Operations on Maps. Mircea V. Diudea. Faculty of Chemistry and Chemical Engineering Babes-Bolyai
|
|
- Sabina Leslie Newton
- 5 years ago
- Views:
Transcription
1 Operations on Maps Mircea V. Diudea Faculty of Chemistry and Chemical Engineering Babes-Bolyai Bolyai University Cluj,, ROMANIA 1
2 Contents Cage Building by Map Operations 1. Capra, Ca/S 1 2. Septupling S Septupling S 2 2
3 Operations on Maps A map map M is a combinatorial representation of a closed surface. 1 Several operations on a map allow its transformation into new maps (convex( polyhedra). Platonic polyhedra: Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron 1. Pisanski, T.; Randić, M. Bridges between Geometry and Graph Theory. In: Geometry at Work, M. A. A. Notes, 2000, 53,
4 Euler Theorem on Polyhedra Any map M and its transforms by map operations will obey the Euler theorem 1 v e + f = χ = 2(1 g) χ = Euler s characteristic v = number of vertices, e = number of edges, f = number of faces, g = genus ; (g = 0 for a sphere; 1 for a torus). 1. L. Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. I. Petropolitanae Comment. Acad. Sci. I. Petropolitanae 1758, 4,
5 Platonic Solids Tetrahedron Cube Octahedron 5
6 Platonic Solids Dodecahedron Icosahedron Dual of a triangulation is always a cubic net. 6
7 Schlegel projection A projection of a sphere-like polyhedron on a plane is called a Schlegel diagram. In a polyhedron, the center of diagram is taken either a vertex,, the center of an edge or the center of a face 7
8 Cube and its dual, Octahedron Schlegel diagrams Cube Du Du Octahedron 8
9 Polygonal P s Capping Capping P s (s = 3, 4, 5) of a face is achieved as follows:1 Add a new vertex in the center of the face.. Put s -3 points on the boundary edges. Connect the central point with one vertex (the end points included) on each edge. The parent face is thus covered by: trigons (s = 3), tetragons (s = 4) pentagons (s = 5). The P 3 operation is also called stellation or (centered) triangulation. When all the faces of a map are thus operated, it is referred to as an omnicapping P s operation. 1. M. V. Diudea, Covering forms in nanostructures, Forma 2004 (submitted) 9
10 The resulting map shows the relations: P s (M ): Maps transformed as above form dual pairs : ; Vertex multiplication ratio by this dualization is always: Since: Polygonal P s Capping v = v + e = se 0 e ( e = se f = s 0 f ( s 3) e0 f 0 = s0 f0 + s 2) Du ( P 3 ( M )) = Le( M ) Du ( P 4 ( M )) = Me( Me( M )) Du ( P 5 ( M )) = Sn( M ) v ( Du) / v = d 0 0 v( Du) = f ( Ps ( M )) = s f = d v
11 P 3 Capping = Triangulation P 3 (Cube) P 3 (Dodecahedron) 11
12 P 4 Capping = Tetrangulation P 4 (Cube) 1 P 4 (Dodecahedron) 1 1. Catalan objects (i.e., duals of the Archimedean solids). 2. B. de La Vaissière, P. W. Fowler, and M. Deza, J. Chem. Inf. Comput. Sci., 2001, 41,
13 P 5 Capping = Pentangulation 1 P 5 (Cube) P 5 (Dodecahedron) 1. For other operation names see \conway_notation.html 13
14 7. Capra 1 Ca (M ) / Septupling Ca (M ) = Tr P 5 (P 5 (M )) Goldberg 2 relation: m = ( (a 2 + ab + b 2 ); a b ; a + b > 0 Le : (1, 1); m = 3 Q : (2, 0); m = 4 Ca : (2, 1); m = 7 1. M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 2003, 48, M. Goldberg, Tohoku Math. J., J 1937, 43,,
15 Capra Goldberg factors m by operations on a 3-valent 3 map Operation Identity Tripling (Leapfrog ) Quadrupling (Chamfering) Septupling (Capra) Symbol I: : (1,0) Le: (1, 1) Q: (2, 0) Ca: (2, 1) m The m factor is also related to the formula giving the volume of truncated pyramid, of height h : V = mh/3 coming from the ancient Egypt. 15
16 Ca (M)) = =Tr P 5 (P 5 (M )), )), examples Square face E 2 (M) P 5 (M) Tr P5 (P 5 (M)) 16
17 Theorem on Ca (M ) The vertex multiplication ratio in Ca (M ) is 2d irrespective of the tiling type of M. Demonstration: : Observe that, for each vertex of M, 2d 0 new vertices result in Ca (M ) and the old vertex is preserved,, thus demonstrating the theorem: 1 v = 2 2d 0 v 0 + v 0 v /v 0 = (2d 0 +1)v 0 /v 0 = 2 2d M. V. Diudea, Forma,, 2004 (submitted). 17
18 Capra, continued The transformed parameters are: 1 Ca (M ): v 1 = (2 = (2d 0 +1)v 0 =v e 0 + s 0 0 f e = 7 7e 0 = 3 3e s 0 0 f f = (s 0 +1)f 0 It involves rotation by π/2s 0 of the parent faces 1. M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 2003, 48,
