The computer program and VRML

Starting with a specified vertex star and a given vertex symbol, the program searches for all combinatorially possible, legitimate vertex adjacencies (adjacency symbols) associated with that vertex star and then combines translated, rotated, and reflected copies of the star (according to each vertex adjacency rule) to try to form a polyhedron. (Though some simpler tests that can eliminate impossible polyhedra are performed first, in order to be more efficient.) For each polyhedron found, a VRML97 file is created that describes a 3-D representation of the polyhedron. While we refer to infinite isogonal polyhedra as sponges, and which are usually considered to be three-dimensional, the two-dimensional planar tilings can also be considered as (flat) polyhedra in three-space. On doing this, the formerly planar tiling now has two sides in space, which leads to more possibilities for adjacency symbols (using flipped tiles). The program can handle these polyhedra as well.

When the program creates the VRML output files, it merely fills the file with multiple copies of the vertex star, overlaying them with all of the included polygons. For example, if the vertex star contained a square, then every visible square in the VRML model would actually be present four times, one each from each of the four vertex stars at the four corners of the square. This is as opposed to converting the four overlapping squares to a single square. There are two reasons for this. One reason is simply that it makes it easier to write the program! The second reason is that when two red sides (or two black sides) face each other instead of red facing black (that is, when the adjacency symbol includes a ^), an observer viewing that polygonal area from any point might see either black or red, depending on which square was drawn last. But there is no reason why one color should be given a higher priority over the other for display purposes. Therefore all squares, with all colors, are supplied in the file. Display software can then choose to blend the colors, display both alternately, or randomly choose. So one consquence is that when these overlaps occur, the display can seem to flicker between colors. Of course, a second consequence is that the VRML file is much larger than it would be otherwise, but the current arrangement allows the viewer to make its own decision, with the full information being supplied, on how to interpret the situation.

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Last updated: April 18, 2019