User Manual for 2dTiler V2.0
Contents
1. Geometriae Dedicata 19 177 196 1985 B Grinbaum P Mani Levitska and G C Shephard Tiling three dimensional space with polyhedral tiles of a given isomorphism type J London Math Soc 2 29 1 181 191 1984 B Griinbaum and G C Shephard Incidence symbols and their applications In Relations between Combinatorics and Other Parts of Mathematics volume 34 pages 199 244 1979 B Griinbaum and G C Shephard Tilings with congruent tiles Bulletin of the AMS 3 951 973 1980 B Griinbaum and G C Shephard A hierarchy of classification methods for patterns Zeitschrift f Kristallographie 154 163 187 1981 B Gr nbaum and G C Shephard Patterns on the 2 sphere Mathematika 28 1 35 1981 B Griinbaum and G C Shephard Spherical tilings with transitivity properties In Geometric Vein Coxeter Festschrift Springer Verlag New York Heidelberg Berlin 1981 B Gr nbaum and G C Shephard Tilings and Patterns W H Freeman and Company New York 1987 H Heesch Uber Raumteilungen Nachr Gesellschaft der Wiss Gottingen pages 35 42 1934 H Heesch Regul res Parkettierungsproblem Westdeutscher Verlag Cologne 1968 D H Huson Die Klassifikation 2 isohedraler Pflasterungen der euklidischen Ebene Master s thesis Bielefeld University 1986 D H Huson Patches Stripes and Net Like Tilings PhD thesis Bielefeld University 1989 D H Huson Crystals tilings and Delaney Dress symbols In Annual Report ZiF 90 912. Dress Presentations of discrete groups acting on simply connected manifolds Adv in Math 63 196 212 1987 A W M Dress N P Dolbilin and D H Huson Two finiteness theorems for periodic tilings of d dimensional euclidean space Discrete and Computational Geometry 20 143 153 1998 A W M Dress and D H Huson On tilings of the plane Geometriae Dedicata 24 295 310 1987 A W M Dress and D H Huson Heaven and hell tilings Revue Topologie Structurale 17 25 42 1991 A W M Dress D H Huson and E Moln r The classification of face transitive 3d tilings Acta Crys tallographica A49 806 817 1993 A W M Dress D H Huson and A Miller Symmetrie und Topologie von Riesenmolekiilen supramolekularen Clustern und Kristallen In Muster des Lebendigen Vieweg Braunschweig Wies baden 1994 A W M Dress and R Scharlau Zur Klassifikation aquivarianter Pflasterungen Mitteilungen aus dem Math Seminar Giessen 164 83 136 1984 W D Dunbar Geometric orbifolds Revista Matematica de la Universidad Complutense de Madrid 1 1 3 68 99 1988 10 34 35 36 37 38 39 40 41 42 43 44 45 46 AT 48 49 50 51 52 53 54 55 R Franz and D H Huson The classification of quasi regular polyhedra of genus 2 Discrete and Computational Geometry 7 347 357 1992 B Griinbaum H D L ckenhoff G C Shephard and A Temesvari The enumeration of normal 2 homeohedral tilings
3. II pages 85 95 National Academy of Science of Ukraine Institute of Mathematics 1998 Proceedings Vol 21 O Delgado Friedrichs and D H Huson Tiling space by platonic solids I Discrete and Computational Geometry 21 299 315 1999 O Delgado Friedrichs and D H Huson Four regular vertex transitive space tilings J Discrete and Computational Geometry 24 279 292 2000 O Delgado Friedrichs D H Huson and E Zamorzaeva The classification of 2 isohedral tilings of the plane Geometriae Dedicata 42 43 117 1992 N P Dolbilin and D H Huson Computing the Delone tiling and Delaney Dress symbol of a periodic point set in euclidean space Preprint FSPM Bielefeld University 1994 N P Dolbilin and D H Huson Periodic Delone tilings Per Math Hung 34 1 2 57 64 1997 Special Volume on Packing Covering and Tiling A Dress A Miiller and W J Orville Thomas editors Topological Aspects of Molecular Structures volume 336 of Journal of Molecular Structure THEOCHEM Elsevier Science B V Netherlands 1995 Special Issue A W M Dress Regular polytopes and equivariant tessellations from a combinatorial point of view In Algebraic Topology pages 56 72 SLN 1172 Gottingen 1984 A W M Dress The 37 combinatorial types of regular heaven and hell patterns in the euclidean plane In H S M Coxeter et al editor M C Escher Art and Science pages 35 43 Elsevier Science Publishers B V North Holland 1986 A W M
4. Journal 24 167 172 1981 K Westphal Zur Konstruktion zweidimensionaler Pflasterungen in allen drei Geometrien Master s thesis Bielefeld University 1991 12 78 79 80 81 E Zamorzaeva The classification of 2 regular tilings for 2 dimensional similarity symmetry groups in Russian Akad Nauk MSSR Institut matematiki s VC 1984 Kishinev E Zamorzaeva On delone sorts of multiregular tilings in Russian Dep v VINITI 22 04 88 No 3132 V88 1988 Kishinev E Zamorzaeva Sorts of biregular tilings of the plane for similarity symmetry groups in Russian The Questions of Discrete Geometry Shtiintsa pages 83 103 1988 Kishinev H Zieschang E Vogt and H Coldewey Surfaces and Planar Discontinuous Groups Springer 1980 13