19 Capra, continued Ca n (M ); Iterative operation d 0 > 3 d 0 = 3 v n = 8 8v n -1 7v n -2 ; n 2 v n = 7 n v 0 e n = 7 n e 0 = 7 n 3v 0 /2 f n = 0 f + (7 n -1) 1) v 0 /2 19
20 Capra, continued Ca insulates any face of M by its own hexagons, which are not shared with any old face (in contrast to Le or Q ). In case of a 3-valent regular map the vertex multiplication ratio by Capra is 7. Maps of d 0 > 3 are not regular anymore. 20
21 Capra - Chiral, continued Since P 5 can be done either clockwise or counter-clockwise, clockwise, it results in an enantiomeric pair: CaS (M ) and CaR (M ), in terms of the sinister/rectus stereochemical isomers Enantiomeric pair of the Ca (C ),, in Schlegel projection 21
22 Capra, examples The sequence CaS (CaS (M )) results in a twisted transform, while CaR (CaS (M )) will straighten the structure. Ca (T ); N = 28 CaR (CaS (T )); N =
23 Capra, examples The sequence Du (Op (Ca (M ))) is just the snub Sn (M), being this transform a chiral one. Ca (C ); N = 56 CaR (CaS (C )); N =
24 Capra, examples Enantiomeric pair of a Ca (SWNT) 24
25 Capra - Open The Open operation Op n can be written as: Op n (Ω (M )) = E n Ω (M ) E n Ω is the n -homeomorphic transformation of the edges bounding the parent-like faces, namely those resulted by the Ω operation. In case Ω (M ) = Ca (M ) the resulting open map has all polygons of the same (6+n) ) size.. It is the only known operation leading to a Platonic tessellation in open/infinite objects. 25
26 Capra Open, Open, continued The transformed parameters are: 1 Op (Ca (M )): )): v Op Op = v + s 0 0 f =v e 0 e Op = 9 9e 0 f Op = f -f 0 = s 0 0 f The open (transformed) maps cannot longer be embedded in a surface of genus zero (e.g.,., in the plane or the sphere). 1. M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 2003, 48,
27 Capra - Open, continued Open polyhedra are embedded in surfaces of negative Euler χ values and g > 1 The genus g of an open Capra object is calculable as: χ( Op( Ca( M ))) = vop eop + f Op = v0 e0 = 2-2g g = ( 2 v 0 + e 0 ) / 2 = f 0 The spherical character of the parent polyhedron involves the Euler relation written as: v e + f = The number of parent faces,, which equals that of open faces,, is: f 0 = 2 v0 + e0 thus demonstrating the g formula,, which can be generalised for any open object. / 2 27
28 Capra - Open, continued Lattices with g > 1 will have negative and consequently negative curvature. The genus of the five Platonic solids transformed by is: 2 (Tetrahedron); 3 (Cube); 4 (Octahedron); 6 (Dodecahedron) and 10 (Icosahedron) Op( Ca( M )) χ( Op( M )) 28
29 Capra - Open, examples Op (Ca (T )); S = N = 40; E = 54; F 7 = 12; g = 2 Op (Ca (C )); S = N = 80; E = 108; F 7 = 24; g = 3 29
30 Capra - Open, examples CaR (Op (CaS (T ))); S = N = 232; E = 330; F = 96; g = 2 CaR (Op (CaS (C ))); S = N = 464; E = 660; F = 192; g = 3; 30
31 Capra - Open, examples Op (Ca (M )) leads to all heptagon covering of negatively curved units Ca (D ); N = 140 Op (Ca (D )); S = N = 200; E = 270; F 7 = 60; g = 6 31
32 Capra - Open, examples The Fowler s phantasmagoric fullerene C 260, covered only by pentagons and heptagons and its Ca transform C 260 ; ( (I h ) CaR(C (C 260 ); ); N =
33 Capra - Open, examples CaR (Op (CaS (D))) N = 1160; E = 1650; F = 480; g = 6; S =
34 Capra - Open The building block Op (Ca (T )), enables construction of a supra-molecular (open) dodecahedron, 1 a multi torus of genus 21,, entirely covered by heptagons F 7 (Platonic tessellation) Supra-D (Op (Ca (T ))); S = N = 620; E = 900; F 7 = 240; g = 21 Ca (Supra-D (Op (Ca (T )))); S = N = 4100; g = M. V. Diudea, Forma,, 2004 (submitted) 34
35 POAV Strain Energy In the POAV1 theory 1,2 the π-orbital axis vector makes equal angles to the three σ-bonds of the sp 2 carbon: θ p = θ σπ - 90 o pyramidalization angle SE = 200(θ p ) 2 strain energy (1/3) Σθij deviation to planarity 1. R.C. Haddon, J. Am. Chem. Soc., 112, 3385 (1990). 2. R.C. Haddon, J. Phys. Chem. A, 105, 4164 (2001). 35
36 Capra - Open - Leapfrog The sequence Le (Op (Ca (M))) enables a covering by disjoint azulenic (5,7) units 1 Le (Op (Ca (T ))) N = 120; E = 174; F = 52; g = 2 Le (Op (Ca (C ))) N = 240; E = 348; F = 104; g = 3 1. M. V. Diudea, Forma,, 2004 (submitted) 36
37 Capra - Open - Leapfrog The sequence Le (Op (Ca (M))) enables a covering by disjoint azulenic (5,7) units 1 Le (Op (Ca (D ))) N = 600; E = 870; F = 260; g = 6 Supra-(azulenic azulenic)-d (Le (Op (Ca(T)))) N = 2400; g = M. V. Diudea, Forma,, 2004 (submitted) 37
