The Datafile PD-CD 4/DATAFILE_PDCD4.iso/education/escher/!!Tilings/!Help

Text File | 1996-04-16 | 15.1 KB | 287 lines

                                   TILINGS

 PURPOSE OF THE PROGRAM

  The familiar tiling of a kitchen or bathroom wall is usually produced by
 placing square or rectangular tiles edge-to-edge and corner-to-corner over
 the wall. Many more interesting tilings can be formed by tiles of more
 complex shapes. Such tilings are produced by placing a shaped tile in a
 number of orientations, repetitively over a plane. For the tiles to fit
 together, only certain orientations of the tiles can be used, and they in
 turn constrain the allowed shapes. In spite of these restrictions, there are
 vast numbers of possible tilings which are really limited only by one's
 imagination. The Dutch artist M C Escher was the great pioneer and exploiter
 of this idea.

  This program allows appropriate shapes to be generated and the associated
 tilings to be produced.


 GENERAL FORM OF THE PROGRAM

  The master program is the application !!Tilings, and double clicking on its
 icon in the Tilings Directory viewer will install it on the iconbar. All
 available facilities - designing a new tile shape, redesigning a previously
 saved design, and displaying saved tilings - can be initiated by dragging an
 appropriate file on to this icon.

  Possible tile shapes are generated from a number of basic tile motifs,
 which can be triangles or quadrilaterals. There are five triangle-based
 systems and eleven based on quadrilaterals, each with a separate application
 directory. Dragging one of these applications to the iconbar icon will allow a
 tile shape to be designed and an associated coloured tiling to be displayed.
 Both the tile design and tiling can be saved.

  Opening a particular system application directory by double clicking on it
 while holding down Shift will reveal, amongst other things, two directories
 - Designs and Tilings - where designs and tilings are automatically put when
 saved. Dragging files or directories from either of these directories to the
 iconbar icon will either allow further operations on a previous design or
 display a saved tiling and its component parts; some designs and tilings can
 be found in the application directories for the systems !Q-02, !Q-08, !Q-09,
 !Q-10, !T-01, and !T-02. How to operate these two options is described further
 below.

  Detailed information on each tiling system is obtained by running it in
 the usual way by double clicking on its application icon. This shows,
 using the  basic motif, how shapes which can be generated are constrained
 by the geometry of the particular system, how the shape will be placed in
 different orientations to form a cluster, and how the tiling is formed.
 The  design program for the system can be run by clicking SELECT whilst this
 information screen is shown; clicking with ADJUST returns you to the Desktop.
 The application directory icon associated with a particular system is a
 simplified form of its basic motif, showing how it can be deformed.

  It is important for the running of the program that the main master
 application !!Tilings and the associated systems applications directories
 are kept within the same directory. Also and unusually, no actions are
 initiated by clicking with SELECT or ADJUST on the iconbar icon; its main
 response is to dragging actions. Clicking on it with the MENU button displays
 a menu with, in addition to the usual Info and Quit items, a Set Mode item.
 The program is designed to operate most effectively with the resolution
 provided in mode 27 and is set up for that mode. If mode 27 is not
 available, Set Mode allows the less satisfactory mode 12 to be used. The
 setting chosen is remembered from one running to another.


 DESIGN PROGRAM DETAILS

 First Screen

  On running the design program, in one of the two ways described, a screen
 is displayed showing an outline of the appropriate basic motif for the
 chosen system. Its form, proportions and position can be altered within the
 constraints of the system by dragging the yellow squares with the mouse
 pointer and holding down ADJUST. The dimensions of the motif as it is being
 altered are shown in graphic units; this can be useful in creating the
 correct proportions when reproducing a design from a drawing. When the
 desired motif is set up, it is fixed by clicking SELECT, and the second
 screen is shown.


 Second Screen

  This displays the main design screen. The chosen motif is reproduced,
 showing its fixed points. A small replica shows with coloured, half
 arrowheads the related pairs of sides or segments; these can be shaped only
 in corresponding ways as indicated by the arrowheads.

