6.  Use Your Imagination

Of course, one of the goals of a comprehensive package is to allow users to try their own motifs. The Motif option to EscherTiling should be a list of lists, each of which contains a single Polygon and Line objects, perhaps with colors added (to force lines to be black, say). Ideally, Disks and Circles should work too, but they are not fully supported by our package now.  And some care might be required to make sure that border points are really on the border, not just within, say, 0.0001 of the border (this can arise when curved borders are constructed using iterators). This section shows some patterns from other motifs.

A Seven-Part Motif

It  is natural to extend Escher’s idea to a 7×7 grid. Figure 29 shows one such seven-part motif, called SevenPartMotif in our package.

Figure 29.  A seven-part motif.

Remarkably, using black for the first three polygons in the 7-list and 3 shades of gray for the remaining 4, and including no special coloring instructions whatsoever, yields a perfectly shaded pattern, one that is reminiscent of native American art! Of course, using automatic coloring works in this case too.

Figure 30.  The pattern for the signaturewhen the seven-part motif is used.

More Escher

Of course, Escher himself considered other motifs. For example, in [Schattschneider 1990, p. 47] we find the following figure, which Escher carved in 1942. This time, we have made the polygons of the motif white.

Figure 31. An early motif used by Escher; it has 6 polygons, colored white.

Notice that beyond the usual rules for a motif, there appear to be two additional rules governing this structure: (1) line segments in the polygons are either parallel or perpendicular, and (2) when line segments are extended to meet the boundaries of the unit square, they meet only on the existing segments of the polygons or at the corners of the square.

EXERCISE.  Exactly where does the motif in the Figure 31 meet the boundary of the unit square? (The solution can, of course, be found in the code in the Escher package.)

Three of the patterns that result from this motif are shown in Figure 32. Because none of the polygons cross, a big tile size of {2, 2} suffices for all, and only one color is needed (the light gray is just background).

Figure 32.  Three patterns (numbers 103, 56, and 146 from our AllSignatures catalog) generated by the motif in Figure 31.

It turns out, though, that Escher colored his with two colors, insisting that “adjacent” regions have different colors (again, see [Schattschneider 1990, p. 47]). Achieving this with our routines presents some difficulty, because we consider the polygons of the motif to be nonintersecting. Well, they are nonintersecting, as our Crossings routine knows:

What we need is to have our MinColor routine believe that “adjacent” regions actually touch. So we’ll fake it! Let’s duplicate the motif, but change its order so our routines will not confuse it for the original motif. Then, in view of the positions of the various polygons (see the labeled motif below), assert that polygon #1 touches #2, #2 touches #3, and so on.

Now we satisfy Escher’s sensibilities and more fully appreciate this remarkable motif.

                                               (a)

                                               (b)

Figure 33.  (a)  Escher’s 1942 motif, colored in a sequence of 6 grays and labeled so that the reader can see how we forced “adjacent” polygons to be colored differently. (b) The patterns of Figure 32 when Escher’s desires for distinct colors have been followed.

Also in [Schattschneider 1990, p. 46], we see two more motifs, dated  c. 1943, nearly along the same lines (if you’ll pardon the pun), but actually with slightly less strict requirements. The first of them has a polygon not intersecting the rim at all. In our package these are called  Escher1943aMotif and Escher1943bMotif. Again, the reader is invited to determine the various parameters involved in constructing the motifs.

Figure 34. Two more early motifs of Escher, which in the package are called Escher1943aMotif and Escher1943bMotif.

Here are the tilings in [Schattschneider 1990] using these motifs. In one case we customized the crossing instructions by setting Crossings[Escher1943aMotif] equal to {{1,5},{2,5},{3,5},{4,5}} . And these images make use of the TilingGridLines option to draw lines between the tiles, as Escher himself did in his notebooks.

Figure 35.  Four patterns using some unusual motifs of Escher. The upper ones use Escher1943aMotif and the signatures , ; the lower pair are from Escher1943bMotif using the signatures , .

Escher’s Curves

Escher also experimented with curved patterns, and our package includes one of his called EscherCurves[n]. The n controls the resolution of the curves. For exploration, use a small value of n for speed; for a smoother image, increase n. Here is an example with good resolution.

Figure 35. A curved motif designed and investigated by Escher. The use of InterpolatingFunction allows the horseshoe shape to be drawn with a resolution that the user can control.

Figure 36. A pattern based on Escher’s curved motif and the 114th signature in the catalog.