# Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn Diagrams

Size: px
Start display at page:

Transcription

1 Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn Diagrams Khalegh Mamakani and Frank Ruskey Dept. of Computer Science, University of Victoria, Canada. Abstract. An n-venn diagram consists of n curves drawn in the plane in such a way that each of the 2 n possible intersections of the interiors and exteriors of the curves forms a connected non-empty region. A Venn diagram is convexly-drawable if it can be drawn with all curves convex and it is simple if at most two curves intersect at any point. A Venn diagram is called polar symmetric if its stereographic projection about the infinite outer face is isomorphic to the projection about the innermost face. We outline an algorithm that shows there are exactly 46 simple convexly drawable polar-symmetric 6-Venn diagrams. Key words: Venn diagram, polar-symmetry Introduction Named after John Venn( ), who used diagrams of overlapping circles to represent propositions [], Venn diagrams are commonly used in set theory to visualize the relationship between different sets. When talking about Venn diagrams, the traditional three circles diagram with 3-fold symmetry often comes to mind (Figure (a)). Although it is not possible to use circles to draw Venn diagrams of more than 3 sets, more than 3 sets can be represented if the curves of the Venn diagram are other simple closed curves. Figure (b) shows a 5- set Venn diagram composed of 5 congruent ellipses which was discovered by Grünbaum[5], and Figure (c) shows a 7-set Venn diagram with 7-fold rotational symmetry called Adelaide which was discovered independently by Grünbaum [7] and Edwards [3]. Intensive research has been done recently on generating and drawing Venn diagrams of more than three sets, particularly in regard to symmetric Venn diagrams, which are those where rotating the diagram by 36/n degrees results in the same diagram. Henderson considered rotationally symmetric Venn diagrams and he showed that they could exist only prime number of curves [9]. Griggs, Killian, and Savage published a constructive method for producing symmetric Venn diagrams with a prime number of curves [8]. Venn diagrams exist for any number of curves and several constructions of them are known [2]. Another type of symmetry, introduced by Grünbaum [6], is called polarsymmetry which can be imagined by first projecting the diagram onto the surface of a sphere with the regions corresponding to the full and empty sets at the

2 2 Mamakani and Ruskey north and south poles, respectively. The Venn diagram is polar-symmetric if it is invariant under polar flips, meaning that the north and south hemispheres are congruent. In other words, for a polar-symmetric Venn diagram on the plane turning the diagram inside-out (the innermost face becomes the outermost face) gives the same Venn diagram. Note that all three Venn diagrams of Figure are polar-symmetric, as well as being rotationally symmetric. The only other attempt that we know of to exhaustively list some interesting class of 6-Venn diagrams occurs in the the work of Jeremy Carroll, who discovered that there are precisedly 26 such Venn diagrams where all curves can be drawn as triangles [2]. He used a brute force search algorithm for all possible face sizes of a Venn diagram. However, the problem of generating polar-symmetric six-set Venn diagrams has not been studied before. In this paper we are restricting our attention to the special (and most studied) class of Venn diagrams that are both simple and monotone (drawable with convex curves). We introduce two representations of these diagrams. Inspired by Carroll s work, an algorithm for generating all possible simple monotone polar-symmetric six-set Venn diagrams is developed an algorithm which determines that there are exactly 46 simple monotone polar-symmetric 6-Venn diagrams. Although our results are oriented towards 6-Venn diagrams, they could in principle be applied to general n-venn diagrams, but the computations will quickly become prohibitive. Nevertheless, we believe that the data structures and ideas introduced here (i.e., the representations) will be useful in further investigations, and in particular to resolving one of the main outstanding open problems in the area of Venn diagrams: is there a simple -Venn diagram? We intend to use the data structures proven useful here to attack a restricted, but natural, version of that problem: is there a simple monotone polar-symmetric -Venn diagram? The study of symmetric Venn diagrams in interesting not only because symmetry is core aspect of mathematical enquiry, but also because we are often led to diagrams of great inherent beauty. Furthermore, the geometric dual of a simple Venn diagram is a maximal planar spanning subgraph of the the hypercube, and so results about Venn diagrams often have equivalent statements as results about the hypercube. The remainder of paper is organized as follows. In the Section 2 we introduce basic definitions. Representations of simple monotone Venn diagrams are explained in Section 3. The generating algorithm and results are explained in the last two sections. 2 Basic Definitions A closed curve in the plane is simple if it doesn t intersect itself. Each simple closed curve decomposes the plane into two connected regions, the interior and the exterior. An n-venn diagram is a collection of n finitely intersecting simple closed curves C = {C, C,..., C n } in the plane, such that there are exactly 2 n nonempty and connected regions of the form X X X n, where

3 Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn 3 {} {,2} {,3} {,2,3} {2} {3} {2,3} (a) (b) (c) Fig.. (a) A 3-Venn diagram whose curves are circles. (b) A 5-Venn diagram whose curves are ellipses. (c) A symmetric 7-Venn known as Adelaide. Xi is either the unbounded open exterior or open bounded interior of curve Ci. Each connected region corresponds to a subset of the set {,,..., n }. Two Venn diagrams are isomorphic if one of them can be changed into the other or its mirror image by a continuous transformation of the plane. A k-region in a diagram is a region that is in the interior of precisely k curves. In an n-venn diagram, each k-region corresponds to a k-element subset of a set with n elements. So, there are nk k-regions. A Venn diagram is monotone if every k-region is adjacent to both some (k )-region (if k > ) and also to some (k + )-region (if k < n). A diagram is monotone if and only if it is drawable with each curve convex []. A simple Venn diagram is one in which exactly two curves cross each other at each intersection point. Figure 2 shows a simple monotone 6-Venn diagram. Consider a Venn diagram as being projected onto the surface of a unit sphere where the empty region of the diagram encloses the north pole of the sphere and the innermost region contains the south pole. A cylindrical projection of the

