 & 3, Fitting Type 2 Type 3 Spanning Tree with One 3-deg Node, with 2, 2 & 3 branch edges . qFit the deg-3 node to V1, WLOG 1 of 8 vertices, and direct the 3 edges branch along V4 (WLOG 1 of 3 directions). We find two different paths, . based on how the two 2-edge branches are placed. Both cuttings yield the same Cube-Net. Thus, 48 such spanning trees are associated withasaa with C 9. Spanning Trees with Two 3-deg Nodes Fitting ◦ This spanning tree is asymmetric, so place the 1-1-1 deg 3 node at V1 (1of 8) and node2 at V4 (WLOG 3 ways). Then the shorter 1 edge branch either goes to V3 or V5 (2 choices) and the 3-edge branch fits neatly in. Thus, this configuration has 8.3.2 = 48 different spanning trees, all leading to the C8 Cube-Net configuration. Only shown: (In other, switch 3 and 5.) Spanning Trees with Two 3-deg Nodes Fitting qA symmetric spanning tree, so place one 1-2-1 deg 3 node at V1 (1of 8) and the other at V4 (WLOG 3 ways). But this is reversible, so count 12 ways. Next at V1 set the 1-edge branch to V6 (or V2, so 2 ◦ c ways). The 2-edge branch at V2 must go to V7, not V3 Thus, 24 variations so far. But in completing the 2nd 1-2-1 node we find two possibilities, as shown on the right, leading to different Cube-Nets, given on the next slide, with 24 spanning trees each embeddable within the cube. Cube-Nets for Spanning Tree with Two 3-deg Nodes: Fitting Spanning Trees with Two 3-deg Nodes, Fitting the last one . q Place one node of this symmetric spanning tree at V1 and the longer branch towards V4 (1 of 3 choices). The other node must end up at V8, so there are 4 ways in the cube that these nodes can be paired. So 12 variations are possible. However, at ◦ . V4 the 3-edge branch can alternatively visit V3 or V5, given another factor of 2, for a total of 24 ways to embed the spanning tree. Surprisingly, both variations lead to Cube-Net configuration C 11. Cube-Stats: 6 Spanning Trees, 384 Variations, and 11 Cube-Nets Tree Type Variation # Cube-Net Diagram Linear: End Nodes Cross Diagonal 24 C 4 Linear: End Nodes Adjacent 24 C 1 Linear: End Nodes Adjacent 24 C 10 One 3-deg Node 1-2-4 branches 48 C 7 One 3-deg Node 1-2-4 branches 48 C 2 One 3-deg Node 1-2-4 branches 48 C 3 One 3-deg Node 2-2-3 branches 48 C 9 Two 3-deg Nodes 1-1-1-3-1 48 C 8 Two 3-deg Nodes 1-2-1-2-1 24 C 5 Two 3-deg Nodes 1-2-1-2-1 24 C 6 Two 3-deg Nodes 1-1-3-1-1 24 C 11 TOTAL 384 Relevant and Related Literature ◦ Turney (1984) forms trees from the Cube-Nets (vertex being a face and edge if two faces are contiguous) and shows how to associate the 11 Cube-Nets with ”paired” 6-node trees.
Recommended publications
  • On the Archimedean Or Semiregular Polyhedra

    On the Archimedean Or Semiregular Polyhedra

    ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces.
  • 1 Lifts of Polytopes

    1 Lifts of Polytopes

    Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe.
  • Geometry © 2000 Springer-Verlag New York Inc

    Geometry © 2000 Springer-Verlag New York Inc

    Discrete Comput Geom OF1–OF15 (2000) Discrete & Computational DOI: 10.1007/s004540010046 Geometry © 2000 Springer-Verlag New York Inc. Convex and Linear Orientations of Polytopal Graphs J. Mihalisin and V. Klee Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA {mihalisi,klee}@math.washington.edu Abstract. This paper examines directed graphs related to convex polytopes. For each fixed d-polytope and any acyclic orientation of its graph, we prove there exist both convex and concave functions that induce the given orientation. For each combinatorial class of 3-polytopes, we provide a good characterization of the orientations that are induced by an affine function acting on some member of the class. Introduction A graph is d-polytopal if it is isomorphic to the graph G(P) formed by the vertices and edges of some (convex) d-polytope P. As the term is used here, a digraph is d-polytopal if it is isomorphic to a digraph that results when the graph G(P) of some d-polytope P is oriented by means of some affine function on P. 3-Polytopes and their graphs have been objects of research since the time of Euler. The most important result concerning 3-polytopal graphs is the theorem of Steinitz [SR], [Gr1], asserting that a graph is 3-polytopal if and only if it is planar and 3-connected. Also important is the related fact that the combinatorial type (i.e., the entire face-lattice) of a 3-polytope P is determined by the graph G(P). Steinitz’s theorem has been very useful in studying the combinatorial structure of 3-polytopes because it makes it easy to recognize the 3-polytopality of a graph and to construct graphs that represent 3-polytopes without producing an explicit geometric realization.
  • Graph Theory

