Computer Science and Artificial Intelligence Laboratory Technical Report MIT-CSAIL-TR-2006-024 April 14, 2006 The Symmetriad: A Journey of Discovery Through the Land of the Polychora Alexey Radul massachusetts institute of technology, cambridge, ma 02139 usa — www.csail.mit.edu The Symmetriad A Journey of Discovery Through the Land of the Polychora by Alexey Radul Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Electrical Engineering and Computer Science at the Massachusetts Institute of Technology June 2005 Copyright 2005 Alexey Radul. All rights reserved. The author hereby grants to M.I.T. permission to reproduce and distribute publicly paper and electronic copies of this thesis and to grant others the right to do so. Author............................................. ................. Department of Electrical Engineering and Computer Science February 23, 2005 Certified by......................................... ................. Gerald Jay Sussman Matsushita Professor of Electrical Engineering Thesis Supervisor Accepted by......................................... ................ Arthur C. Smith Chairman, Department Committee on Graduate Students 2 The Symmetriad A Journey of Discovery Through the Land of the Polychora by Alexey Radul Submitted to the Department of Electrical Engineering and Computer Science February 23, 2005 In Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Electrical Engineering and Computer Science ABSTRACT I devised and implemented a method for constructing regular and semiregular geo- metric objects in n-dimensional Euclidean space. Given a finite reflection group (a Coxeter group) G, there is a standard way to give G a group action on n-space. Reflecting a point through this group action yields an object that exhibits the sym- metries specified by G. If the point is chosen well, the object is guaranteed to be regular or semiregular, and many interesting regular and semiregular objects arise this way. By starting with the symmetry group, I can use the group structure both to simplify the actual graphics involved with displaying the object, and to illustrate various aspects of its structure. For example, subgroups of the symmetry group (and their cosets) correspond to substructures of the object. Conversely, by displaying such symmetric objects and their various substructures, I find that I can elucidate the structure of the symmetry group that gives rise to them. I have written The Symmetriad, the computer system whose name this document has inherited, and used it to explore 3- and 4-dimensional symmetric objects and their symmetry groups. The 3-dimensional objects are already well understood, but they serve to illustrate the techniques used on the 4-dimensional objects and make them more comprehensible. Four dimensions offers a treasure trove of intriguing structures, many of which have no ready 3D analogue. These are what I will show you here. Thesis Supervisor: Gerald Jay Sussman Title: Matsushita Professor of Electrical Engineering 3 4 Acknowledgments The Symmetriad owes a great debt to Rebecca Frankel, who originally started the project, and gave unfaltering support and sage advice as I continued it. I would never have gotten through the thesis without the aid and encouragement of my the- sis supervisor, Gerry Sussman. My thanks also go to all the people who read and commented on the various drafts of this thesis, and to all the people with whom I discoursed about it. 5 6 Contents 1 Introduction 13 1.1 Objective ................................. 13 1.2 Contribution................................ 14 1.3 PriorWork ................................ 14 1.4 ThesisOverview.............................. 14 2 The Math 17 2.1 ReflectionGroups............................. 17 2.1.1 Basics ............................... 18 2.1.2 Chambers ............................. 19 2.1.3 RootSystems ........................... 21 2.1.4 FundamentalSystems . 22 2.1.5 Coxeter Groups, Systems, and Diagrams . 23 2.1.6 ClassificationTheorem . 24 2.2 SymmetricObjects ............................ 25 2.3 Notation.................................. 27 2.4 Exploration ................................ 31 2.4.1 Objects That Are Not Fully Articulated . 35 3 A Case Study 39 3.1 FullyArticulatedSolid .......................... 43 3.2 NotFullyArticulatedSolid . 57 4 System Structure 69 4.1 SystemOverview ............................. 69 4.2 MoreDetailedDescription . 70 4.3 Numerics ................................. 72 5 Picture Gallery 75 6 Conclusion 103 6.1 Summary ................................. 103 6.2 FutureWork................................ 103 6.3 Closing................................... 104 7 A Mathematical Details 105 A.1 CellsandCosets ............................. 