4 4 Mamakani and Ruskey Fig. 2. A simple monotone 6-Venn diagram. Venn diagram can be obtained by mapping the surface of sphere to a 2π by 2 rectangle on the plane, where the equator of the sphere maps to a horizontal line of length 2π and the north and south pole of the sphere are mapped to the top and bottom sides of the rectangle respectively. In this representation, the top of cylinder represents the empty region and bottom of cylinder represents the innermost region. A Venn diagram is said to be polar symmetric if it is invariant under polar flips. In the cylindrical representation the polar flip is equivalent to turning the cylinder upside-down. For a Venn diagram on the plane, a polar flip is equivalent to turning the diagram inside-out, with the innermost face becoming the outermost. Figure 3 shows the cylindrical representation of the polar symmetric 6-Venn diagram shown in Figure 2. It is known that there are exactly 6 simple monotone 7-Venn diagrams with rotational and polar symmetry [4],[3]. Fig. 3. Cylindrical representation of the 6-Venn diagram of Figure 2. A simple Venn diagram can be viewed as a planar graph where the intersection points of the Venn diagram are the vertices of the graph and the sections of

5 Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn 5 the curves that connect the intersection points are the edges of the graph. For a planar graph with f faces, v vertices and e edges, Euler s formula states that f + v = e + 2. A graph of an n-venn diagram has 2 n faces. In a simple Venn diagram each vertex of this graph has degree 4; i.e. e = 2v, so a simple n-venn diagram has 2 n 2 vertices. 3 Representing Venn Diagrams In this section we introduce two representations for simple monotone Venn diagrams. First we discuss the binary matrix representation where each in the matrix represents an intersection point of the corresponding Venn diagram. Having the matrix representation of a diagram, it is easy to check if it is a Venn diagram or not. In the second part we show how to represent simple monotone Venn diagrams using compositions. We use this representation to find all candidate diagrams. Then we filter non-venn diagrams using the matrix representation. 3. Matrix representation For a simple monotone n Venn, every -region is adjacent to the empty region. So the empty region surrounds a ring of ( n ) -regions. Since each region is started by one intersection point and ended by another one, there are ( n ) intersection points in the first ring. Similarly, every 2-region is adjacent to at least one -region. So, there are ( n 2) 2-regions that form a second ring surrounded by the first ring and which contains ( n 2) intersection points. In general, in a simple monotone n Venn diagram, there are n rings of regions, where all regions in a ring are enclosed by the same number of curves and every region in ring i, i n, is adjacent to at least one region in ring i and also to at least one region in ring i +. The number of intersection points in the i th ring is the same as the number of regions, which is ( n i). The rings have different colors in Figure (c). Thus a simple monotone n Venn diagram can be represented by a n by m binary matrix, m 2 n 2, where each in the matrix represents an intersection point in the Venn diagram. A Venn matrix has the following properties : There are ( n i) s in the i th row of the matrix, i n. There are no two adjacent s in any row or column of the matrix. A valid matrix must represent exactly 2 n 2 distinct regions of the corresponding Venn diagram. The two other regions are the outermost and the innermost regions. Figure 4 shows the matrix representation of the Venn diagram of Figures 2 and 3. The rank of a region of a Venn diagram is defined by n i= 2i X i where x i = if curve i encloses the region and x i = otherwise. Given a matrix representation P, we need to check that no two ranks are the same to check if it represents a valid Venn diagram.

6 6 Mamakani and Ruskey Fig. 4. Matrix representation of the 6-Venn diagram of Figures 2 and 3. For a given matrix P, suppose vector C = [c, c,, c n ] represents the curve labels of the corresponding diagram in cylindrical representation where c is the label of the outmost curve and c n is the label of the innermost curve. Then a region at ring i, i n, is enclosed by curves c,, c i and the rank of the region is i k= 2k c k. To get the curve labels for each region we need to update C based on the entries of matrix P at each column. For a given column j of the matrix, each entry of represents an intersection point. So for each row i, if p ij = then we need to exchange c i and c i+ to get the next curve labels. Starting from left to right with C = [,,, n ] as the initial curve labels, then we can compute the rank of each region. Matrix P represents a valid Venn diagram if we get exactly 2 n 2 regions with distinct ranks and C = [,,, n ] at the end. 3.2 Compositions In this part we introduce the other representation we use to generate Venn diagrams. In this representation, we use a sequence of non-negative integers to show the size of faces in each ring and and also to specify the position of intersection points of the next ring. Definition. Let a, a 2,, a k be non-negative integers such that : k a i = n i= Then (a, a 2,, a k ) is called a composition of n into k parts or a k composition of n. In a simple monotone n-venn diagram there are ( n i+) intersection points at ring i + that are distributed among ( n i) intersection points at ring i. So if we pick a particular point at ring i as the reference point, then we can specify the exact location of points at ring i + using a composition of ( ( n i+) into n i) parts. Definition 2. Let C(n, k) be the set of all compositions of n into k parts. For a simple monotone n-venn diagram V, starting from an arbitrary point of the first ring, we label the intersection points of V from to 2 n 2 in clockwise