38 Retro-Capra M 0 = P -5 (Tr Tr -(P5 (P5)(M (M 1 )) = E -2 (P -5a 5a(M 1 ) ) Delete the smallest faces and continue with E -2 Ca (D); N = 140 P -5a 5a(D); N = 80 38
39 Septupling Operations Formulas S M ) = TrP ( P ( M )) = ( Ca( M ) S + 2 = J ( 1, 3) ( E4 ( M )) 39
40 S 1 & S 2 Operations P 5 (C ) S 1 (C ) Op(S 1 (C )) E 4 (C ) S 2 (C ) Op 2a (S 2 (C )) 40
41 Lattice counting by the Septupling-Open sequence of map operations Parameter Operation 1 v = v0 + 2e0 + s0 f 0 = (2d 0 + 1) v e = 7e 0 f = s f f0 0 S 1 ( M ) 2 v Op = v + s0 f 0 = ( 3d 0 + 1) v e Op = 9e 0 f Op = f f = 0 s0 f0 0 Op( S 1 ( M )) 3 v = ( 2d 0 + 1) v e = 7e 0 f = s f f0 0 S 2 ( M ) 4 v Op = ( 4d + v e Op = 11e 0 f Op = s 0 f 0 0 1) 0 Op 2a ( S 2 ( M )) 41
42 Septupling S 2 S 2 (T ); N = 28 S 2 Op 2a (T ); N = 52 42
43 Septupling S 2 - Iterative SS 2 (T ); N = 196 = 4 x 7 2 SSS 2 (T ); N = 1372 = 4 x
44 Septupling S 2 S 2 (C ); N = 56 = 8 x 7 S 2 (O ); N = 54 = 6 x 9 44
45 Septupling S 2 - Racemic S 2 -Op S (C ); N = 104 S 2 -Op R (C ); N =
46 Iterative lattice counting by Septupling S ( M i, n ) Case: d 0 = 3 v e n n = = n 7 v 0 n 7 e 0 f n n = 7 1) ( s0 f 0 ) / 6 + ( f 0 46
47 Septupling S 2 Iterative Fractalization S 2 S 2 (C ); N = 392 = 8 x 7 2 S 2 S 2 S 2 (C ); N = 2744 = 8 x
48 Septupling S 2 Iterative Fractalization S 2 (D); N = 140 = 20 x 7 S 2 S 2 (D); N = = 20 x 7 2 =
49 Septupling S 2 Iterative Fractalization S 2 S 2 S 2 (D); N = 6860 = 20 x 7 = 6860 = 20 x 7 2 Two-fold symmetry S 2 S 2 S 2 (D); N = 6860 = 20 x 7 = 6860 = 20 x 7 3 Five-fold symmetry 49
50 Septupling S 2 Iterative Fractalization S 2 (I ); N = 132 = 12 x 11 S 2 S 2 (I ); N = 972 = 12 x
51 Septupling S 2 Iterative Fractalization S 2 S 2 S 2 (I ); N = 6852 Two-fold symmetry S 2 S 2 S 2 (I ); N = 6852 Five-fold symmetry 51
52 Platonic and Archimedean polyhedra (derived from Tetrahedron) Symbol Polyhedron Formula 1 T Tetrahedron - 2 O Octahedron Me(T ) 3 C Cube (hexahedron) Du(O)=Du(Me(T )) 4 I Icosahedron Sn(T ) 5 D Dodecahedron Du(I ) = Du(Sn(T )) 1 TT Truncated tetrahedron Tr(T ) 2 TO Truncated octahedron Tr(O) = Tr(Me(T )) 3 TC Truncated cube Tr(C) = Tr(Du(Me(T ))) 4 TI Truncated icosahedron Tr(I) = Tr(Sn(T)) 5 TD Truncated dodecahedron Tr(D) = Tr(Du(Sn(T ))) 6 CO Cuboctahedron Me(C) = Me(O)= Me(Me(T )) 7 ID Icosidodecahedron Me(I) = Me(D) = Me(Sn(T )) 8 RCO Rhombicuboctahedron Me(CO) = Me(Me(C)) = Du(P 4 (C )) 9 RID Rhombicosidodecahedron Me(ID) = Me(Me(I)) = Du(P 4 (I)) 10 TCO Truncated cuboctahedron Tr(CO) = Tr(Me(Me(T ))) 11 TID Truncated icosidodecahedron Tr(ID) = Tr(Me(Sn(T ))) 12 SC Snub cube Sn(C) = Du(P 5 (C)) = Du(Op(Ca(C ))) 13 SD Snub dodecahedron Sn(D) = Du(P 5 (D)) = Du(Op(Ca(D))) 52
Fantasmagoric Fulleroids Revisited
Leonardo Electronic Journal of Practices and Technologies ISSN 1583-1078 Issue 9, July-December 2006 p 193-202 Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, 400028, Cluj, Romania
More informationMa/CS 6b Class 9: Euler s Formula
Ma/CS 6b Class 9: Euler s Formula By Adam Sheffer Recall: Plane Graphs A plane graph is a drawing of a graph in the plane such that the edges are noncrossing curves. 1 Recall: Planar Graphs The drawing
More informationDate: Wednesday, 18 January :00AM. Location: Barnard's Inn Hall
Wallpaper Patterns and Buckyballs Transcript Date: Wednesday, 18 January 2006-12:00AM Location: Barnard's Inn Hall WALLPAPER PATTERNS AND BUCKYBALLS Professor Robin Wilson My lectures this term will be
More informationMap-colouring with Polydron
Map-colouring with Polydron The 4 Colour Map Theorem says that you never need more than 4 colours to colour a map so that regions with the same colour don t touch. You have to count the region round the
More informationEuler Characteristic
Euler Characteristic Rebecca Robinson May 15, 2007 Euler Characteristic Rebecca Robinson 1 PLANAR GRAPHS 1 Planar graphs v = 5, e = 4, f = 1 v e + f = 2 v = 6, e = 7, f = 3 v = 4, e = 6, f = 4 v e + f
More informationPlatonic Solids and the Euler Characteristic
Platonic Solids and the Euler Characteristic Keith Jones Sanford Society, SUNY Oneonta September 2013 What is a Platonic Solid? A Platonic Solid is a 3-dimensional object with flat faces and straight edges
More informationLecture 19: Introduction To Topology