5. Window item starts a second copy of the program but only works if the program is called 2dt and is in your path e The File Open item opens a modified tiling file ends on 2dt e The File Open Collection item opens a collection of tilings file ends on tgs 7 2 The The File Save item saves the current tiling in its current state as a modified tiling to a file ending on 2dt The File Export PS item produces a postscript picture of the current tiling and exports it to a file The File To Geomview items saves the current tiling in Geomview format The File Commands item open the command line interface to the program Unfortunately the commands are currently undocumented One is of particular interest dim 2 or dim 3 change the dimension of the space in which the tiling is draw For example use dim 2 to see the stereographic projection of a spherical tiling or use dim 3 to see the hyperboloid picture of a hyperbolic tiling The File Quit item terminates the program Edit menu Edit menu contains the following commands The Edit Undo item is not implemented The Edit gt Cut item is not implemented The Edit Copy to XFig item copies the current tiling to the paste buffer of the xFig program For this to work you must have pstoedit installed freely obtainable on the web The Edit Paste item is not implemented The Edit Line Select item contains a submenu that allows you to select all select all v
6. to an OpenGL window By selecting the Modify Mutate menu item the user can open an additional Mutate window For a symmetry group containing rotations or dihedral symmetries this windows allows modification of the rotational orders and thus of the symmetry group of the displayed tiling All menus are tear off menus and this can also be considered as individual windows 6 Files The program works with four types of files Files ending on tgs contain a collection of symbols one for each tiling and no additional information These files are opened using the File Open Collection menu item Files ending on 2dt each contain precisely one modified tiling that is a tiling and additional information including all modifications interactively made by the user The files are opened using the File Open menu item Files ending on 2tc contain a color palette which is used to color tilings These files are opened and saved using the Tile Colors Open and Tile Colors Save inthe Tile Colors submenu of the Edit menu Additionally the menu item File Create PS produces a ps file containing a postscript picture of the current tiling 7 Menus Note that all menus can be torn off opened as a separate window by selecting the dotted line at the top of the menu In the following we discuss each menu and the menu items it contains 7 1 File menu The File menu of the Controls window contains the following menu items e The File New
7. 1 56 D H Huson 2dTiler a program for systematically generating and visualizing periodic 97 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 tilings of the plane sphere and hyperbolic plane 1998 Available from ftp ftp uni bielefeld de pub math tiling 2dtiler S Levy T Munzner M Phillips C Fowler and N Thurston Geomview The Geometry Center University of Minnesota Minneapolis USA 1 5 edition 1995 A M Macbeath The classification of non euclidean plane crystallographic groups Canad J Math 19 1192 1205 1967 W Meeks and P Scott Finite group actions on 3 manifolds Invent Math 86 287 346 1986 E Molnar Symmetry breaking of the cube tiling and the spatial chess board by d symbols Beitr ge zur Algebra und Geometrie 35 2 205 238 1994 E Molnar and I Prok Classification of solid transitive simplex tilings in simply connected 3 spaces part I Colloquia Math Soc J Bolyai pages 311 362 1994 63 Intuitive Geometry Szeged Hungary E Molnar and I Prok Classification of solid transitive simplex tilings in simply connected 3 spaces part II Metric realizations To appear in Studia Sci Math Hungar 1998 J M Montesinos Classical Tessellations and Three Manifolds Springer Verlag New York Heidelberg Berlin 1987 D Schattschneider Visions of Symmetry Notebooks Periodic Drawings and Related Work of M C Esche
8. User Manual for 2dTiler V2 0 Daniel H Huson July 28 2005 ZdTiler Viewer ZdTiler Viewer Contents Contents 1 Overview 2 Installation 3 Getting started 4 Mathematical background 5 Windows 6 Files 7 Menus TL Ple ment ocoosorcscrcss ee ee ee ee ee T2 Ban MONI oe se oh ee dae ee ee he et ae a eee bah oO er le Tea Tilngs menu scores ERG EH SE eae he Ee EES 6 Ta Layowt IN 6 ee a kk ee Se eA eee a oe Pe ae eS eee 6 76 Help menu i424464 484 2a gb e eee a Ra oe ee 7 8 Mouse interaction 7 9 Keyboard shortcuts 7 10 Bugs and unusual features 9 References 9 1 Overview 2dTiler is an interactive viewer for periodic tilings of all three two dimensional geometries written by Daniel Huson and Klaus Westphal Input is a file of records each describing a Delaney Dress symbol a data structure that prescribes the topology and symmetries of a peri odic tiling and output is the picture of that tiling The geometric shape of the tiles and the specific symmetry group can be interactively modified 2 Installation To install the program under Linux download the archive file http www ab informatik uni tuebingen de software 2dtiler 2dtiler20 zip and unpack it using the command unzip 2dtiler20 zip this will produce a directory 2dtiler containing the following e a file manual pdf containing a user manual for the program e a script 2dt to start the program from outside of the 2dtiler directory the e
