Part III: The Tapered Star

Islamic Geometric Ornament: Construction of the Twelve Point Islamic Star

One of the changes which affects the appearance of the Islamic star most is taper in the arms of the stars. The choice of parallel or tapered arms is the artist’s choice. Both pattern types are very common.

A single point was moved in the definition of the pattern to convert the parallel layout above to the tapered layout. The layout, rules and procedures are then exactly the same.

Alan D Adams, Holland, New York, March 2013. License: Creative Commons -Attribution 3.0 Unported (CC BY 3.0) Text, photos and drawings.

All twelve point stars share the same divided circle definition. The two patterns at the chapter header also share the same tiling. Both patterns share the same decisions designed to maximize symmetry. The last decision in the layout of the parallel arm star is the placement of the point (g).

The difference in the tapered layout is that the ends of the star arms do not follow the dodecagon. In almost all historic examples of tapered stars, it moves inward toward the center of the major star. When the arm end moves from the dodecagon, the same construction is used to maintain the best possible symmetry in the minor five point star. The point (g), where the arm side intersects the end, remains on the bisector (o’ k) to produce this tapered pattern.

An infinite range of tapers are possible but some intersection should be used to define (g) so that the layout is reproducible. The minor layout circle and bisector construction is only drawn once for each pattern, but if the pattern is to be repeated or scaled up, using an identifiable intersection is prudent. A guess at a pleasing taper is made and an intersection on the layout circle was chosen in this case, as shown on the left.

A very small change in the position of point (g) will produce a pronounced taper in the star arm. The original parallel arm is outlined in yellow. The real effect of the taper is only seen after the completed layout. The remaining layout procedure is identical to the parallel arm star. One additional layout circle is now required through point (g), circle (o g). The arm end no longer follows the dodecagon

The star polygon is completed by connecting points on circles (o c) and (o b), as before. When the arms of the polygon are extended out, they now terminate on circle (o g) and the arm ends are drawn in from points (a) to the intersections on circle (o g).

The small change in position of point (g) results in a very noticeable taper in the arm of the star. Tiling this layout yields the chapter heading figure. The taper of the major star has also produced changes in the minor five point star; it is not independent but produced by the tiling. The five point star is also a tapered arm star now. The maximum symmetry of this minor star has been maintained.

The taper can be chosen to suit the artist, or they are sometimes dictated by the tiling, especially in mixed star tilings as will be seen in the 12 plus 8 and 12 plus 9 point mixed tiling. The parallel arm star is of course, simply a special case of the taper; the taper is zero. There two other commonly seen special cases of tapered stars.

In the second special case of parallel arm stars the end of the star arm follows a layout hexagon. In this case the outside boundaries of each adjacent pair of arms are parallel, or every fourth arm side.

Notice that the minor polygon formed by the tiling is a perfect hexagon in this figure; it shows the maximum possible symmetry for all elements.

This versatile star is found in several special layouts but it is often difficult to spot. In the case below it is found under a four arm star. The alternate parallel faces can be used easily to form figures with the four, six or twelve fold symmetry emphasized. This complex tiling and others will be considered later under octagonal mixed tilings. It is a preview of how much the figure can change with alternate tilings.

The final special case of the layout is a case where points (a), (g) and (c) lie on a straight line. It is not initially obvious that this will produce a pleasing figure, but it does and the resulting figure is very common in both twelve and ten point layouts. If the point (g) is moved along the bisector until (a), (g) and (c) lie on a straight line, the bisector is no longer needed. Points (a) and (c) define the pattern.

Completing the pattern leads to something which has little resemblance to the original parallel arm star but shares exactly the same layout; one point has moved and all other operations are the same.

The parts of the completed unit star are not related to the original parallel arm layout in the way one might at first suspect. The almond shaped elements, with their points at (a), do not correspond to the almond shape element in the parallel arm star. These almonds correspond to the cell represented by the elongated hexagonal arm of the original parallel arm star.

There is now a single row, or order, of cells around a central star polygon.

The minor hexagonal polygon is still present in this figure, as a very small almost triangular hexagon. The five point star has filled outward to a six sided figure. The points defining three of the arms of the five point star have also become co-linear.

Frequently a very small or very large polygon looks out of scale and distracts from the figure. Both of the oddly shaped hexagons corresponding to the five pointed star and the minor polygon formed in tiling are less symmetric than ideal figures. It would be possible to simply straighten the layout lines and alter the figures, but this is seldom done in historic figures. It is more common to see a change of layout or a change of lacing patterns to fix a problematic figure.

These figure can be re-tiled to improve balance and symmetry, but this belongs in a later discussion. There are multiple alternate layouts of this figure but the problem polygons can also be eliminated by alternate lacing of the patterns without changing the rules.

The lacing is turned back into the pattern instead of extending to the tiling polygon. This is a common way to modify a pattern to change symmetry or improve appearance, as detailed above.

The pattern becomes an interlaced 6 point star and infinite connecting bands. It is remarkable how different this pattern has become while retaining an identical construction method.

There are an infinite number of tapered arm 12 point star pattern. These examples show a typical tapered arm star case and the three special cases, but there are an enormous number of variations to be found in the historic patterns. Some are determined by other parameters of the pattern, some are simply different tapers selected to suit some other aspect of a composition. Taper in the arms is one of the few parameters which is often under the free control of the artist in these infinite tilings.

Most historic examples follow very similar rules of construction and proportion, geometrically indistinguishable from the system used here. The layout used to determine the arm proportions is remarkably successful at reproducing the proportions and angles of historic patterns.

The next issue to investigate is alternate tilings.