7 Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn 7 direction. Let l i denote the label of the starting point at ring i. The composition representation P of V is a set of n 2 pairs of form l i, c i, where ( ) (( ) ( )) n n n l =, l i = l i +, c i C, i i + i Figure 5 shows the composition representation of the 6-Venn diagram of Figure 2 and Fig. 5. Composition representation of the 6-Venn diagram of Figures 2 and 3. We now list several observations that will help us cut down on the size of the search space. Observation For any simple monotone n-venn diagram V, the largest part of c i in the composition representation is n i. Proof. A region at ring i is enclosed by i curves above its starting and ending points. Since the size of a region is at most n and no two edges belong to the same curve [], at most n i remaining curves can be used to shape the region. As shown in Figure 6, to put p intersection points between the to end points of the region on the next ring, we need p + curves, p curves for the bottom side and two curves for the left and right sides. So, p n i. Enclosed by i curves p intersection points Fig. 6. Largest part of the composition at level i.

8 8 Mamakani and Ruskey Observation 2 In the composition representation of any simple monotone Venn diagram with more than 3 curves, there are no two non-adjacent s in c. Proof. Suppose, there is a such Venn diagram V, then the first ring of the Venn diagram will be like Figure 7, where regions A and D correspond to non-adjacent s in the composition and A D. Then A D = which contradicts the assumption that V is a Venn diagram. So in the first ring composition there are at most two s which must be adjacent. F A B C D E AF AB CD DE Fig. 7. Non-Adjacent s in the first ring composition. Observation 3 There are no two faces of size 3 adjacent to another face of size 3 in a simple monotone n Venn diagram V. Proof. There are only two cases, shown in Figure 8, that two faces of size 3 could be adjacent to a single face of size 3. However, both cases result in a two part disconnected region(the shaded region) which contradicts the fact that V is a Venn diagram. Observation 4 There are no two consecutive s in c 2 for the composition representation of any simple monotone Venn diagram. Proof. By observation 3 Definition 3. Let r, r 2 C(n, k) be two composition of of n into k parts. r and r 2 are rotationally distinct if it is not possible to get r 2 from any rotation of r or its reversal. Theorem ( 5. Let F n denote the set of all rotationally distinct compositions of n ) 2 into n parts such that for any r Fn there are no two non-adjacent parts of and all parts are less than or equal to n 2. Then all simple monotone n-venn diagrams can be generated from composition representations where the first composition c F n. Proof. Given a simple monotone n-venn diagram, suppose we get the composition representation P of V by picking a particular intersection point x in the first ring as the reference point. Now let P be another representation of V using

9 Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn 9 any other intersection point different than x as the reference point. It is clear that c in P is a rotation of c in P. Also for any composition representation P of the mirror of V the first composition c in P is a rotation of the reversal of c. By the observations and 2 the largest part of c is n 2 and there are no two non-adjacent s in c. Therefore, there is a composition c F n which is rotationally identical to c. A B C A B A C AB BC AC BC AB AC BC (a) (b) Fig. 8. Possible cases for two faces of size 3 being adjacent to a single face of size 3. 4 Generating Algorithm Given the upper/lower half of the cylindrical representation of a polar symmetric Venn diagram, one can generate the whole diagram by creating a copy of the given half, turning it upside down and rotating it until the two parts match together. So, to generate a polar symmetric monotone Venn diagram, we need only to generate the first n 2 2 compositions. Two halves of the diagram can match only if gluing them using the intersection points doesn t create any faces of size 2. Given the last composition of the upper half, for each positive part a j there are a j edges that bound the corresponding face from the bottom and there is a gap between two faces corresponding to two consecutive parts of the composition. So, we can map the composition to a bit-string where each represents a bounding edge of a face and each represents the gap between two faces. The length of bit-string is the same as the sum of all parts of the composition. In other words the composition (a, a 2,, a k ) is mapped to the following bit-string. a bits a 2 bits a k bits {}}{{}}{{}}{ We can find all matching of the two halves by computing the bitwise and of the bit-string and its reverse for all left rotations of the reverse bit-string. Any result other than means that there is at least one face of size 2 in the middle. Then for each matching we compute the matrix representation of the resulting

10 Mamakani and Ruskey diagram. The matrix can be obtained by sweeping the compositions from left to right and computing the position of each intersection point. Checking each resulting matrix for all compositions gives us all possible polar symmetric Venn diagram. Algorithm : GenPolarSymSixVenn begin foreach composition (a, a 2,, a 6 ) F 6 do foreach composition (b, b 2,, b 5 ) C(2, 5) do create the corresponding upper and lower halves for i to 2 do glue the upper and lower halves if there are no parallel edges in the diagram then compute matrix X representing the diagram if isv enn(x) then Print(X) rotate lower half one point to the left 5 results Using the exhaustive search we found 46 simple monotone polar symmetric 6-Venn diagrams. Table shows the number of Venn diagrams for each particular composition of the first level. Figure 9 shows one Venn diagram for each of those 29 compositions that have at least one Venn diagram. Except for three cases, for each pair of the first and second level compositions there is at most one Venn diagram. Each of the other three pairs give exactly two Venn diagrams. Renderings of each of the 46 diagrams may be found at the website References. B. Bultena, B. Grünbaum, F. Ruskey, Convex Drawings of Intersecting Families of Simple Closed Curves, th Canadian Conference on Computational Geometry, 999, J. Carroll, Drawing Venn triangles, Technical Report HPL-2-73, HP Labs (2). 3. A. W. F. Edwards, Seven-set Venn Diagrams with Rotational and Polar Symmetry, Combinatorics, Probability, and Computing, 7 (998) T. Cao, K. Mamakani and F. Ruskey, Symmetric Monotone Venn Diagrams with Seven Curves, Fifth internationl conference on Fun with Algorithms, Ischia Island Italy, 2. LNCS 699, B. Grünbaum, Venn Diagrams and Independent Families of Sets, Mathematics Magazine, Jan-Feb 975, B. Grünbaum, Venn Diagrams I,Geombinatorics, Volume I, Issue 4, (992) 5 2.