Chris Tralie, Duke University 3/24/2016 Announcements Group Assignment 2 Due Wednesday 3/30 First project milestone Friday 4/8/2016 Welcome to unit 3! Table of Contents The Euler Characteristic Spherical
More informationOne simple example is that of a cube. Each face is a square (=regular quadrilateral) and each vertex is connected to exactly three squares.
Berkeley Math Circle Intermediate I, 1/23, 1/20, 2/6 Presenter: Elysée Wilson-Egolf Topic: Polygons, Polyhedra, Polytope Series Part 1 Polygon Angle Formula Let s start simple. How do we find the sum of
More informationof Nebraska - Lincoln
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
More informationExample: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices:
11.1: Space Figures and Cross Sections Polyhedron: solid that is bounded by polygons Faces: polygons that enclose a polyhedron Edge: line segment that faces meet and form Vertex: point or corner where
More informationWeek 7 Convex Hulls in 3D
1 Week 7 Convex Hulls in 3D 2 Polyhedra A polyhedron is the natural generalization of a 2D polygon to 3D 3 Closed Polyhedral Surface A closed polyhedral surface is a finite set of interior disjoint polygons
More informationQuestion. Why is the third shape not convex?
1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third
More informationThe Construction of Uniform Polyhedron with the aid of GeoGebra
The Construction of Uniform Polyhedron with the aid of GeoGebra JiangPing QiuFaWen 71692686@qq.com 3009827@qq.com gifted Department High-school northeast yucai school northeast yucai school 110179 110179
More informationRightStart G Learning Materials: Learning Goals/Performance Objectives: Learning Activities:
RightStart G Class Description: RightStartmath.com says "Learn intermediate mathematics hands-on and visually through geometry. With a tool set consisting of a drawing board, T-square, triangles, compass,
More informationEdge-transitive tessellations with non-negative Euler characteristic
October, 2009 p. Edge-transitive tessellations with non-negative Euler characteristic Alen Orbanić Daniel Pellicer Tomaž Pisanski Thomas Tucker Arjana Žitnik October, 2009 p. Maps 2-CELL EMBEDDING of a
More informationNon-flat tilings with flat tiles
Non-flat tilings with flat tiles Rinus Roelofs Sculptor Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: rinus@rinusroelofs.nl www.rinusroelofs.nl Abstract In general a tiling is considered to be a
More informationMath 311. Polyhedra Name: A Candel CSUN Math
1. A polygon may be described as a finite region of the plane enclosed by a finite number of segments, arranged in such a way that (a) exactly two segments meets at every vertex, and (b) it is possible
More information11.4 Three-Dimensional Figures
11. Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called
More informationPolyhedra. Kavitha d/o Krishnan
Polyhedra Kavitha d/o Krishnan Supervisor: Associate Professor Helmer Aslaksen Department of Mathematics National University of Singapore Semester I 2001/2002 Abstract Introduction The report focuses on
More informationWe 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
More informationON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA
ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA arxiv:math/0505488v1 [math.gt] 4 May 005 Mark B. Villarino Depto. de Matemática, Universidad de Costa Rica, 060 San José, Costa Rica May 11, 005 Abstract We
More informationZipper Unfoldings of Polyhedral Complexes. Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit
Zipper Unfoldings of Polyhedral Complexes Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit 1 Unfolding Polyhedra Durer 1400 s Durer, 1498 snub cube 2 Unfolding Polyhedra Octahedron all
More informationZipper Unfoldings of Polyhedral Complexes
Zipper Unfoldings of Polyhedral Complexes Erik D. Demaine Martin L. Demaine Anna Lubiw Arlo Shallit Jonah L. Shallit Abstract We explore which polyhedra and polyhedral complexes can be formed by folding
More informationAnswer Key: Three-Dimensional Cross Sections
Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection
More informationSection 9.4. Volume and Surface Area. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 9.4 Volume and Surface Area What You Will Learn Volume Surface Area 9.4-2 Volume Volume is the measure of the capacity of a three-dimensional figure. It is the amount of material you can put inside