  In designing a tile, three modes of operation are available - Shape,
 Decorate, and Erase. These are chosen from the menu which is always shown
 in one of its two states - active, dark red, or passive, light grey. When
 passive, the current mode is shown in black. The menu can be activated
 virtually at any time by clicking the MENU button, and a new mode or other
 option chosen using the SELECT button. The grid and basic motif outline can
 be toggled on and off, and for convenience this can also be done with
 ADJUST which will leave the menu active. The design can be saved at any
 time as a datafile in the Designs Directory; if no name is entered for the
 design and RETURN is simply pressed, the design will not be saved and the
 program will return to the active menu. The Show Cluster option moves the
 program to its third screen, described below.

  Of the three modes, Shape is used to produce the shaped tile outline,
 Decorate allows internal decoration of the shape, and Erase allows curves
 to be deleted.

 Shaping
  When shaping, the program must know which segment is being worked on, so
 that the corresponding segment to be shaped simultaneously is defined. The
 segment is chosen by clicking SELECT with the mouse pointer on the
 appropriate arrowhead on the replica motif. Any shape can then be drawn
 between the extremities of the chosen segment of the basic motif; at the
 same time the corresponding shape will be drawn by the program between the
 extremities of the corresponding segment.

  Drawing the shape is done through a number of Bezier curves joining the
 two extremities of the segment. The endpoints of these joined Bezier curves
 are created by clicking SELECT with the mouse pointer approximately where
 you wish  each one to be; the first must be at one segment extremity, the
 last at the other. On SELECTing the last, the control points for all the
 curves appear. These, as well as the endpoints - except those at the
 segment extremities - can then be be dragged around using the ADJUST button
 to produce the desired shape, which is fixed by clicking SELECT. The shape
 can be edited at any later time simply by SELECTing the segment or its
 corresponding partner in the replica motif when in the Shape mode.

 Decorating
  Decoration is carried out in a similar way. But only one curve is drawn
 at a time, and there is no need to choose segments. In this case the start
 of the whole curve, defined by the first click on SELECT, can be anywhere;
 the finish can likewise be anywhere and is defined by clicking ADJUST after
 the finish point has been SELECTed. This reveals the control points and
 allows curve shaping to be carried out as before, except that in this case
 the start and finish points are also moveable. Clicking on ADJUST outside
 this routine, as long as you are in Decorate mode, toggles startpoint
 markers of the various decoration curves on and off. When on, clicking
 SELECT on one of them allows the associated curve to be edited. New curves
 can still be drawn whilst the editing option is on.

 Erasing
  If editing segment or decoration shapes is insufficient, as a final resort
 any segment pair or decoration curve can be erased. To avoid this being
 done inadvertently, the separate Erase mode must be entered. Then, on
 SELECTing any segment in the replica motif, the current shape of it and
 its partner will be erased. SELECTing the "D" which appears in the centre
 of the replica motif in this mode, reveals the startpoint markers on each
 decoration curve; clicking SELECT on any of these will erase the associated
 curve.

  On completion of the design, or at any other stage, Show Cluster can be
 selected from the menu. This produces the third screen.


 Third Screen

  In any of the tiling systems, a number of tiles, each in a different
 orientation, form a cluster. This cluster is such that the tiling is
 generated by placing the cluster repetitively over the whole plane in a
 single orientation. The appropriate cluster for the system being worked on
 is drawn on the third screen. In some systems, this is larger than is
 required geometrically, so that colourings can be achieved in which adjacent
 tiles are of different colours.

  A menu with characteristics similar to that on the second screen is also
 present. It provides colouring and tiling options as well as repositioning
 of the cluster and return to the previous screen for redesign of the tile.

  To colour the tiles in the cluster before forming the tiling, the colour
 option should be chosen. In this case a colouring palette is revealed, from
 which colours can SELECTed and each tile flood-filled with SELECT. Filling
 can also be done with ADJUST; this leaves the mouse pointer in the cluster,
 which is convenient if the same colour is to be used on another tile.

  On selecting the tiling option, a scale bar reveals itself with the cursor
 on the number one. The cluster size to be used in the tiling can be scaled by
 moving this cursor with the ADJUST button. The tiling can then be generated by
 clicking SELECT.