11 Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn Fig simple monotone polar symmetric 6-Venn diagrams, representing all possible compositions for the outermost ring.

12 2 Mamakani and Ruskey Composition Venn Diagrams Composition Venn Diagrams Table. Number of polar symmetric 6-Venn diagrams for F 6 7. B. Grünbaum, Venn Diagrams II, Geombinatorics, Volume II, Issue 2, (992) J. Griggs, C. E. Killian and C. D. Savage, Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice, Electronic Journal of Combinatorics, Volume (no. ), #R2, (24). 9. D. W. Henderson, Venn diagrams for more than four classes, American Mathematical Monthly, 7 (963) J. Venn, On the diagrammatic and mechanical representation of propositions and reasonsings, Philosophical Magazine and Journal of Science, Series 5, vol., No. 59, 88.. K. B. Chilakamarri, P. Hamburger and R. E. Pippert, Venn diagrams and planar graphs, Geometriae Dedicata, 62 (996) F. Ruskey and M. Weston, A survey of Venn diagrams, The Electronic Journal of Combinatorics, 997. Dynamic survey, Article DS5 (online). Revised 2, 25.

### Which n-venn diagrams can be drawn with convex k-gons?

Which n-venn diagrams can be drawn with convex k-gons? Jeremy Carroll Frank Ruskey Mark Weston Abstract We establish a new lower bound for the number of sides required for the component curves of simple

### Symmetric Monotone Venn Diagrams with Seven Curves

Symmetric Monotone Venn Diagrams with Seven Curves Tao Cao, Khalegh Mamakani, and Frank Ruskey Dept. of Computer Science, University of Victoria, Canada. Abstract. An n-venn diagram consists of n curves

### arxiv: v1 [cs.cg] 27 Jul 2012

A New Rose : The First Simple Symmetric 11-Venn Diagram Khalegh Mamakani and Frank Ruskey Dept. of Computer Science, University of Victoria, Canada. arxiv:1207.6452v1 [cs.cg] 27 Jul 2012 Abstract. A symmetric

### Generating Simple Convex Venn Diagrams

Generating Simple Convex Venn Diagrams Khalegh Mamakani a, Wendy Myrvold a, Frank Ruskey a a Department of Computer Science, University of Victoria, BC, Canada Abstract In this paper we are concerned with

### New Roses: Simple Symmetric Venn Diagrams with 11 and 13 Curves

Discrete Comput Geom (2014) 52:71 87 DOI 10.1007/s00454-014-9605-6 New Roses: Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh Mamakani Frank Ruskey Received: 27 July 2012 / Revised: 28 May

### THE SEARCH FOR SYMMETRIC VENN DIAGRAMS

GEOMBINATORICS 8(1999), pp. 104-109 THE SEARCH FOR SYMMETRIC VENN DIAGRAMS by Branko Grünbaum University of Washington, Box 354350, Seattle, WA 98195-4350 e-mail: grunbaum@math.washington.edu Given a family

### 2D and 3D Animation Using Rotations of a Jordan Curve Peter Hamburger Western Kentucky University Department of Mathematics

2D and 3D Animation Using Rotations of a Jordan Curve Peter Hamburger Western Kentucky University Department of Mathematics e-mail: peter.hamburger@wku.edu Edit Hepp Artist e-mail: hamburger1edit@gmail.com

### Venn s Diagrams: Maximum sets drawing

Venn s Diagrams: Maximum sets drawing Aureny Magaly Uc Miam 01107794@academ01.ccm.itesm.mx ITESM Campus Ciudad de México ABSTRACT In this paper, I tried to describe the maximum number of sets which can

### Half-Simple Symmetric Venn Diagrams

Half-Simple Symmetric Venn Diagrams Charles E. Killian ckillian@cs.duke.edu Frank Ruskey Carla D. Savage Mark Weston mweston@cs.uvic.ca savage@csc.ncsu.edu September 9, 2004 Abstract A Venn diagram is

### Venn Diagrams with Few Vertices

Venn Diagrams with Few Vertices Bette Bultena and Frank Ruskey abultena@csr.csc.uvic.ca, fruskey@csr.csc.uvic.ca Department of Computer Science University of Victoria Victoria, B.C. V8W 3P6, Canada Submitted:

### 40 What s Happening in the Mathematical Sciences

11-set Doily. Peter Hamburger s11-set doily. Top, theentire rotationally symmetric Venn diagram. Bottom, one of the 11 individual curves that makes up the diagram. (The other 10 curves are all, of course,

