Part V: Expanded Tiling Units.

Islamic Geometric Ornament: Construction of the Twelve Point Islamic Star

As the tiling polygon of the classic parallel arm 12 point Islamic star is expanded, the space outside the major star expands. Some remarkably complex looking figures can be constructed by simple expansion of the tiling polygon with absolutely no other changes to the construction rules. The figure above from the Blue Mosque of Aqsunqur, 747 AH / 1347 AD, is constructed in an expanded tiling hexagon exactly as for the first parallel arm 12 Point star. No new rules, extra layout or design techniques need to be introduced. The complexity is apparent, not real.

While the construction of the figure is the same as what has already been presented, construction of the required tiling polygon requires somewhat more work in each case. As the tiling polygon expands, the star often rotates inside it to maintain the proper tiling relationships.

The tiling polygon must be constructed with the desired relation to the base layout circle of the Islamic star in spacing and angle.

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

The expansion of the tiling polygon requires extending the star layout if an interlacing pattern is to result. In the earlier layouts, some lines defining the ends of the arms of the star were extended to intersect the tiling polygon where they do not meet it directly, in the corners of the tiling hexagon. In the stars above, all of the arm ends are extended to intersect beyond the major star at the new intersections (e) and (h) resulting in new orders of petals around the center star.

The common variations of the Islamic star can be classed by these extensions. An extension to close off the gaps between the second order of hexagonal petals results in the center third order figure, with new third order of arrowhead shaped petals intersecting at point (e). If the extension is continued to the next intersection (h) the fourth order figure on the right results. The figure on the right gives rise to the chapter head figure, but some ground work is required to work up to it.

Drawing a new star which tiles at the tips of the third order of petals requires two changes. The new arm ends are rotated 15° from the previous order of petals; the “top” of the pattern moved. A new tiling hexagon is needed through those endpoints. The relationship of the star base layout circle and the tiling polygon needs to be re-evaluated.

As always, the first step of the layout must be the tiling polygon, so that it is possible to draw an exact size pattern; design starts with the tiling polygon and works inward. The tiling polygon and first tier of layout get a new color here. This is to draw attention to the fact that they serve a different purpose here.

The figure on the left is absolutely identical to the figure developed in part I for the simplest standard, hexagonal tiled, parallel arm star and the construction is identical. The purpose of these three hexagons has, however, changed here. The outermost green hexagon remains the tiling polygon. The two layout hexagons in light blue still serve to divide the circle into 24 parts. In the standard star, the circle in light blue would be the major layout circle for the star. For this expanded tiling polygon star the dark blue circle, drawn through the intersections of the layout hexagons, becomes the major layout circle. This change has moved the star inward, away from the tiling boundary.

The star is also rotated it 15°, the arm ends at point (a) are now on the dark blue radii. In this layout, the tiling hexagon is not related to the star arms, at point (a). The tiling hexagon through (a) was used to locate the minor layout circle to define the arm geometry in the standard layout. In this case a more general, equivalent,

definition is used. The minor layout circle is defined on the line between points (x) and (x’); these points define a line through point (a) perpendicular to the radius on which (a) lies.The minor layout circle for the arm is drawn through point (o’) on the light blue radius and the bisector for the angle formed by the line (a x’) and that radius is drawn. This minor layout circle and the dark blue base layout circle will define a star identical to the standard parallel arm star.

The same definition used for the original example is used for the star. The line from point (a) following the light blue layout dodecagon intersects the bisector (o’ k) at the point defined as (g). The line (g b c) defines the arm of the star. The layout points for the star are transferred around the radii by circles (o b) and (o c). The star polygon is drawn in by connecting the radii and inter-radii at circles (o b) and (o c).

The star polygon sides are extended outward, to the layout dodecagon in light blue, and inward until they intersect each other. The arm ends are drawn in along the layout dodecagon, but in this expanded star these

ends continue until they intersect outside the major layout circle. This results in a star where six arrowhead petals intersect at the green layout hexagon and six intersect inside it. This result is very similar to the simplest star, and those arms which do not intersect at the tiling hexagon are treated as they were for the simple star: the arm ends are extended to intersect the tiling hexagon.

All of the rules are followed here and the figure should tile perfectly, and it does.