9. ects the contract edge mode in which clicking a selected edge will contract that edge if possible 7 6 Help menu The Help menu only contains one item e The Help About item reports the version number 8 Mouse interaction Dragging the mouse in the Viewer window will either zoom rotate or translate the current tiling depending on the current mode see Clicking on an edge or vertex will select it if it was unselected and is selectable If it is selected it will either allow reshaping by dragging it will truncate a vertex or it will contract an edge depending on the modification mode 9 Keyboard shortcuts In the Controller window the menus can be navigated using Alt key short cuts using the underlined letters in the menus The Viewer window recognizes the following keyboard short cuts spacebar key move to next tiling a key b key B key c key gt C key d key D key f key G key h key IT key 1 key m key o key 0 key Q key r key R key s key t key T key x key zZ key gt key key lt key gt key select all edges move to previous tiling show hide chamber system barycentric subdivision show hide colors change modification mode to contract edge mode randomly change colors dualize show hide fundamental domain save tiling in Geomview
10. format color in black and white heaven and hell mode command line interface read command from stdin draw less tiles hyperbolic case draw more tiles hyperbolic case reset layout orientate tiling if possible quit change drag mode to rotate mode change modification mode to reshape mode straighten all edges change drag mode to translate mode change modification mode to truncate vertex Mode compute tiling with maximal symmetry change drag mode to zoom mode move to first tiling in file move to last tiling in file move to previous tiling in file move to next tiling in file 0 key to 9 key change width of selected edges to this number 10 Bugs and unusual features Before showing a tiling the program computes an internal display list of all objects If you have zoomed a Euclidean tiling to a very small size or have requested that the program shows very many hyperbolic ones then it might take minutes before the picture appears Unfortunately there is currently no mechanism in place to interrupt this without killing the program The program can not display any spherical tiling with trivial symmetry group Rendering of spherical tilings is very poor For nice pictures of them use the Geomview Many others References N 12 13 L Balke and D H Huson The hierarchy of plane equivariant tilings In preparation 1994 L Balke and D H Huson Two dimensional groups o
11. he appearance of tilings Alternatively to reshape tilings press the a key while in the Viewer window This will select and highlight all movable vertices and movable edge centers Click and drag these to reshape Use the Modify Mutate item to modify the symmetry group of a tiling Modified tilings can be saved in their current state using the File Save item 4 Mathematical background The program is based on combinatorial tiling theory In this approach due to Andreas Dress each tiling is encoded in terms of its Delaney Dress symbol which is essentially a finite connected edge colored and vertex labeled graph that precisely captures the topology and symmetries of a tiling The input files of the program consist of systematically enumerated symbols The program reads a symbol from the input file constructs a fundamental domain for the tiling computes generators forthe symmetry group and then applies these to the fundamental domain to obtain the whole tiling See the references for more details 5 Windows The program has two main windows the Viewer window and the Controls window All drawing takes place in the Viewer window whereas all user menus are contained in the Controls window The reason why the menus are not attached to the Viewer window is that the Window is imple mented using OpenGL whereas the user menus are written in tcl tk and when the program was written it was not possible to attach a tcl tk menu bar