### Searching For Simple Symmetric Venn Diagrams

Searching For Simple Symmetric Venn Diagrams by Abdolkhalegh Ahmadi Mamakani B.Sc., Isfahan University of Technology, 1994 M.Sc., Amirkabir University of Technology, 1998 A Dissertation Submitted in Partial

### Face-balanced, Venn and polyvenn diagrams. Bette Bultena B.Sc., University of Victoria, 1995 M.Sc., University of Victoria, 1998

Face-balanced, Venn and polyvenn diagrams by Bette Bultena B.Sc., University of Victoria, 1995 M.Sc., University of Victoria, 1998 A Dissertation Submitted in Partial Fulfillment of the Requirements for

### Planar Graphs. 1 Graphs and maps. 1.1 Planarity and duality

Planar Graphs In the first half of this book, we consider mostly planar graphs and their geometric representations, mostly in the plane. We start with a survey of basic results on planar graphs. This chapter

### Doodles and doilies, non-simple symmetric Venn diagrams

Discrete Mathematics 257 (2002) 423 439 www.elsevier.com/locate/disc Doodles and doilies, non-simple symmetric Venn diagrams Peter Hamburger Department of Mathematical Sciences, Indiana University Purdue

### Elementary Planar Geometry

Elementary Planar Geometry What is a geometric solid? It is the part of space occupied by a physical object. A geometric solid is separated from the surrounding space by a surface. A part of the surface

### CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS

CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1. Define Graph. A graph G = (V, E) consists

### Planar Graphs with Many Perfect Matchings and Forests

Planar Graphs with Many Perfect Matchings and Forests Michael Biro Abstract We determine the number of perfect matchings and forests in a family T r,3 of triangulated prism graphs. These results show that

### Three applications of Euler s formula. Chapter 10

Three applications of Euler s formula Chapter 10 A graph is planar if it can be drawn in the plane R without crossing edges (or, equivalently, on the -dimensional sphere S ). We talk of a plane graph if

### Course Number: Course Title: Geometry

Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line

### EULER S FORMULA AND THE FIVE COLOR THEOREM

EULER S FORMULA AND THE FIVE COLOR THEOREM MIN JAE SONG Abstract. In this paper, we will define the necessary concepts to formulate map coloring problems. Then, we will prove Euler s formula and apply

### INTRODUCTION TO GRAPH THEORY. 1. Definitions

INTRODUCTION TO GRAPH THEORY D. JAKOBSON 1. Definitions A graph G consists of vertices {v 1, v 2,..., v n } and edges {e 1, e 2,..., e m } connecting pairs of vertices. An edge e = (uv) is incident with

### Prime Time (Factors and Multiples)

CONFIDENCE LEVEL: Prime Time Knowledge Map for 6 th Grade Math Prime Time (Factors and Multiples). A factor is a whole numbers that is multiplied by another whole number to get a product. (Ex: x 5 = ;

### Matching and Planarity

Matching and Planarity Po-Shen Loh June 010 1 Warm-up 1. (Bondy 1.5.9.) There are n points in the plane such that every pair of points has distance 1. Show that there are at most n (unordered) pairs of

### Exercise set 2 Solutions

Exercise set 2 Solutions Let H and H be the two components of T e and let F E(T ) consist of the edges of T with one endpoint in V (H), the other in V (H ) Since T is connected, F Furthermore, since T

### PERFECT FOLDING OF THE PLANE

SOOCHOW JOURNAL OF MATHEMATICS Volume 32, No. 4, pp. 521-532, October 2006 PERFECT FOLDING OF THE PLANE BY E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN Abstract. In this paper we introduced the concept of

### 1 Appendix to notes 2, on Hyperbolic geometry:

1230, notes 3 1 Appendix to notes 2, on Hyperbolic geometry: The axioms of hyperbolic geometry are axioms 1-4 of Euclid, plus an alternative to axiom 5: Axiom 5-h: Given a line l and a point p not on l,

### An Investigation of the Planarity Condition of Grötzsch s Theorem

Le Chen An Investigation of the Planarity Condition of Grötzsch s Theorem The University of Chicago: VIGRE REU 2007 July 16, 2007 Abstract The idea for this paper originated from Professor László Babai

### 2. Draw a non-isosceles triangle. Now make a template of this triangle out of cardstock or cardboard.

Tessellations The figure at the left shows a tiled floor. Because the floor is entirely covered by the tiles we call this arrangement a tessellation of the plane. A regular tessellation occurs when: The

### Introduction to Transformations. In Geometry

+ Introduction to Transformations In Geometry + What is a transformation? A transformation is a copy of a geometric figure, where the copy holds certain properties. Example: copy/paste a picture on your

### Chapter 6. Planar Orientations. 6.1 Numberings of Digraphs

Chapter 6 Planar Orientations In this chapter we will focus on algorithms and techniques used for drawing planar graphs. The algorithms we will use are based on numbering the vertices and orienting the

### 2 Geometry Solutions

2 Geometry Solutions jacques@ucsd.edu Here is give problems and solutions in increasing order of difficulty. 2.1 Easier problems Problem 1. What is the minimum number of hyperplanar slices to make a d-dimensional

### TWO CONTRIBUTIONS OF EULER

TWO CONTRIBUTIONS OF EULER SIEMION FAJTLOWICZ. MATH 4315 Eulerian Tours. Although some mathematical problems which now can be thought of as graph-theoretical, go back to the times of Euclid, the invention