  On occasions it may be found that the cluster as drawn on the screen will
 go off-screen at the top, bottom or left, or extend beyond the white work
 area to the right. To avoid the imperfect tiling which will result from
 such an occurrence, the reposition option should be used. This allows a
 vector to be set up with the ADJUST button from the origin of the small
 set of axes which is displayed. When the vector is SELECTed, the cluster
 will be redrawn, displaced by the length, and in the direction of the
 vector. In the unlikely event that the cluster cannot be moved far enough
 in one displacement, the process can be repeated as many times as required.
 To avoid wasted effort, repositioning be should done before any colouring is
 carried out.


 Fourth Screen

  This screen is completely filled with the tiling. No menu is shown unless
 called for by clicking on the MENU button. Options for changing the colours
 in the tiling, saving the tiling, returning to the cluster screen for
 recolouring or rescaling, making a hardcopy of the tiling, killing the
 menu, or quitting are available.

  When Recolour is selected, a colour editor appears, in which a copy of
 the colour of the tiling at the mouse pointer is shown. SELECTing the
 colour to be changed allows it to be altered using the ADJUST button on the
 red, green, and blue slider bars. The modified colour is fixed by clicking
 SELECT on the OK box; clicking SELECT on the Cancel box cancels the
 adjustment and returns you to the original colour. Clicking MENU when no
 colour is selected returns you to the Tiling Menu.

  The Save Tiling option operates similarly to that for a design save, with
 the tiling saved as a compressed data file, called Array, in the Tilings
 Directory. The associated cluster and basic tile shape are also saved in
 the same way; before the latter is saved, it can be coloured.

  When Hardcopy is selected, the tiling is saved to the Escherplus file as a
 sprite-file which can then be printed out as hardcopy in any appropriate way.


 DESIGNS AND TILINGS

  As mentioned before, within each system application directory are two
 directories - Designs and Tilings - containing previously saved designs and
 tilings. Designs are present as data files and can be displayed in the main
 design screen of the program by dragging the appropriate file to the
 Tilings icon.

  The Tilings directory holds a number of other directories, each of which
 contains three compressed data files from which can be shown a tiling
 (array), the associated cluster and basic tile. Dragging a directory to
 the Tilings icon allows the basic tile, cluster and tiling to be displayed
 in turn. SELECT moves the display on and ADJUST moves it back. Moving on
 from the tiling or back from the basic tile returns you to the Desktop.

  Any one of these three displays can also be shown simply by opening the
 directory and dragging the appropriate data file to the Tilngs icon. In
 this case, clicking any mouse button returns you to the Desktop. If MENU is
 used, you will be given the opportunity to make a hardcopy if you wish,
 before returning to the Desktop.

  Some already saved Designs and Tilings can be found in the application
 directories for the systems !Q-02, !Q-08, !Q-09, !Q-10,!T-01, and !T-02.


 TILING SYSTEM DETAILS

  Detailed information on any tiling system is obtained by running it, ie by
 double clicking on its icon. This produces a screen showing the basic tile
 motif, with its sides marked by coloured half-arrows. A valid tile for the
 system can be formed only if sides with the same colour of arrow are shaped
 in similar ways, indicated by the direction and handedness of the
 half-arrows; the program ensures that this takes place. Related sides can
 be formed from one another either by a simple translation, in which case
 they are labelled with a T, by a rotation, when the labelling is in
 degrees - eg 60 - or by a reflection in one of two orthogonal axes, always
 arranged to be vertical or horizontal, with the labelling Rv or Rh.

  Also shown is how the tiles fit together by rotations and reflections to
 form a cluster which can be used to form the tiling simply by repetitively
 fitting it to itself over the plane with no rotations or reflections. In
 some cases the cluster is larger than required simply on geometrical
 grounds; this is done to allow tilings to be generated with no adjacent
 tiles having the same colour.

  Finally, an array is shown which indicates how the final tiling is
 obtained.


 PATTERNS

  If the tile edges are blanked out from an array of decorated tiles, what
 is left is often referred to as a pattern, as opposed to a tiling. This
 pattern consists of an array formed by the original tile decorations
 regularly placed over the plane in a number of different orientations. For
 example, the tile decorations in the array just referred to in the previous
 paragraph would form a pattern of dogs.

  Such patterns can be generated by appropriately decorating the basic motif
 of a tiling system and leaving the edges untouched.






 Happy tiling!                                   Tom P McLean,
                                                 Kinlochard,
                                                 Purlieu Wood,
                                                 MALVERN WR14 4DJ