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Filling the Plane

Equipped with a taxonomy of typically Islamic motifs that can be inscribed in regular polygons, we are now ready to create complete periodic designs. We start with a periodic tiling containing regular polygons, with irregular polygons thrown in as needed to fill gaps. For each regular $n$-gon with $n>4$, we choose an $n$-fold star, rosette or extended rosette to place in it and replicate that motif everywhere the $n$-gon appears in the tiling. The motif is placed so that its points bisect the edges of the $n$-gon.

The result is a design like that of Figure 5(b). There are still large gaps where motifs were not placed, corresponding here to to the squares in the original tiling. Each square edge is adjacent to an edge of an octagon, and so a vertex of the chosen motif is incident to it. The presence of these vertices suggests a technique for filling the gaps in a natural way, by extending the line segments that terminate on the boundary of the region until they meet other extended segments in the region's interior. Except for degenerate cases, following this procedure guarantees that the resulting design will admit an interlacing.

Figure 5: Given the octagon and square tiling shown in (a), we decide to place 8-fold rosettes in the octagons and let the system infer geometry for the squares. The rosette is copied to all octagons in (b), and lines from unattached tips are extended into the interstitial spaces until they meet in (c). The construction lines are removed, resulting in the final design shown in (d).

\includegraphics [width=1.4in]{figures/pattern_initial.eps} \includegraphics [width=1.4in]{figures/pattern_no_infer.eps} \includegraphics [width=1.4in]{figures/pattern_infer.eps} \includegraphics [width=1.4in]{figures/pattern_final.eps}
(a) (b) (c) (d)

Figure 5(c) shows the design with the free rosette tips extended into the gaps. Here, the natural extension creates regular octagons in the interstitial regions. To complete the construction, the original tiling is removed, resulting in the design in Figure 5(d), a well-known Islamic star pattern [3, plate 48].

Figure 6: Some alternative patterns based on the octagon-square tiling that can be constructed by varying the motif placed in the octagons.
\includegraphics [width=1.2in]{figures/pattern_alt1.eps} \includegraphics [width=1.2in]{figures/pattern_alt2.eps} \includegraphics [width=1.2in]{figures/pattern_alt3.eps}

Given a tiling containing regular polygons and gaps, we can now construct a wide range of different designs by choosing different motifs for the regular polygons. Even when restricted to the octagon-square tiling used above, many different designs can be created. Three alternative designs appear in Figure 6. Of course, we can expand the range of this technique in the other dimension by also encoding a large number of different tilings.

The implementation currently encodes fourteen tilings from which Islamic star patterns may be produced. Some are familiar regular or semi-regular tilings [8, Section 2.1]. Some are derived by examination of well-known Islamic patterns. The remaining tilings were discovered by experimentation and lead to novel Islamic designs shown in Section 5.

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Craig Kaplan 2000-08-16