Several aspects of this figure are remarkable. This central star is identical to the standard parallel arm star. The troublesome five pointed stars are, however, completely gone. All star arms end in arrowhead darts and all of the small interstitial polygons are regular hexagons. A new minor polygon is created by the tiling which has

appealing symmetry and a good balance in size. The central stars no longer tile arm tip to arm tip so the emphasis in the figure is centered on the isolated star. The pattern also has a pronounced vertical emphasis which was not found in the simplest pattern. This pattern is reasonably common in historic patterns and suggests an obvious extension, to square repeat tiling.

The square tiling polygon is easy to identify; it has the dimensions of the inscribed circle of the tiling hexagon. The hexagon will still be part of the layout. Changes to the lacing pattern will be required. Square repeats of radial symmetry patterns always leave empty void space in the corners to be filled. In this case, the simplest rules of extending the lacing until it intersects the tiling polygon, gives less than ideal results.

The result places too much emphasis on the square symmetry elements around the central star. This tiling is seen in historic patterns, but it is not common. In this case, where the standard rules produce a legitimate tiling but unsatisfying result, it is best to go back to the tiled figure on the left and consider how the pattern can be

extended, still within simple rules, to avoid the unsatisfactory result. The problem, shape of the void and the standard rules usually suggest a solution.

The pattern lines which caused the problem in the tiling were a part of a perfect hexagon in the hexagonal tiling. Re-introducing that hexagon, highlighted in yellow, is easy since four sides already exist in the square layout. If the lines stop, to form a closed hexagon, a large void remains in the pattern. Continuing the lines on to a further intersection is the obvious solution, and the third pattern below is the result. This is a very successful and common square repeat tiling of the parallel arm star.

This layout uses the common rules of tiling which were discussed earlier with no significant modification. The artist may choose one of two options for a layout line at a tiling boundary. the pattern line may cross a tiling boundary, ending at the tiling boundary in the isolated unit layout. This causes a change of direction at that tiling boundary when it is tiled. It may also “reflect” at the tiling boundary. A reflection means that the layout line crosses the boundary and continues straight. This requires it to re-enter the layout at the same point.

The layout above uses both. The design is constructed by regarding the triangular void in the square tile as a tiling boundary. The new side of the hexagon is then a “line which has changed direction at a tiling boundary.” This treatment is equivalent to a layout constructed from an irregular octagonal tile and a rhombus tile. This method of construction, which will not be treated further here, is often more useful for analyzing than it is for constructing figures. It was extensively developed by Hanbury Hankin and is found in his publications.4

The layout of the new hexagon structures in the figure uses the existing two sides of that small hexagon to identify the layout circle, as above. To define the second new pattern line, a reflected line is constructed where the new hexagon side meets the tiling polygon. Constructing a reflected line requires some layout work, or as we have here, two existing parallel lines. A new layout circle is drawn and the reflected line added.

The new layout lines can be transferred to the extended layout of the star arm with the two circles shown. One transfers the new hexagon layout point on the tiling edge and one locates the re-entering layout line on the parallel line.

A number of other modifications could be made to this extended tiling, and it will be re-used again in mixed star tilings. Few of the options would introduce new ideas beyond the procedures already shown. The next step to introduce new patterns is a further expansion of an order of petals, and the construction of the pattern from the Blue Mosque of Aqsunqur.

References:

4) E Hanbury Hankin has a completely different approach to construction of Islamic star patterns. His method, frequently referred to as “Polygons in Contact,” is most practical for full size drawing with templates. It is very likely that a method like it was used for large construction layouts.

a) Hankin, E. Hanbury, The Drawing of Geometric Patterns in Saracenic Art, Memoirs of the Archeological Survey of India, Nr. 15, Government of India Central Publication Branch, Calcutta, 1925. Reprinted 1998. Brings together most of the material below with some original material.

b) Hankin, E. Hanbury, On Some Discoveries of the Methods of Design Employed in MohammedanArt. Journal of the Society of Arts, 53, 17 March, London, George Bell & Sons, 1905, p461-472.Also republished in two parts in- The American Architect, 87, Nr. 1534, 20 May 1905, 159-161 and Pt II, Nr. 1535, 27 May 1905, 167-170.

c) Hankin, E. Hanbury, Examples of Methods of Drawing Geometrical Arabesque Patterns. TheMathematical Gazette, 12 (176), 1925, p370–373 DOI: 10.2307/3604213

d) Hankin, E. Hanbury, Some difficult Saracenic designs II, The Mathematical Gazette, 18 (229), 1934,p165-168. DOI:10.2307/3606813

e) Hankin, E. Hanbury, Some difficult Saracenic designs III". The Mathematical Gazette, 20 (241), 1936, p318–319. DOI:10.2307/3607312