More informationDual Models: One Shape to Make Them All
Bridges Finland Conference Proceedings Dual Models: One Shape to Make Them All Mircea Draghicescu ITSPHUN LLC mircea@itsphun.com Abstract We show how a potentially infinite number of 3D decorative objects
More informationLocal Mesh Operators: Extrusions Revisited
Local Mesh Operators: Extrusions Revisited Eric Landreneau Computer Science Department Abstract Vinod Srinivasan Visualization Sciences Program Texas A&M University Ergun Akleman Visualization Sciences
More informationINTRODUCTION 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
More informationClass Generated Review Sheet for Math 213 Final
Class Generated Review Sheet for Math 213 Final Key Ideas 9.1 A line segment consists of two point on a plane and all the points in between them. Complementary: The sum of the two angles is 90 degrees
More informationMath 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi
Math 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi 1. Euler Characteristic of Surfaces Leonhard Euler noticed that the number v of vertices, the number e of edges and
More information1. CONVEX POLYGONS. Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D.
1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third
More informationFive Platonic Solids: Three Proofs
Five Platonic Solids: Three Proofs Vincent J. Matsko IMSA, Dodecahedron Day Workshop 18 November 2011 Convex Polygons convex polygons nonconvex polygons Euler s Formula If V denotes the number of vertices
More informationAbstract Construction Projects and the Imagination
Abstract Construction Projects and the Imagination Hands-on projects for understanding abstract mathematical concepts through the use of polyhedral models and planar designs The 3-dimensional projects
More informationComputer Graphics using OpenGL, 3 rd Edition F. S. Hill, Jr. and S. Kelley
Computer Graphics using OpenGL, 3 rd Edition F. S. Hill, Jr. and S. Kelley Chapter 6.1-3 Modeling Shapes with Polygonal Meshes S. M. Lea University of North Carolina at Greensboro 2007, Prentice Hall 3D
More informationChapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings
Chapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 12: Planar Graphs Math 184A / Fall 2017 1 / 45 12.1 12.2. Planar graphs Definition
More informationExplore Solids
1212.1 Explore Solids Surface Area and Volume of Solids 12.2 Surface Area of Prisms and Cylinders 12.3 Surface Area of Pyramids and Cones 12.4 Volume of Prisms and Cylinders 12.5 Volume of Pyramids and
More informationTomáš Madaras and Pavol Široczki
Opuscula Math. 34, no. 1 (014), 13 138 http://dx.doi.org/10.7494/opmath.014.34.1.13 Opuscula Mathematica ON THE DIMENSION OF ARCHIMEDEAN SOLIDS Tomáš Madaras and Pavol Široczki Communicated by Mariusz
More informationClassifying 3D Shapes
Classifying 3D Shapes Middle School Texas Essential Knowledge and Skills (TEKS) Math 5.4B Algebraic reasoning The student applies mathematical process standards to develop concepts of expressions and equations.
More informationgorithm exists for generating topologies of fullerenes and other trivalent polyhedral cages with prescribed properties 6. The stability of a fullerene
Isoperimetric Quotient for Fullerenes and Other Polyhedral Cages Tomaz Pisanski, Matjaz Kaufman, Drago Bokal, Institut za matematiko, ziko in mehaniko, Univerza v Ljubljani Jadranska 19 1000 Ljubljana,
More informationResearches on polyhedra, Part I A.-L. Cauchy
Researches on polyhedra, Part I A.-L. Cauchy Translated into English by Guy Inchbald, 2006 from the original: A.-L. Cauchy, Recherches sur les polyèdres. Première partie, Journal de l École Polytechnique,
More informationConvex Hulls (3D) O Rourke, Chapter 4
Convex Hulls (3D) O Rourke, Chapter 4 Outline Polyhedra Polytopes Euler Characteristic (Oriented) Mesh Representation Polyhedra Definition: A polyhedron is a solid region in 3D space whose boundary is
More informationMath 462: Review questions
Math 462: Review questions Paul Hacking 4/22/10 (1) What is the angle between two interior diagonals of a cube joining opposite vertices? [Hint: It is probably quickest to use a description of the cube
More information7. The Gauss-Bonnet theorem
7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationREGULAR TILINGS. Hints: There are only three regular tilings.