    Graph Theory

    1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently.
  • Archimedean Solids

    Archimedean Solids

    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat).
  • Graph Mining: Social Network Analysis and Information Diffusion

    Graph Mining: Social Network Analysis and Information Diffusion

    Graph Mining: Social network analysis and Information Diffusion Davide Mottin, Konstantina Lazaridou Hasso Plattner Institute Graph Mining course Winter Semester 2016 Lecture road Introduction to social networks Real word networks’ characteristics Information diffusion GRAPH MINING WS 2016 2 Paper presentations ▪ Link to form: https://goo.gl/ivRhKO ▪ Choose the date(s) you are available ▪ Choose three papers according to preference (there are three lists) ▪ Choose at least one paper for each course part ▪ Register to the mailing list: https://lists.hpi.uni-potsdam.de/listinfo/graphmining-ws1617 GRAPH MINING WS 2016 3 What is a social network? ▪Oxford dictionary A social network is a network of social interactions and personal relationships GRAPH MINING WS 2016 4 What is an online social network? ▪ Oxford dictionary An online social network (OSN) is a dedicated website or other application, which enables users to communicate with each other by posting information, comments, messages, images, etc. ▪ Computer science : A network that consists of a finite number of users (nodes) that interact with each other (edges) by sharing information (labels, signs, edge directions etc.) GRAPH MINING WS 2016 5 Definitions ▪Vertex/Node : a user of the social network (Twitter user, an author in a co-authorship network, an actor etc.) ▪Edge/Link/Tie : the connection between two vertices referring to their relationship or interaction (friend-of relationship, re-tweeting activity, etc.) Graph G=(V,E) symbols meaning V users E connections deg(u) node degree:
  • Steinitz's Theorem Project Report §1 Introduction §2 Basic Definitions

    Steinitz's Theorem Project Report §1 Introduction §2 Basic Definitions

    Steinitz's Theorem Project Report Jon Hillery May 17, 2019 §1 Introduction Looking at the vertices and edges of polyhedra gives a family of graphs that we might expect has nice properties. It turns out that there is actually a very nice characterization of these graphs! We can use this characterization to find useful representations of certain graphs. §2 Basic Definitions We define a space to be convex if the line segment connecting any two points in the space remains entirely inside the space. This works for two-dimensional sets: and three-dimensional sets: Given a polyhedron, we define its 1-skeleton to be the graph formed from the vertices and edges of the polyhedron. For example, the 1-skeleton of a tetrahedron is K4: 1 Jon Hillery (May 17, 2019) Steinitz's Theorem Project Report Here are some further examples of the 1-skeleton of an icosahedron and a dodecahedron: §3 Properties of 1-Skeletons What properties do we know the 1-skeleton of a convex polyhedron must have? First, it must be planar. To see this, imagine moving your eye towards one of the faces until you are close enough that all of the other faces appear \inside" the face you are looking through, as shown here: This is always possible because the polyedron is convex, meaning intuitively it doesn't have any parts that \jut out". The graph formed from viewing in this way will have no intersections because the polyhedron is convex, so the straight-line rays our eyes see are not allowed to leave via an edge on the boundary of the polyhedron and then go back inside.
  • Edge Energies and S S of Nan O Pred P I Tatt

    Edge Energies and S S of Nan O Pred P I Tatt

    SAND2007-0238 .. j c..* Edge Energies and S s of Nan o pred p i tatt dexim 87185 and Llverm ,ultIprogramlabora lartln Company, for Y'S 11 Nuclear Security Admini 85000. Approved for Inallon unlimited. Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government, nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, make any warranty, express or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefidness of any information, apparatus, product, or process disclosed, or represent that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government, any agency thereof, or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof, or any of their contractors. Printed in the United States of America. This report has been reproduced directly from the best available copy. Available to DOE and DOE contractors from U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN 3783 1 Telephone: (865) 576-8401 Facsimile: (865) 576-5728 E-Mail: reDorts(cL.adonls.os(I.oov Online ordering: htto://www.osti.ao\ /bridge Available to the public from U.S.
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm

    15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm

    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
  • What Is a Polyhedron?