### Lattice Polygon s and Pick s Theorem From Dana Paquin and Tom Davis 1 Warm-Up to Ponder

Lattice Polygon s and Pick s Theorem From Dana Paquin and Tom Davis http://www.geometer.org/mathcircles/pick.pdf 1 Warm-Up to Ponder 1. Is it possible to draw an equilateral triangle on graph paper so

### Connected Components of Underlying Graphs of Halving Lines

arxiv:1304.5658v1 [math.co] 20 Apr 2013 Connected Components of Underlying Graphs of Halving Lines Tanya Khovanova MIT November 5, 2018 Abstract Dai Yang MIT In this paper we discuss the connected components

### DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI

DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI Department of Computer Science and Engineering CS6702 - GRAPH THEORY AND APPLICATIONS Anna University 2 & 16 Mark Questions & Answers Year / Semester: IV /

### Pick s Theorem and Lattice Point Geometry

Pick s Theorem and Lattice Point Geometry 1 Lattice Polygon Area Calculations Lattice points are points with integer coordinates in the x, y-plane. A lattice line segment is a line segment that has 2 distinct

### MATH 890 HOMEWORK 2 DAVID MEREDITH

MATH 890 HOMEWORK 2 DAVID MEREDITH (1) Suppose P and Q are polyhedra. Then P Q is a polyhedron. Moreover if P and Q are polytopes then P Q is a polytope. The facets of P Q are either F Q where F is a facet

### On the null space of a Colin de Verdière matrix

On the null space of a Colin de Verdière matrix László Lovász 1 and Alexander Schrijver 2 Dedicated to the memory of François Jaeger Abstract. Let G = (V, E) be a 3-connected planar graph, with V = {1,...,

### XVIII. AMC 8 Practice Questions

XVIII. AMC 8 Practice Questions - A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures? (A) (B) 3 (C) 4 (D) 5 (E)

### 6th Bay Area Mathematical Olympiad

6th Bay Area Mathematical Olympiad February 4, 004 Problems and Solutions 1 A tiling of the plane with polygons consists of placing the polygons in the plane so that interiors of polygons do not overlap,

### Simultaneously flippable edges in triangulations

Simultaneously flippable edges in triangulations Diane L. Souvaine 1, Csaba D. Tóth 2, and Andrew Winslow 1 1 Tufts University, Medford MA 02155, USA, {dls,awinslow}@cs.tufts.edu 2 University of Calgary,

### Glossary of dictionary terms in the AP geometry units

Glossary of dictionary terms in the AP geometry units affine linear equation: an equation in which both sides are sums of terms that are either a number times y or a number times x or just a number [SlL2-D5]

### 8. The triangle is rotated around point D to create a new triangle. This looks like a rigid transformation.

2.1 Transformations in the Plane 1. True 2. True 3. False 4. False 5. True 6. False 7. True 8. The triangle is rotated around point D to create a new triangle. This looks like a rigid transformation. 9.

### Curriki Geometry Glossary

Curriki Geometry Glossary The following terms are used throughout the Curriki Geometry projects and represent the core vocabulary and concepts that students should know to meet Common Core State Standards.

### Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling Welcome to the lectures on computer graphics. We have

### 4. Simplicial Complexes and Simplicial Homology

MATH41071/MATH61071 Algebraic topology Autumn Semester 2017 2018 4. Simplicial Complexes and Simplicial Homology Geometric simplicial complexes 4.1 Definition. A finite subset { v 0, v 1,..., v r } R n

### On Graphs Supported by Line Sets

On Graphs Supported by Line Sets Vida Dujmović, William Evans, Stephen Kobourov, Giuseppe Liotta, Christophe Weibel, and Stephen Wismath School of Computer Science Carleton University cgm.cs.mcgill.ca/

### Unit 1, Lesson 1: Moving in the Plane

Unit 1, Lesson 1: Moving in the Plane Let s describe ways figures can move in the plane. 1.1: Which One Doesn t Belong: Diagrams Which one doesn t belong? 1.2: Triangle Square Dance m.openup.org/1/8-1-1-2

### The search for symmetric Venn diagrams

Geombinatorics 8(1999), 104-109. The search for symmetric Venn diagrams by Branko Grünbaum University of Washington, Box 354350, Seattle, WA 98195-4350 e-mail: grunbaum@math.washington.edu Given a family

### CLASSIFICATION OF SURFACES

CLASSIFICATION OF SURFACES YUJIE ZHANG Abstract. The sphere, Möbius strip, torus, real projective plane and Klein bottle are all important examples of surfaces (topological 2-manifolds). In fact, via the

### NESTED AND FULLY AUGMENTED LINKS

NESTED AND FULLY AUGMENTED LINKS HAYLEY OLSON Abstract. This paper focuses on two subclasses of hyperbolic generalized fully augmented links: fully augmented links and nested links. The link complements

### On the Minimum Number of Convex Quadrilaterals in Point Sets of Given Numbers of Points

On the Minimum Number of Convex Quadrilaterals in Point Sets of Given Numbers of Points Hu Yuzhong Chen Luping Zhu Hui Ling Xiaofeng (Supervisor) Abstract Consider the following problem. Given n, k N,

### A graph is finite if its vertex set and edge set are finite. We call a graph with just one vertex trivial and all other graphs nontrivial.