REGULAR TILINGS Description: A regular tiling is a tiling of the plane consisting of multiple copies of a single regular polygon, meeting edge to edge. How many can you construct? Comments: While these
More information1 The Platonic Solids
1 The We take the celebration of Dodecahedron Day as an opportunity embark on a discussion of perhaps the best-known and most celebrated of all polyhedra the Platonic solids. Before doing so, however,
More informationNets and Tiling. Michael O'Keeffe. Introduction to tiling theory and its application to crystal nets
Nets and Tiling Michael O'Keeffe Introduction to tiling theory and its application to crystal nets Start with tiling in two dimensions. Surface of sphere and plane Sphere is two-dimensional. We require
More informationPlatonic Polyhedra and How to Construct Them
Platonic Polyhedra and How to Construct Them Tarun Biswas June 17, 2016 The platonic polyhedra (or platonic solids) are convex regular polyhedra that have identical regular polygons as faces They are characterized
More informationPolygons and Convexity
Geometry Week 4 Sec 2.5 to ch. 2 test Polygons and Convexity section 2.5 convex set has the property that any two of its points determine a segment contained in the set concave set a set that is not convex
More informationMath 366 Lecture Notes Section 11.4 Geometry in Three Dimensions
Math 366 Lecture Notes Section 11.4 Geometry in Three Dimensions Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and is hollow. A sphere is the set of all points at a
More informationEuclid forgot to require that the vertices should be the same, so his definition includes the deltahedra.
2 1. What is a Platonic solid? What is a deltahedron? Give at least one example of a deltahedron that is t a Platonic solid. What is the error Euclid made when he defined a Platonic solid? Solution: A
More informationGeometro: Developing Concepts for Math, Science and O&M with Students who are Visually Impaired October 5, 2012
Texas School for the Blind and Visually Impaired Outreach Programs www.tsbvi.edu 512-454-8631 1100 W. 45 th St. Austin, Texas 78756 Geometro: Developing Concepts for Math, Science and O&M with Students
More informationWeek 9: Planar and non-planar graphs. 7 and 9 November, 2018
(1/27) MA284 : Discrete Mathematics Week 9: Planar and non-planar graphs http://www.maths.nuigalway.ie/ niall/ma284/ 7 and 9 November, 2018 1 Planar graphs and Euler s formula 2 Non-planar graphs K 5 K
More informationThe 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.
More informationGrade VIII. Mathematics Geometry Notes. #GrowWithGreen
Grade VIII Mathematics Geometry Notes #GrowWithGreen Polygons can be classified according to their number of sides (or vertices). The sum of all the interior angles of an n -sided polygon is given by,
More informationSonobe Origami for enriching understanding of geometric concepts in three dimensions. DONNA A. DIETZ American University Washington, D.C.
Sonobe Origami for enriching understanding of geometric concepts in three dimensions DONNA A. DIETZ American University Washington, D.C. Donna Dietz, American University Sonobe Origami for enriching understanding
More informationLeonardo s Elevated Polyhedra - Models
Leonardo s Elevated Polyhedra - Models Rinus Roelofs Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: rinus@rinusroelofs.nl www.rinusroelofs.nl Information Rinus Roelofs was born in 1954. After studying
More information168 Butterflies on a Polyhedron of Genus 3
168 Butterflies on a Polyhedron of Genus 3 Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-2496, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/
More informationCourse: Geometry Year: Teacher(s): various
Course: Geometry Year: 2015-2016 Teacher(s): various Unit 1: Coordinates and Transformations Standards Essential Questions Enduring Understandings G-CO.1. Know 1) How is coordinate Geometric precise definitions
More informationTHE CLASSIFICATION OF FOOTBALL PATTERNS
THE CLASSIFICATION OF FOOTBALL PATTERNS V. BRAUNGARDT AND D. KOTSCHICK ABSTRACT. We prove that every spherical football is a branched cover, branched only in the vertices, of the standard football made
More informationConnected Holes. Rinus Roelofs Sculptor Lansinkweg AL Hengelo The Netherlands
Connected Holes Rinus Roelofs Sculptor Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: rinus@rinusroelofs.nl www.rinusroelofs.nl Abstract It is possible to make interwoven structures by using two
More informationTriangles and Squares David Eppstein, ICS Theory Group, April 20, 2001
Triangles and Squares David Eppstein, ICS Theory Group, April 20, 2001 Which unit-side-length convex polygons can be formed by packing together unit squares and unit equilateral triangles? For instance
More informationTiling of Sphere by Congruent Pentagons
Tiling of Sphere by Congruent Pentagons Min Yan September 9, 2017 webpage for further reading: http://www.math.ust.hk/ mamyan/research/urop.shtml We consider tilings of the sphere by congruent pentagons.
More informationSHAPE AND STRUCTURE. Shape and Structure. An explanation of Mathematical terminology
Shape and Structure An explanation of Mathematical terminology 2005 1 POINT A dot Dots join to make lines LINE A line is 1 dimensional (length) A line is a series of points touching each other and extending
More informationarxiv: 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
More information3.D. The Platonic solids
3.D. The Platonic solids The purpose of this addendum to the course notes is to provide more information about regular solid figures, which played an important role in Greek mathematics and philosophy.
