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Escher | 2D

Welcome to the Escher | 2D page.  This pure Java applet demonstrates in 2D the powerful concept of Geometric Substitutions.   For a backgrounder on this topic, please read my Escher intro page.  The applet is loading just below.  Dont forget to read the User Guide + Tutorial section below the applet window.


Microsoft IE 4.0 (and above) or Netscape Communicator 4.04 (and above).  Older versions of these browsers may not host this demo properly. 

Screen resolution of 1024 | 768 @ 16 bit colors


Houston, We need a Java enabled Browser


This demo has two windows.  In the Bottom Window, you draw a pattern or choose from a list of pre-defined patterns.  The applet applies the current pattern on the selected polygon and displays the result in the Top Window. 

If the applet has loaded properly, you should see a filled Square in the Top window.  You can replace it with a different Polygon by selecting it from the Choose Polygon combo box.  The following polygons are available for your perusal:  Square, Pentagon, Hexagon, Heptagon, Octagon, Nanogon, Decagon and Oval.  You can change the color and the drawing style of the polygon by choosing the desired value from Polygon Color and Drawstyles combo boxes.

Three scrollbars sorround the displayed polygon in the Top Window.  The scroll bar on the right allows you to Zoom In and Out of the polygon.  The scroll bar at the bottom allows you to pan left and right about the polygon.  The scroll bar at the left allows you to pan top and bottom about the polygon.  The Reset button sets the viewing parameters to their defaults.

There are two ways to model a pattern on the Square.  First, you could draw the pattern over the white reference triangle (the Board) in the Applet Window using Create Triangle, Move Vertex and Delete Triangle tools.  The other (easy) way is to choose one from the list of pre-defined patterns.  So, click the Choose Pattern combo box and select the  pattern named Wheel-I.  You will now see the pattern modeled on the Square in the Top window.   If you now select Pentagon from the Choose Polygon combo box, this pattern will be modeled on the Pentagon as well.

From the Apply Pattern combo box, you tell the applet how many times to recursively model the pattern on the polygon.  By default, the applet models the pattern Once.  Twice is another alternative.  If you choose Zero Times, the pattern will not be modeled at all.   Note that after every application, the number of triangles in the polygon increases sharply.  Coming back to our example,   boldly apply Wheel-I Once, Twice and Zero times on the Square and see what happens.

When you select a pattern from the Choose Pattern combo box, the applet discards any pattern that was already  applied on the polygon.   That is, it applies the selected pattern on the polygon afresh.  You might instead want to apply Wheel-I first and follow it up with Benz-I.  In that case, you use the New Pattern Layer button.  You select the pattern Wheel-I, hit the New Pattern Layer button and select Benz-I.    What would you do if you wanted to apply Wheel-I twice and follow it up with Benz-I ?  You first bang the Reset button to kill existing layers, choose Wheel-I, choose Twice from the Apply Pattern combo box, bang the New Pattern Layer button and choose Benz-I. 

Any pattern you select from the Choose Pattern combo box, is copied on the Board and is available for editing.  Three tools help you do this : Create Triangle, Move Vertex and Delete Triangle.  Note that the selected pattern is not modified, only a copy of it is modified.  So you can revert to the original pattern as often as you like.   There is no Delete All Triangles tool, instead select the pattern Blank from the Choose Pattern combo box. 

As you modify the triangles in the pattern, you will notice that changes on the polygon are performed real-time in the  Top Window.  However, this could consume a lot of CPU cycles when the initial polygon contains a lot of triangles.  You could stall real-time update by selecting Zero Times from the Apply Pattern combo box.  When you have the   pattern you like, you could select Once or Twice from the Apply Pattern combo box to apply it on the polygon.  


If you leave your mouse inside the Applet Window while the applet is loading, the cursor leaves its mark there.  The workaround is to select a pattern from the Choose Pattern combo box.

If a vertex of your triangle lies exactly on the same line as the apex of the reference triangle (Board), it is not included in the pattern and hence not applied on the polygon


Best viewed in Netscape 4.x or IE 4.x @ 1024 | 768 | 16-bit colors
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Copyright (c) 1998 Jaganathan Sampath
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This page was last updated on: December 9, 1998
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