2301-670 Graph theory 1.1 What is a graph? 1 st semester 2550 1 1.1. What is a graph? 1.1.2. Definition. A graph G is a triple (V(G), E(G), ψ G ) consisting of V(G) of vertices, a set E(G), disjoint from

### Combinatorics: The Fine Art of Counting

Week Three Solutions Note: in these notes multiplication is assumed to take precedence over division, so 4!/2!2! = 4!/(2!*2!), and binomial coefficients are written horizontally: (4 2) denotes 4 choose

### Jordan Curves. A curve is a subset of IR 2 of the form

Jordan Curves A curve is a subset of IR 2 of the form α = {γ(x) : x [0,1]}, where γ : [0,1] IR 2 is a continuous mapping from the closed interval [0,1] to the plane. γ(0) and γ(1) are called the endpoints

### LAMC Advanced Circle October 9, Oleg Gleizer. Warm-up

LAMC Advanced Circle October 9, 2016 Oleg Gleizer prof1140g@math.ucla.edu Warm-up Problem 1 Prove that a straight line tangent to a circle is perpendicular to the radius connecting the tangency point to

### MATH DICTIONARY. Number Sense. Number Families. Operations. Counting (Natural) Numbers The numbers we say when we count. Example: {0, 1, 2, 3, 4 }

Number Sense Number Families MATH DICTIONARY Counting (Natural) Numbers The numbers we say when we count Example: {1, 2, 3, 4 } Whole Numbers The counting numbers plus zero Example: {0, 1, 2, 3, 4 } Positive

### INTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES

INTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES RACHEL CARANDANG Abstract. This paper provides an overview of the homology groups of a 2- dimensional complex. It then demonstrates a proof of the Invariance

### Math 302 Introduction to Proofs via Number Theory. Robert Jewett (with small modifications by B. Ćurgus)

Math 30 Introduction to Proofs via Number Theory Robert Jewett (with small modifications by B. Ćurgus) March 30, 009 Contents 1 The Integers 3 1.1 Axioms of Z...................................... 3 1.

### Discrete Applied Mathematics. A revision and extension of results on 4-regular, 4-connected, claw-free graphs

Discrete Applied Mathematics 159 (2011) 1225 1230 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam A revision and extension of results

### Mathematics Background

Finding Area and Distance Students work in this Unit develops a fundamentally important relationship connecting geometry and algebra: the Pythagorean Theorem. The presentation of ideas in the Unit reflects

### Assignment 4 Solutions of graph problems

Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of

### Paperclip graphs. Steve Butler Erik D. Demaine Martin L. Demaine Ron Graham Adam Hesterberg Jason Ku Jayson Lynch Tadashi Tokieda

Paperclip graphs Steve Butler Erik D. Demaine Martin L. Demaine Ron Graham Adam Hesterberg Jason Ku Jayson Lynch Tadashi Tokieda Abstract By taking a strip of paper and forming an S curve with paperclips

### The radius for a regular polygon is the same as the radius of the circumscribed circle.

Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.

### Geometry 10 and 11 Notes

Geometry 10 and 11 Notes Area and Volume Name Per Date 10.1 Area is the amount of space inside of a two dimensional object. When working with irregular shapes, we can find its area by breaking it up into

### Lecture 2 September 3

EE 381V: Large Scale Optimization Fall 2012 Lecture 2 September 3 Lecturer: Caramanis & Sanghavi Scribe: Hongbo Si, Qiaoyang Ye 2.1 Overview of the last Lecture The focus of the last lecture was to give

### The Graphs of Triangulations of Polygons

The Graphs of Triangulations of Polygons Matthew O Meara Research Experience for Undergraduates Summer 006 Basic Considerations Let Γ(n) be the graph with vertices being the labeled planar triangulation

### Discrete mathematics , Fall Instructor: prof. János Pach

Discrete mathematics 2016-2017, Fall Instructor: prof. János Pach - covered material - Lecture 1. Counting problems To read: [Lov]: 1.2. Sets, 1.3. Number of subsets, 1.5. Sequences, 1.6. Permutations,

### 3. Voronoi Diagrams. 3.1 Definitions & Basic Properties. Examples :

3. Voronoi Diagrams Examples : 1. Fire Observation Towers Imagine a vast forest containing a number of fire observation towers. Each ranger is responsible for extinguishing any fire closer to her tower

### Math 443/543 Graph Theory Notes 5: Planar graphs and coloring

Math 443/543 Graph Theory Notes 5: Planar graphs and coloring David Glickenstein October 10, 2014 1 Planar graphs The Three Houses and Three Utilities Problem: Given three houses and three utilities, can

### We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.

Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the

### Geometry Practice. 1. Angles located next to one another sharing a common side are called angles.

Geometry Practice Name 1. Angles located next to one another sharing a common side are called angles. 2. Planes that meet to form right angles are called planes. 3. Lines that cross are called lines. 4.