More informationEuler Characteristic
Euler Characteristic Face Classification set_view(gl_render); set_scene(gl_render); glgetdoublev(gl_modelview_matrix, modelview_matrix1); glgetdoublev(gl_projection_matrix, projection_matrix1); glgetintegerv(gl_viewport,
More informationPlatonic Solids. Jennie Sköld. January 21, Karlstad University. Symmetries: Groups Algebras and Tensor Calculus FYAD08
Platonic Solids Jennie Sköld January 21, 2015 Symmetries: Groups Algebras and Tensor Calculus FYAD08 Karlstad University 1 Contents 1 What are Platonic Solids? 3 2 Symmetries in 3-Space 5 2.1 Isometries
More informationTorus Knots with Polygonal Faces
Proceedings of Bridges 214: Mathematics, Music, Art, Architecture, Culture Torus Knots with Polygonal Faces Chern Chuang Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 2139,
More informationProjection of Polyhedra onto Coxeter Planes Described with Quaternions
SQU Journal for Science, 214, 19(2) 77-9 214 Sultan Qaboos University of Polyhedra onto Coxeter Planes Described with Quaternions Mudhahir Al-Ajmi*, Mehmet Koca and Hashima Bait Bu Salasel Department of
More informationChapter 1. acute angle (A), (G) An angle whose measure is greater than 0 and less than 90.
hapter 1 acute angle (), (G) n angle whose measure is greater than 0 and less than 90. adjacent angles (), (G), (2T) Two coplanar angles that share a common vertex and a common side but have no common
More informationFace-regular 3-valent two-faced spheres and tori
BBBW Research Memorandum 976 2006/01/12 Face-regular 3-valent two-faced spheres and tori Mathieu DUTOUR Rudjer Bo sković Institute, Zagreb and Institute of Statistical Mathematics, Tokyo Michel DEZA LIGA,
More informationSTELLATION OF POLYHEDRA, AND COMPUTER IMPLEMENTATION
STELLATION OF POLYHEDRA, AND COMPUTER IMPLEMENTATION CHRISTOPHER J. HENRICH Abstract. The construction and display of graphic representations of stellated polyhedra requires a painstaking and close analysis
More informationMathematics Concepts 2 Exam 1 Version 4 21 September 2018
Mathematics Concepts 2 Exam 1 Version 4 21 September 2018 Name: Permissible Aides: The small ruler distributed by the proctor Prohibited: Class Notes Class Handouts Study Guides and Materials The Book
More informationDefinitions. Topology/Geometry of Geodesics. Joseph D. Clinton. SNEC June Magnus J. Wenninger
Topology/Geometry of Geodesics Joseph D. Clinton SNEC-04 28-29 June 2003 Magnus J. Wenninger Introduction Definitions Topology Goldberg s polyhedra Classes of Geodesic polyhedra Triangular tessellations
More informationDepartment of Physics, College of Science, Sultan Qaboos University P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman
Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions II Mehmet Koca a), Mudhahir Al Ajmi b) and Saleh Al- Shidhani c) Department of Physics, College of Science, Sultan
More informationMath 213 Student Note Outlines. Sections
Math 213 Student Note Outlines Sections 9.2 9.3 11.1 11.2 11.3 9.2 KEY IDEAS, page 1 of 2 Polygon Vertex Angles Sum Regular Polygons: Vertex Angles Congruence Definition Regular Polygons Definition Tessellation
More informationDoes it Look Square? Hexagonal Bipyramids, Triangular Antiprismoids, and their Fractals
Does it Look Square? Hexagonal Bipyramids, Triangular Antiprismoids, and their Fractals Hideki Tsuiki Graduate School of Human and Environmental Studies Kyoto University Yoshida-Nihonmatsu, Kyoto 606-8501,
More informationState if each pair of triangles is similar. If so, state how you know they are similar (AA, SAS, SSS) and complete the similarity statement.
Geometry 1-2 est #7 Review Name Date Period State if each pair of triangles is similar. If so, state how you know they are similar (AA, SAS, SSS) and complete the similarity statement. 1) Q R 2) V F H
More informationWeek 9: Planar and non-planar graphs. 1st and 3rd of November, 2017
(1/26) MA284 : Discrete Mathematics Week 9: Planar and non-planar graphs http://www.maths.nuigalway.ie/~niall/ma284/ 1st and 3rd of November, 2017 1 Recall... planar graphs and Euler s formula 2 Non-planar
More informationNon-extendible finite polycycles
Izvestiya: Mathematics 70:3 1 18 Izvestiya RAN : Ser. Mat. 70:3 3 22 c 2006 RAS(DoM) and LMS DOI 10.1070/IM2006v170n01ABEH002301 Non-extendible finite polycycles M. Deza, S. V. Shpectorov, M. I. Shtogrin
More informationMultiply using the grid method.
Multiply using the grid method. Learning Objective Read and plot coordinates in all quadrants DEFINITION Grid A pattern of horizontal and vertical lines, usually forming squares. DEFINITION Coordinate
More informationThe Volume of a Platonic Solid
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-007 The Volume of a Platonic Solid Cindy Steinkruger
More informationJunior Math Circles March 3, D Geometry I
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Junior Math Circles March 3, 2010 3D Geometry I Opening Problem Max received a gumball machine for his
More informationToday we will be exploring three-dimensional objects, those that possess length, width, and depth.