12. isible or select all invisible lines or to deselect all lines A line is invisible if you have given it width none A selected edge is highlighted in red The Edit Line Width item is used to set the line width of all currently selected edges to the selected number or to none The Edit Straighten item pulls all selected or all edges straight If Always is on then the program will alway straighten a tiling when first drawing it The Edit Tile Colors submenu contains a number of different items The Tile Colors Show item toggles between displaying colors or not The Tile Colors Color item sets to color mode the Tile Colors Black amp White item sets to black and white mode and the Tile Colors Random item selects new random colors The Tile Colors Open and Tile Colors Save items are used to open or save a color palette a 2tc file 7 3 Tilings menu The Tilings menu is used to move through a collection of tilings e The Tilings Next item moves to next tiling in file e The Tilings First item moves to first tiling in file e The Tilings Previous item moves to previous tiling in file e The Tilings Last item moves to last tiling in file e The Tilings Search item is not implemented 7 4 Layout menu The Layout menu controls layout related aspects of the program e The Layout Show F Domain items highlight the fundamental domain e The Layout Show Chambers item shows the chamber system The next
13. pages 45 51 ZiF Bielefeld University Germany 1992 D H Huson The generation and classification of tile k transitive tilings of the euclidean plane the sphere and the hyperbolic plane Geometriae Dedicata 47 269 296 1993 D H Huson RepTiles Ein mathematisch fundiertes Computerprogramm zur Erzeugung von Pflasterun gen In V Reiss editor Pressedienst Forschung number 6 pages 1 4 Bielefeld University 1993 See also Seltsame Kacheln Die Zeit Easter 1993 and RepTiles Ein Computerprogramm zur Erzeugung von Pflastersteinen Neue Z richer Zeitung 23 6 1993 D H Huson Tile transitive partial tilings of the plane Contributions to Algebra and Geometry 34 1 87 118 1993 D H Huson A four color theorem for periodic tilings Geometriae Dedicata 51 47 61 1994 D H Huson DeloneTiles a program package for computing 3D periodic Delone tilings and their Delaney symbols 1995 Available from ftp ftp uni bielefeld de pub math tiling delone D H Huson FunTiles a program for systematically generating and visualizing periodic tilings of the plane sphere and hyperbolic plane 1996 Available from ftp ftp uni bielefeld de pub math tiling funtiles D H Huson Ribbon tilings from spherical ones Geometriae Dedicata 63 147 152 1996 D H Huson Visualization of periodic tilings In Hans Christian Hege and Konrad Polthier editors Visualization and Mathematics pages 135 139 Springer Verlag Heidelberg 1997 1
14. r W H Freeman and Co New York 1990 P Schmidt n homeohedral types of tilings Geometriae Dedicata 32 319 327 1989 A Schoenflies Krystallsysteme und Krystallstruktur Teubner Leipzig 1923 Reprinted by Springer Berlin 1984 E Schulte Nontiles and nonfacets for the euclidean space spherical complexes and convex polytopes J Reine Angw Math 352 161 183 1984 M Senechal Crystalline Symmetries An informal mathematical introduction Adam Hilger Bristol Philadelphia and New York 1990 J V Smith Topochemistry of zeolites and related materials 1 Topology and geometry Chem Rev 88 149 182 1988 J V Smith and W J Dytrych Nets with channels of unlimited diameter Nature 309 607 608 1984 W P Thurston The Geometry and Topology of Three Manifolds Princeton University Princeton 1980 W P Thurston Three Dimensional Geometry and Topology volume I Princeton University Press Princeton 1997 J Tits A local approach to buildings In Geometric Vein Coxeter Festschrift pages 519 547 Springer Verlag New York Heidelberg Berlin 1981 E B Vinberg editor Geometry II volume 29 of Encyclopedia of Mathematical Sciences Springer Verlag Berlin Heidelberg 1993 G F Voronoi Re cherches sur les parall lo dres primitifs I II J reine angew Math 134 136 198 287 67 181 1908 1909 D F Watson Computing the n dimensional Delaunay tesselation with application to Voronoi polytopes The Computer
15. rbifolds and tilings Geometriae Dedicata 60 89 106 1996 J H Conway O Delgado Friedrichs D H Huson and W P Thurston On three dimensional space groups Contributations to Geometry and Algebra 42 2 475 507 2001 J H Conway and D H Huson The orbifold notation for two dimensional groups Structural Chemistry 13 3 4 247 257 2002 L Danzer B Gr nbaum and G C Shephard Does every type of polyhedron tile three space Structural Topology 8 3 14 1983 With a French translation O Delgado Friedrichs Die automatische Konstruktion periodischer Pflasterungen Master s thesis Bielefeld University 1990 O Delgado Friedrichs Euclidicity Criteria for Three Dimensional Branched Triangulations PhD thesis Bielefeld University 1994 O Delgado Friedrichs A W M Dress and D H Huson RepTiles Ein Programm zur interaktiven Erzeugung periodischer Pflasterungen In Computeralgebra in Deutschland pages 261 262 Fachgruppe Computeralgebra der GI DMV und GAMM Passau Heidelberg 1993 O Delgado Friedrichs A W M Dress and D H Huson An algorithmic approach to tilings In C J Colbourn and E S Mahmoodian editors Combinatorics Advances pages 111 119 Kluwer 1995 O Delgado Friedrichs A W M Dress and D H Huson Tilings and symbols a report on the uses of symbolic calculation in tiling theory In Computer Algebra in Science and Engineering pages 273 286 World Scientific 1995 O Delgado Friedrichs A W M D