### The Fibonacci hypercube

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 187 196 The Fibonacci hypercube Fred J. Rispoli Department of Mathematics and Computer Science Dowling College, Oakdale, NY 11769 U.S.A. Steven

### PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES

UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,

### Rubber bands. Chapter Rubber band representation

Chapter 1 Rubber bands In the previous chapter, we already used the idea of looking at the graph geometrically, by placing its nodes on the line and replacing the edges by rubber bands. Since, however,

### CLASSIFICATION OF SURFACES

CLASSIFICATION OF SURFACES JUSTIN HUANG Abstract. We will classify compact, connected surfaces into three classes: the sphere, the connected sum of tori, and the connected sum of projective planes. Contents

### Apollonian ring packings

Apollonian ring packings Dedicated to Ron Graham Adrian Bolt Steve Butler Espen Hovland Abstract Apollonian packings are a well-known object in mathematics formed by starting with a set of three mutually

### An Eternal Domination Problem in Grids

Theory and Applications of Graphs Volume Issue 1 Article 2 2017 An Eternal Domination Problem in Grids William Klostermeyer University of North Florida, klostermeyer@hotmail.com Margaret-Ellen Messinger

### MC 302 GRAPH THEORY 10/1/13 Solutions to HW #2 50 points + 6 XC points

MC 0 GRAPH THEORY 0// Solutions to HW # 0 points + XC points ) [CH] p.,..7. This problem introduces an important class of graphs called the hypercubes or k-cubes, Q, Q, Q, etc. I suggest that before you

### Coxeter Decompositions of Hyperbolic Polygons

Europ. J. Combinatorics (1998) 19, 801 817 Article No. ej980238 Coxeter Decompositions of Hyperbolic Polygons A. A. FELIKSON Let P be a polygon on hyperbolic plane H 2. A Coxeter decomposition of a polygon

### Lofting 3D Shapes. Abstract

Lofting 3D Shapes Robby Prescott Department of Computer Science University of Wisconsin Eau Claire Eau Claire, Wisconsin 54701 robprescott715@gmail.com Chris Johnson Department of Computer Science University

### arxiv: v1 [cs.cg] 2 Jul 2016

Reversible Nets of Polyhedra Jin Akiyama 1, Stefan Langerman 2, and Kiyoko Matsunaga 1 arxiv:1607.00538v1 [cs.cg] 2 Jul 2016 1 Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan

### Crossings between curves with many tangencies

Crossings between curves with many tangencies Jacob Fox,, Fabrizio Frati, János Pach,, and Rom Pinchasi Department of Mathematics, Princeton University, Princeton, NJ jacobfox@math.princeton.edu Dipartimento

### ON SWELL COLORED COMPLETE GRAPHS

Acta Math. Univ. Comenianae Vol. LXIII, (1994), pp. 303 308 303 ON SWELL COLORED COMPLETE GRAPHS C. WARD and S. SZABÓ Abstract. An edge-colored graph is said to be swell-colored if each triangle contains

### Topology of Surfaces

EM225 Topology of Surfaces Geometry and Topology In Euclidean geometry, the allowed transformations are the so-called rigid motions which allow no distortion of the plane (or 3-space in 3 dimensional geometry).

### Moore Catholic High School Math Department

Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during

### TILING PROBLEMS: FROM DOMINOES, CHECKERBOARDS, AND MAZES TO DISCRETE GEOMETRY

TILING PROBLEMS: FROM DOMINOES, CHECKERBOARDS, AND MAZES TO DISCRETE GEOMETRY BERKELEY MATH CIRCLE 1. Looking for a number Consider an 8 8 checkerboard (like the one used to play chess) and consider 32

### Bellman s Escape Problem for Convex Polygons

Bellman s Escape Problem for Convex Polygons Philip Gibbs philegibbs@gmail.com Abstract: Bellman s challenge to find the shortest path to escape from a forest of known shape is notoriously difficult. Apart

### A Genus Bound for Digital Image Boundaries

A Genus Bound for Digital Image Boundaries Lowell Abrams and Donniell E. Fishkind March 9, 2005 Abstract Shattuck and Leahy [4] conjectured and Abrams, Fishkind, and Priebe [1],[2] proved that the boundary

### Math 311. Trees Name: A Candel CSUN Math

1. A simple path in a graph is a path with no repeated edges. A simple circuit is a circuit without repeated edges. 2. Trees are special kinds of graphs. A tree is a connected graph with no simple circuits.

### K 4,4 e Has No Finite Planar Cover

K 4,4 e Has No Finite Planar Cover Petr Hliněný Dept. of Applied Mathematics, Charles University, Malostr. nám. 25, 118 00 Praha 1, Czech republic (E-mail: hlineny@kam.ms.mff.cuni.cz) February 9, 2005

### Construction of planar triangulations with minimum degree 5

Construction of planar triangulations with minimum degree 5 G. Brinkmann Fakultät für Mathematik Universität Bielefeld D 33501 Bielefeld, Germany gunnar@mathematik.uni-bielefeld.de Brendan D. McKay Department

### Some Open Problems in Graph Theory and Computational Geometry

Some Open Problems in Graph Theory and Computational Geometry David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science ICS 269, January 25, 2002 Two Models of Algorithms Research

### Interactive Math Glossary Terms and Definitions

Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Addend any number or quantity being added addend + addend = sum Additive Property of Area the

### Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines, a line which passes through the point P

Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines, a line which passes through the point P and parallel to l, can be drawn. A triangle can be

### arxiv: v1 [math.co] 15 Apr 2018

REGULAR POLYGON SURFACES IAN M. ALEVY arxiv:1804.05452v1 [math.co] 15 Apr 2018 Abstract. A regular polygon surface M is a surface graph (Σ, Γ) together with a continuous map ψ from Σ into Euclidean 3-space

### Introduction III. Graphs. Motivations I. Introduction IV

Introduction I Graphs Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Graph theory was introduced in the 18th century by Leonhard Euler via the Königsberg