Lesson 22 Lesson 22, page 1 of 13 Glencoe Geometry Chapter 11.1 3-D figures & Polyhedra Today we will be exploring three-dimensional objects, those that possess length, width, and depth. In Euclidean,
More informationResearch is what I am doing when I don t know what I m doing. Wernher von Braun
Research is what I am doing when I don t know what I m doing. Wernher von Braun School of the Art Institute of Chicago Geometry of Art and Nature Frank Timmes ftimmes@artic.edu flash.uchicago.edu/~fxt/class_pages/class_geom.shtml
More informationtetrahedron octahedron icosahedron cube dodecahedron (Fire) (Air) (Water) (Earth) (Universe)
Platonic Solids A regular polyhedron is one whose faces are identical regular polygons. The solids as drawn in Kepler s Mysterium Cosmographicum: tetrahedron octahedron icosahedron cube dodecahedron (Fire)
More informationPatterned Polyhedra: Tiling the Platonic Solids
Patterned Polyhedra: Tiling the Platonic Solids B.G. Thomas* and M.A. Hann School of Design, University of Leeds Leeds, LS2 9JT, UK b.g.thomas@leeds.ac.uk Abstract This paper examines a range of geometric
More informationPolyhedron. A polyhedron is simply a three-dimensional solid which consists of a collection of polygons, joined at their edges.
Polyhedron A polyhedron is simply a three-dimensional solid which consists of a collection of polygons, joined at their edges. A polyhedron is said to be regular if its faces and vertex figures are regular
More informationChapter 11 Part 2. Measurement of Figures and Solids
Chapter 11 Part 2 Measurement of Figures and Solids 11.5 Explore Solids Objective: Identify Solids Essential Question: When is a solid a polyhedron? Using properties of polyhedra A is a solid that is bounded
More informationRectangular prism. The two bases of a prism. bases
Page 1 of 8 9.1 Solid Figures Goal Identify and name solid figures. Key Words solid polyhedron base face edge The three-dimensional shapes on this page are examples of solid figures, or solids. When a
More informationMATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY. Timeline. 10 minutes Exercise session: Introducing curved spaces
MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY Timeline 10 minutes Introduction and History 10 minutes Exercise session: Introducing curved spaces 5 minutes Talk: spherical lines and polygons 15 minutes
More informationReady To Go On? Skills Intervention 10-1 Solid Geometry
10A Find these vocabulary words in Lesson 10-1 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 10-1 Solid Geometry face edge vertex prism cylinder pyramid cone cube net cross
More informationThe Jitterbug Motion
The By 80-4 Poplar St. Rochester, NY 460 Copyright, September 00 09-9-00 Introduction We develop a set of equations which describes the motion of a triangle and a vertex of the Jitterbug. The Jitterbug
More information1 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,
More informationCARDSTOCK MODELING Math Manipulative Kit. Student Activity Book
CARDSTOCK MODELING Math Manipulative Kit Student Activity Book TABLE OF CONTENTS Activity Sheet for L.E. #1 - Getting Started...3-4 Activity Sheet for L.E. #2 - Squares and Cubes (Hexahedrons)...5-8 Activity
More informationLESSON. Bigger and Bigger. Years 5 to 9. Enlarging Figures to Construct Polyhedra Nets
LESSON 4 Bigger and Bigger Years 5 to 9 Enlarging Figures to Construct Polyhedra Nets This lesson involves students using their MATHOMAT to enlarge regular polygons to produce nets of selected polyhedra,
More informationTHE PLATONIC SOLIDS BOOK DAN RADIN
THE PLATONIC SOLIDS BOOK DAN RADIN Copyright 2008 by Daniel R. Radin All rights reserved. Published by CreateSpace Publishing 3-D renderings were created on a thirteen-year-old Macintosh computer using
More informationFAMILY NAME: (Print in ink) GIVEN NAME(S): (Print in ink) STUDENT NUMBER: SIGNATURE: (in ink) (I understand that cheating is a serious offense)
TITLE PAGE FAMILY NAME: (Print in ink) GIVEN NAME(S): (Print in ink) STUDENT NUMBER: SIGNATURE: (in ink) (I understand that cheating is a serious offense) INSTRUCTIONS TO STUDENTS: This is a 2 hour exam.
More informationTomaz Pisanski, University of Ljubljana, Slovenia. Thomas W. Tucker, Colgate University. Arjana Zitnik, University of Ljubljana, Slovenia
Eulerian Embeddings of Graphs Tomaz Pisanski, University of Ljubljana, Slovenia Thomas W. Tucker, Colgate University Arjana Zitnik, University of Ljubljana, Slovenia Abstract A straight-ahead walk in an
More informationTwo Connections between Combinatorial and Differential Geometry
Two Connections between Combinatorial and Differential Geometry John M. Sullivan Institut für Mathematik, Technische Universität Berlin Berlin Mathematical School DFG Research Group Polyhedral Surfaces
More information