    What Is a Polyhedron?

    3. POLYHEDRA, GRAPHS AND SURFACES 3.1. From Polyhedra to Graphs What is a Polyhedron? Now that we’ve covered lots of geometry in two dimensions, let’s make things just a little more difficult. We’re going to consider geometric objects in three dimensions which can be made from two-dimensional pieces. For example, you can take six squares all the same size and glue them together to produce the shape which we call a cube. More generally, if you take a bunch of polygons and glue them together so that no side gets left unglued, then the resulting object is usually called a polyhedron.1 The corners of the polygons are called vertices, the sides of the polygons are called edges and the polygons themselves are called faces. So, for example, the cube has 8 vertices, 12 edges and 6 faces. Different people seem to define polyhedra in very slightly different ways. For our purposes, we will need to add one little extra condition — that the volume bound by a polyhedron “has no holes”. For example, consider the shape obtained by drilling a square hole straight through the centre of a cube. Even though the surface of such a shape can be constructed by gluing together polygons, we don’t consider this shape to be a polyhedron, because of the hole. We say that a polyhedron is convex if, for each plane which lies along a face, the polyhedron lies on one side of that plane. So, for example, the cube is a convex polyhedron while the more complicated spec- imen of a polyhedron pictured on the right is certainly not convex.
  • Graphs of Polyhedra and the Theorem of Steinitz by António Guedes De Oliveira*

    Graphs of Polyhedra and the Theorem of Steinitz by António Guedes De Oliveira*

    Graphs of polyhedra and the theorem of Steinitz by António Guedes de Oliveira* The theorem of Steinitz characterizes in simple terms the graphs of the polyhedra. In fact, the characteristic properties of such graphs, according to the theorem, are not only simple but “very natural”, in that they occur in various different contexts. As a consequence, for exam- ple, polyhedra and typical polyhedral constructions can be used for finding rectangles that can be decomposed in non-congruent squares (see Figure 1). The extraordi- nary theorem behind this relation is due to Steinitz and is the main topic of the present paper. Steinitz’s theorem was first published in a scientific encyclopaedia, in 1922 [16], and later, in 1934, in a book [17], after Steinitz’s death. It was ignored for a long time, but after “being discovered” its importance never ceased to increase and it is the starting point for active research even to our days. In the middle of the last century, two very important books were published in Polytope The- ory. The first one, by Alexander D. Alexandrov, which was published in Russian in 1950 and in German, under the title “Konvexe Polyeder” [1], in 1958, does not men- Figure 1.— Decomposition of a rectangle in tion this theorem. The second one, by Branko Grünbaum, non-congruent squares “Convex Polytopes”, published for the first time in 1967 (and dedicated exactly to the “memory of the extraordi- nary geometer Ernst Steinitz”), considers this theorem as the “most important and the deepest of the known results about polyhedra” [9, p.
  • 1 Bipartite Matching and Vertex Covers

    1 Bipartite Matching and Vertex Covers

    ORF 523 Lecture 6 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a [email protected]. In this lecture, we will cover an application of LP strong duality to combinatorial optimiza- tion: • Bipartite matching • Vertex covers • K¨onig'stheorem • Totally unimodular matrices and integral polytopes. 1 Bipartite matching and vertex covers Recall that a bipartite graph G = (V; E) is a graph whose vertices can be divided into two disjoint sets such that every edge connects one node in one set to a node in the other. Definition 1 (Matching, vertex cover). A matching is a disjoint subset of edges, i.e., a subset of edges that do not share a common vertex. A vertex cover is a subset of the nodes that together touch all the edges. (a) An example of a bipartite (b) An example of a matching (c) An example of a vertex graph (dotted lines) cover (grey nodes) Figure 1: Bipartite graphs, matchings, and vertex covers 1 Lemma 1. The cardinality of any matching is less than or equal to the cardinality of any vertex cover. This is easy to see: consider any matching. Any vertex cover must have nodes that at least touch the edges in the matching. Moreover, a single node can at most cover one edge in the matching because the edges are disjoint. As it will become clear shortly, this property can also be seen as an immediate consequence of weak duality in linear programming. Theorem 1 (K¨onig). If G is bipartite, the cardinality of the maximum matching is equal to the cardinality of the minimum vertex cover.