16. ress and D H Huson Discrete geometry approaches to molecular topology Theory software and visualization In R Corriu and P Jutzi editors Tailor made Silicon Oxygen Compounds From Molecules to Materials pages 317 328 Vieweg 1996 O Delgado Friedrichs A W M Dress D H Huson J Klinowski and A L Mackay Systematic enu meration of crystalline nets Nature 400 644 647 1999 Letters to Nature O Delgado Friedrichs and D H Huson RepTiles a program for systematically generating and interactively designing periodic tilings of the plane 1992 Available from ftp ftp uni bielefeld de pub math tiling reptiles 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 O Delgado Friedrichs and D H Huson Examples of tilings produced by the program RepTiles In Spektrum der Wissenschaft pages 96 97 December 1993 See also Spektrum der Wissenschaft Special Edition Faltpuzzle 1 1995 O Delgado Friedrichs and D H Huson Recognizing the crystallographic group from the triangulation of a 3 dimensional orbifold Manuscript 1995 O Delgado Friedrichs and D H Huson Orbifold triangulations and crystallographic groups Per Math Hung 34 1 2 29 55 1997 Special Volume on Packing Covering and Tiling O Delgado Friedrichs and D H Huson A combinatorial theory of tilings In P Engel and H Syta editors Voronoi s Impact on Modern Science Book
17. three items determine the drag mode which determines the effect of dragging the mouse in the Viewer window e The Layout Zoom mode item switches to zoom mode e The Layout Rotate mode item switches to rotate mode e The Layout Translate item switches to translate mode The next two items effect only hyperbolic tilings e The Layout Show More Tiles item draws twice as many tiles e The Layout Show Less Tiles item draws half as many tiles e The Layout Reset Layout item reverts to the default layout 7 5 Modify menu The Modify menu contains items that can be used to alter the current tiling and the corre sponding Delaney Dress symbol e The Modify Mutate item opens the Mutate window e The Modify Dualize item is used to switch to the dual tiling e The Modify Max Symmetry item shows the tiling with mazimal symmetry e The Modify Orientate item can be used to remove all orientation reversing symmetries from the symmetry group if present The next three items determine the modification mode effect of the program that is what happens when the mouse is clicked on a selected edge or selected vertex e The Modify Reshape mode item selects the reshape mode in which vertices and edge centers can be dragged and repositioned e The Modify Truncate Vertex Mode item selects the truncate verter mode in which clicking a selected vertex will truncate that vertex e The Modify Contract Edge Mode item sel
18. xecutable program 2dt exe a directory lib containing a compiled version of TCL TK 8 0 5 and a directory tilings containing a number of input files for the program To start the program cd 2dtiler and then type 2dt exe If you would like to be able to start the program from outside of the 2dtiler directory then please set an enviroment variable TILERDIR to point at the directory e g setenv TILERDIR 2dtiler and then copy the file 2dtiler 2dt to your bin directory Then you can use 2dt to start the program Important The program requires TCL TK version 8 0 5 Precisely this version not an ear lier or later one A compiled version of this is provided with the program and should work as long as you start the program as described above If this does not work please download TCL 8 0 5 from ftp www scriptics com pub tcl tcl8_0 tcl8 0 5 tar gz and TK 8 0 5 from ftp www scriptics com pub tcl tcl8_0 tk8 0 5 tar gz and install both packages on your computer 3 Getting started Type 2dt to start the program This opens two windows the Viewer and Controls windows Use the File Open Collection item of the Controls window and select a file from the tilings directory The first tiling of many will appear in the Viewer window Use the Tilings menu to navigate through the file Alternatively you can press the space bar or b key to move forward or backward in the file Use the items in the Edit and Layout menus to modify t
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