Islamic Geometric Ornament: The 12 Point Islamic Star. 1: Basic Structure of the Islamic Star

Part I: Basic Structure of the Islamic Star Islamic Geometric Ornament: Construction of the Twelve Point Islamic Star

The twelve pointed Islamic star is a good example to demonstrate the general method for constructing the Islamic star. It is not the simplest or the most complex, or the most symmetric. It is one of the most common and versatile. Many complex tilings of this star are common across the Islamic world so it makes an excellent example case. The first example will construct the infinite tiling above. -This is a very common, parallel arm, star. -This is the simplest, general, infinite tiling of this star, -This tiling is the best symmetry which can be obtained with the 12 point star.

Alan D Adams, Holland, New York, 4 July 2014. License: Creative Commons -Attribution 3.0 Unported (CC BY 3.0) Text, photos and drawings.

The large expanses of Islamic geometric art are frequently infinite repeating patterns created from geometric tiles, polygon elements which combine to completely cover the plane without gaps or overlaps; a perfect tiling. There are relatively few families of perfect polygon tilings. Only three tilings exist based on a single regular polygon: a polygon with all sides and angles equal. These three regular tilings are familiar from everyday floor and wall tiles. Square, triangular and hexagonal tiles can cover a surface perfectly with no gaps. For an Islamic star tiling, the symmetry of the star should be related to the tiling polygon. Twelve sided polygons, dodecagons, do not yield a perfect tiling. The 12 point star also has 6 fold, 4 fold and 3 fold symmetry. Any of these three regular polygon tilings can be used to tile the 12 point star.

In this first case six fold symmetry, a hexagonal tiling, is used and the relation of the 12 point symmetry to the tiling symmetry is obvious, at the top left. The three fold symmetry equilateral triangle tiling, on the right, is identical to the hexagon tile: six equilateral triangles make a hexagon. Generally, the layout tile to use is the one which has the most obvious relationship to the complete structure of the star. The is hard to use. The hexagon encompasses the radially symmetric star, therefore it is the natural layout polygon. As for many of the infinite tilings of the Islamic star, the 12 point star tiling includes minor figures beyond the main star. In this tiling, a five pointed star and a minor polygon appear. These minor elements are a result of tiling, they are not independently constructed. A large part of the art of the Islamic star is to maximize the symmetry and select pleasing proportions for these tiling elements. The design problem is to define the major star while recognizing that decisions will affect the minor elements formed in the tiling.

The design layout should fulfill several conditions: -This is a parallel arm star, a very specific and very common condition. -A defined size should be easily constructed. -The minor elements of the pattern should be controlled to give the best possible symmetry and proportions. -To allow scaling the pattern, or to lay out multiple repeats, layout should avoid the introduction of measurement other than compass and straight edge. The last point is key. Divided rulers, straight edges with marked units of measure, are a recent commodity. They were not generally available to the historic artists. Even when they are available, introducing measurement invites imprecision in the layouts. It is best avoided. When the pattern are applied to non planar surfaces, measurement even becomes impractical. Defining all points and intersections geometrically is key for repeating and scaling patterns and is still most convenient even in computer generated drawings. The Hexagonal Tiling Layout of a Parallel Arm 12 Point Islamic Star Step one, which is already addressed, is to identify the tile: the figure which will contain and define the star. It is not always as easy as it is here. Here the tiling figure is a regular hexagon. Islamic Stars are radial symmetry figures; their symmetry is based on the circle. A divided circle is needed to construct the star. It does not appear in the final figure, but it is the foundation. For the 12 point star, we need to construct a layout circle with a set size which is precisely related to a tiling hexagon. For the 12 point star, the relationship of the layout circle and the tiling figure is again very simple. The layout circle is the inscribed circle in the tiling hexagon.

