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Stars and Rosettes

In our method, a regular $n$-gon is filled with a figure of symmetry type $d_n$ (which has all the symmetries of the $n$-gon). In practice, these figures belong to a small number of families which we describe below.

For $n\ge 3$, let the unit circle be parameterized via $\gamma(t) = (\cos{(2\pi t/n)},\sin{(2\pi t/n)})$. We construct the $n$-pointed star polygon $(n/d)$ by drawing, for $0\le i<n$, the line segment $\sigma_i$ connecting $\gamma(i)$ and $\gamma(i+d)$. Note that $d<n/2$ and that $(n/1)$ is the regular $n$-gon. For some values of $k\ne i$, $\sigma_i$ will intersect $\sigma_k$, dividing $\sigma_i$ into a number of subsegments. We often choose to draw only the first $s$ subsegments at either end of $\sigma_i$, which we indicate with the extended notation $(n/d)s$. Figure 1 shows the different stars that are possible when $n=8$.

Our implementation generalizes this construction, allowing $d$ to take on any real value in $[1,n/2)$. When $d$ is not an integer, point $P$ is computed as the intersection of line segments $\overline{\gamma(i)\gamma(i+d)}$ and $\overline{\gamma(i+\ensuremath{\left\lfloor d \right\rfloor}-d)\gamma(i+\ensuremath{\left\lfloor d \right\rfloor})}$, and $\sigma_i$ is replaced by the two line segments $\overline{\gamma(i)P}$ and $\overline{P\gamma(i+\ensuremath{\left\lfloor d \right\rfloor})}$. An example of this generalization is given in Figure 2.

Figure 1: The six possible eight-pointed stars when $d$ is an integer.
\includegraphics [width=0.75in]{figures/star_811.eps} \includegraphics [width=0.75in]{figures/star_821.eps} \includegraphics [width=0.75in]{figures/star_822.eps}
$(8/1)1$ $(8/2)1$ $(8/2)2$
\includegraphics [width=0.75in]{figures/star_831.eps} \includegraphics [width=0.75in]{figures/star_832.eps} \includegraphics [width=0.75in]{figures/star_833.eps}
$(8/3)1$ $(8/3)2$ $(8/3)3$

Figure 2: An $(n/d)s$ star for non-integral $d$.
\includegraphics [width=1.7in]{figures/star_83.62.eps}


Figure 3: An arrangement of sixfold stars can be reinterpreted as rosettes. The pattern is one of the oldest in the Islamic tradition.
\includegraphics [width=3.0in]{figures/hex_rosette_rep.eps}

Figure 4: The construction of a ten-pointed rosette.
\includegraphics [width=3in]{figures/rosette_schematic.eps}

When sixfold stars are arranged as on the left side of Figure 3, a higher-level structure emerges: every star is surrounded by a ring of regular hexagons. The pattern can be regarded as being composed of these surrounded stars, or rosettes. Placing copies of the rosette in the plane will leave behind gaps, which in this case happen to be more sixfold stars.

The rosette, a central star surrounded by hexagons, appears frequently in Islamic art. They do not only appear in the sixfold variety, meaning that we must generalize the construction of the rosette to handle arbitrary $n$. The construction given by Lee [11] yields an $n$-fold rosette for any $n\ge 5$ while preserving most of the symmetry of the hexagons. Each hexagon has four edges not adjacent to the central star; all four edges are congruent. Moreover, the outermost edges lie on the regular $n$-gon joining the rosette's tips, and the two ``radial'' edges are parallel.

A diagram of Lee's construction process is shown in Figure 4. To begin, inscribe a regular $n$-gon in the unit circle and draw the $n$-gon whose vertices bisect its edges. Let $A$ and $B$ be adjacent vertices of the outer $n$-gon and $C$ and $D$ be adjacent vertices of the inner $n$-gon with $D$ bisecting $\overline{AB}$. The key is to then identify point $E$, computed as the intersection of $\overline{CD}$ with the bisector of $\angle OAB$. Then $F$ is the intersection of $\overline{OA}$ with the line through $E$ parallel to $\overline{OD}$. The rest of the rosette follows through application of symmetry group $d_n$: edges $\overline{DE}$ and $\overline{EF}$ lead to the outer edges of the hexagons, while copies of $F$ become the points of the inner star, which can be completed with the construction given earlier.

By sliding $E$ along the bisector of $\angle OAB$, we can continuously vary the shape of the rosette while preserving the congruence of the four outer hexagonal edges.

\includegraphics [width=1.1in]{figures/extended_rosette.eps}

Some Islamic designs feature a motif slightly more complicated than a basic rosette, where opposing limiting edges from adjacent tips of the rosette are joined up. The resulting object has the same symmetries and number of outer points as the rosette, but with an additional layer of geometry on its outside. We refer to these as ``extended rosettes''. A ninefold extended rosette appears on the right.

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Craig Kaplan 2000-08-16