105 A.2 SubnotationandSubgeometry . 107 A.3 Degeneracy ................................ 109 8 List of Figures 2-1 Areflectionintwodimensions . 19 2-2 Association of chambers of B2 with elements thereof . 20 2-3 The reflection lines and the roots of A2.................. 21 2-4 A fundamental system for A2....................... 22 2-5 Two Coxeter-Dynkin diagrams, and their Coxeter systems ...... 24 2-6 Some 2D reflection groups, with Coxeter-Dynkin diagrams....... 25 2-7 H3 :1, 0.3, 0.1, with red, green, and blue color coded edges . 28 2-8 H3 :1, 1, 1, with red, green, and blue color coded edges . 30 2-9 Two objects built on B2 ......................... 31 2-10 Association of chambers of B2 with elements thereof . 32 2-11 An object built on B2,withcosetsofsubgroups . 33 2-12 H3111,withbluehighlighteddecagons . 34 2-13 The other kinds of faces of H3111..................... 34 2-14 The effect of one zero in an H3 objectdiagram. 36 2-15 The effect of two zeros in an H3 objectdiagram . 37 3-1 Thetesseract,edgeon .......................... 41 3-2 Thetesseract,turnedslightly . 42 3-3 B41111,edgeon.............................. 44 3-4 B41111,turnedslightly. .. .. 45 3-5 B41111,cornerview............................ 45 3-6 B41111, with the B31113-cellshighlighted . 47 3-7 B4 :6, 1, 1, 1, with the B31113-cellshighlighted . 47 3-8 B41111, with the A31113-cellshighlighted . 48 3-9 B4 :1, 1, 1, 6, with the A31113-cellshighlighted . 48 3-10 B41111, with the hexagonal prism 3-cells highlighted . .. 49 3-11 B4 :1, 1, 6, 1, with the hexagonal prism 3-cells highlighted . 49 3-12 B41111, with the octagonal prism 3-cells highlighted . ... 50 3-13 B4 :1, 6, 1, 1, with the octagonal prism 3-cells highlighted . 50 3-14 A B3111 3-cell with neighboring A31113-cells ............. 53 3-15 A B3111 3-cell with neighboring hexagonal prisms . 53 3-16 A B3111 3-cell with neighboring octagonal prisms . 54 3-17 An A3111 3-cell with neighboring B31113-cells. .. 54 3-18 An A3111 3-cell with neighboring hexagonal prisms . 55 3-19 An A3111 3-cell with neighboring octagonal prisms . 55 3-20 A hexagonal prism with neighboring octagonal prisms . ....... 56 9 3-21 All four kinds of 3-cells of B41111atonevertex . 56 3-22 B41101,edge-on.............................. 57 3-23 B41101,fromacorner .......................... 58 3-24 B4 :5, 1, 0, 1, with the B31013-cellshighlighted . 59 3-25 B4 :1, 1, 0, 5, with the A31103-cellshighlighted . 59 3-26 B41101, with two of the hexagonal prisms highlighted . 60 3-27 A B3101 3-cell of B41101 and its neighboring A31103-cells . 60 3-28 A B3101 3-cell of B41101 and its neighboring hexagonal prisms . 61 3-29 B41001,edge-on.............................. 62 3-30 B41001,fromacorner .......................... 62 3-31 B4 :3, 0, 0, 1, with the B30013-cellshighlighted . 63 3-32 B4 :1, 0, 0, 3, with the A31003-cellshighlighted . 63 3-33 An A3100 3-cell of B41001 and its neighboring B30013-cells . 64 3-34 A triangular prism of B41001 and its neighboring square prisms . 65 3-35 An A3100 3-cell of B41001 and its neighboring square prisms . 65 3-36 Thetesseract,turnedslightly . .. 66 3-37 The16-cell,turnedslightly. .. 67 3-38The16-cell,fromacorner . 67 4-1 OverviewofTheSymmetriad . 71 5-1 Simplex................................... 81 5-2 OneAmongEquals............................. 82 5-3 A40110,uniformlyblue. ......................... 83 5-4 Heart. ................................... 84 5-5 A41001,uniformlyblue. ......................... 85 5-6 Balls..................................... 86 5-7 SteepledHands............................... 87 5-8 Eggshell................................... 88 5-9 Twenty-FourCell.............................. 89 5-10DiamondsareForever. 90 5-11 D41100 ≈ B41100 again. The rotation is contained in the x,y,w space. 91 5-12CubesinRed................................ 92 5-13MoreCubesinRed............................. 93 5-14FlyingCubes. ............................... 94 5-15SmallFlyingCubes. ........................... 95 5-16SmallJaws. ................................ 96 5-17GreatJaws. ................................ 97 5-18CubicRing. ................................ 98 5-19Untitled................................... 99 5-20Untitled................................... 100 5-21ThePlanetsareAligned. 101 5-22DanceofWorlds. ............................. 102 10 List of Tables 2.1 Irreducible Coxeter systems in four or fewer dimensions ........ 26 3.1 The varieties of 3-cell of B41111...................... 46 3.2 3-cell intersection patterns for B41111.................. 52 5.1 Semiregular objects in four dimensions . ... 80 11 12 Chapter 1 Introduction Symmetry has fascinated humans since the dawn of recorded history. Of its