Islamic Geometric Ornament: The 12 Point Islamic Star. VII: 9 Plus 12 Point Star Tiling

7/28/2019 Islamic Geometric Ornament: The 12 Point Islamic Star. VII: 9 Plus 12 Point Star Tiling

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Part VII: Twelve Point plus Nine Point Star Tiling

Islamic Geometric Ornament:

Construction of the Twelve Point Islamic Star

The figure above is a particularly fascinating case of a pattern which should be very complex, but is in fact very

simple. It contains the nine point Islamic star. There is no exact compass and straight edge construction of a

nine sided polygon, as there is no exact compass and straight edge trisection of an angle. Yet, the figure above

is constructed easily by compass and straight edge rules.

This is also the first case of an arbitrarily, but precisely, spaced tiling which is presented here.

Alan D Adams, Holland, New York, 6 June 2013. License: Creative Commons -Attribution 3.0 Unported (CC BY 3.0) Text, photos and drawings.



This figure is very common in historic patterns. It was very popular in Nasrid Spain, from whence the colors of

the chapter heading derive.6 The pattern is unusual in several respects.

--It is the first approximate pattern presented here. There is no exact compass and straight edge construction of

the nine point star but this is a very accurate approximation. All seven, nine and eleven fold stars in Islamic

pattern must be approximations; it is proven that the required angles cannot be constructed with compass and

straight edge.7

--It is the first pattern presented where the 12 point star does not tile to an adjacent 12 point star at any point.

--It is also the first pattern which tiles with a triangle as the tiling polygon.

The pattern in the chapter heading was almost certainly discovered rather than designed. It arises as a surprise

from exploration of the tiling of the 12 point star. Earlier construction chapters explored tilings generated by

predictably expanding the star inside the tiling polygon. An expansion of the tiling polygon to 1.5 times the

layout radius, without extending the star layout, gives the figures below.

Nothing very interesting has happened yet. This is just a normal start for a tiling of the 12 point star with a

tiling polygon defined as 1.5 times as large as the pattern. Rather than expanding the star layout and then

determining the tiling polygon for that star, this type of exploration takes the tiled star and explores layouts

which can combine them. For scaling and laying out repeat patterns, a unit radius is usually chosen, in this

case, the stars are separated by one layout radius. This method can be used to explore how the stars can

interconnect without predefining the case. New and surprising interlaces can be discovered which would have

been difficult to find by generating the unit figure first.



Something very interesting happens when the radii and inter-radii of the 12 point layouts are drawn in. Drawing

a circle through the points used to define the minor layout circles, as shown, defines an inscribed nonagon, a

nine sided polygon, inside the three star layouts.

Constructing a regular nonagon must be done by trial and error approximation with compass and straight edge.

It is tedious and seldom very accurate except in very large figures. Careful measurement shows that this is not a

regular nonagon. It is, however, and excellent approximation. The maximum angle error is 0.2°. That is very

small indeed. Since this nonagon derives from an exact geometric layout, it will scale and reproduce very

accurately.

If the nonagon is treated as a normal layout polygon, the shared minor layout circle can be used to construct a

parallel arm star. The layout points are transferred around the radii and inter-radii as usual. The bisectors are

constructed in the minor layout circles as they would be for any shared star arms. [See App. II] There are angle

errors in the layout, but they are distributed symmetrically around the layout. The star appears symmetric. The

star polygon is constructed by connecting the layout circles on the radii and inter-radii; the arm sides are

extended inward to intersect and outward to the layout polygon. It is almost impossible to tell that the resulting

star is not an exact symmetry star.

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The decision to set the smaller, nine point, star as a parallel arm star determines the layout of the 12 point star.

As dissimilar stars, the 12 point star must be tapered if the 9 point is chosen as parallel. Extending the arm endof the nine point star to intersect the minor layout circle bisector sets the 12 point star layout.

Three layout circles are required for the 12 point star since it is a tapered star. The layout circles are transferred

to the three 12 point layouts and the star is drawn in as usual. The only issue left to determine is how the

remaining arm ends extend out to terminate. In this case, the strict rules based extensions result in entirely

satisfactory results. All decisions in this layout protocol are directed at maintaining the best possible symmetry

for the minor elements, and this minor five point star shows very pleasing symmetry.



The free arm ends from the nine point star extend to cross beyond the line which will become the tiling edge.

This case was dealt with previously. The line is reflected back into the layout with the small layout circle in the

upper right corner. This case, where a star extends slightly outside of its tiling polygon, is common in cases

constructed as this figure was. The structure of the nine point star was not allowed to determine the tiling

polygon. The decisions made in the layout of the 12 point star determined the tiling polygon, which will be a

triangle. The layout could be modified to remove this small overlap; it results in a tapered nine point star and a

different 12 point star taper. The effect is not offensive and it is found as drawn here in most historic examples

of this pattern.

Several tiling polygons can be drawn for this figure but the true unit tile, containing only one complete star, is

centered on the nine point star and contains three partial 12 point stars of one sixth each. This is also the first

case where one tile does not suffice to construct the entire pattern. Triangle tilings are repeated by both

translation, adding identical tiles, and rotation. A second tile rotated 180° is needed for tiling.



This resulting figure as well as examples with tapered arm stars for both nine and 12 point stars are found from

Nasrid Spain to Mughal India over almost ten centuries.6,8 The figure was constructed without making any

modifications to the layout method for the stars and only a minor intellectual change to the approach to tiling.



References:

6) See a Nasrid ceiling panel in the Aga Khan collection, 14-15th CE. Access Nr AKM00730

See David Wade’s http://www.patterninislamicart.com - images SPA1003, SPA1008, SPA1542x from the Alhambra. Most of theAlhambra examples are modifications with parallel arms for both nine and 12 point stars. See also Wade’s SPA1821 for another variant.

In addition, www.tilingsearch.org reports this pattern as occurring in the Alhambra in several places. Their index: T01161, data160/

J43C.

“Two adjacent mosaic panels in each of the four corners of the Mirador de Lindaraja.... window grilles round the lantern of the Sala delos Abencerrajes; as a wooden ceiling over the north part of the Sala del Mexuar; and as the middle of three window grilles over thearch linking the Patio de los Arrayanes and the Sala de la Barca.”

Examples are found with most possible combinations of tapered and parallel arm layouts;A ceiling of the Sultan Oljaytu Mausoleum, Ilkhanid Iran 14th cent. Wade’s image IRA2701;Parallel arm 12 Point star, reversed taper nine point star- The doors at the Khanqah of Baybars al-Gashaqkir, Mamluk Cairo. AH 709;A double tapered Mughal example is Wade’s image IND0729 from the Masjid Jami, Fatehpur Sikri.

7) Polygons with 3, 4, 5, 6, 8, 10, 12, 15, 16 or 17 sides, or even multiples of these, are constructible with compass and straight edge.Others are approximations. The regular 17 sided polygon construction was only published in the 19th century.

8) For a variation with taper arms in both nine and twelve point stars, see; B. Lynn Bodner, A Nine- and Twelve-Pointed Star PolygonDesign of the Tashkent Scrolls, Bridges, 2011, 147-154. Bodner uses a very different approach to pattern construction to generate adifferent approximate construction.

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Islamic Geometric Ornament: The 12 Point Islamic Star. VII: 9 Plus 12 Point Star Tiling