The exact order of operations depends on the orientation of the tiling hexagon. For a “horizontal” tiling hexagon, R2 is used as the right - left repeat spacing. R2 is the circumscribed circle defining the tiling hexagon. For a “vertical” layout hexagon, R1 is used as the repeat spacing equal to the distance across the points of the star arms. R1 is both the inscribed circle of the tiling hexagon and the circumscribed circle defining the layout

hexagon. The size of the area to be covered by the repeating pattern is divided into the desired number of repeats and the radius R2 is determined. There is a strong preference in historic examples for patterns which have half or quarter patterns at corners and sides of the field. As a result, the field is usually an even multiple of R2 in one, R1 in the other dimension. The chapter head figure is a pleasing ratio, 4 times R1 deep. It shows a quarter star pattern at each corner and a half pattern on each side. The width of the figure is not defined by R1 but by R2. Each pattern is 2 x R2 across the points of the tiling hexagon and each tiling hexagon is spaced by one side, also equal to R2. The pattern is therefore 6 times R2 across. This is usually the only “Measurement” you will use in a pattern construction. The Divided Layout Circle, The layout Hexagon For the figure at the chapter heading, the easiest case was used. This case is constructed from the outside in. The hexagon has a remarkable relationship to the circumscribed circle, drawn through its vertices. The length of the sides of the hexagon is exactly equal to the radius of the circle. We simply draw a circle of Radius R2 - the repeat spacing for the pattern. From the indicated point (a), we draw arcs of radius R2 to define the sides of the hexagon. Repeating the arcs from point (a’) yields a hexagon of exact known width, with sides of length R2.

The blue hexagon is the tiling polygon, the limits of the tile which repeats to form the pattern. For this example, the layout of the star lies inside that hexagon touching the center of each side. For this definition, the red circle of radius R1, from the origin to point (c), is the base layout circle for the star. This circle will need to be divided into 24 parts to define a 12 point star. The new radius R1 is used and the exact hexagon construction used above is repeated to inscribe a layout hexagon. The first side is (d - e2). The polygon is completed exactly as for the tiling hexagon to divide the circle into six parts. A second hexagon is needed for 24 divisions, so the process is repeated.

The second layout hexagon is constructed rotated 90° from the first, beginning at points (c) and (c’). Completing the figure results in three hexagons; a tiling hexagon defining the repeat and two layout hexagons defining the division of the circle into twelve parts, where each vertex touches the layout circle.

The dark blue lines connecting the vertices of the layout hexagons are “radii” which divide the circle into 12 parts. The division of the circle into 24 parts is finished with the second set of light blue line, “inter-radii,” connecting the intersections of the layout hexagons. This layout uses a total of two circles and six arcs. All points are geometrically defined and nothing is measured with a ruler. The second case, for a vertically oriented tiling hexagon, proceeds identically but starting from the inside. It is described in Appendix I [Direct Link to I on scribd.com]. The repeated pattern field. At this point, the layout will define one star. To repeat the star, a repeated series of layouts is needed. The division of the larger space to be covered by the layouts was already determined when R2 was chosen. Fortunately, defining the hexagon repeat is very easy. The repeat is easily laid out with each new center defined by R1, not the layout radius for the tiling hexagon, but the layout radius for the star.

The repeat centers are identified and the repeated figures are laid out at each center. The centers are located by one of two equivalent methods. The tiling is edge to edge, so the new hexagon is located from the face of the first tiling hexagon. The sides of the tiling hexagon can be extended to intersect, locating point (o’) or the radii which cross the center of the faces, from the vertices of the layout hexagon, can be extended and point (o’’) located by striking an arc. Extending the tiling hexagon sides is somewhat neater since the lines will be used in the layout of the divided circle in any case. The layout can be extended as far as desired.

The full base layout for the chapter heading figure. For obvious reasons, this is drawn very lightly but with a hard, sharp pencil.

Defining the Twelve Point Islamic Star. To this point, we do not have anything which looks like a star. All of the preliminary work is done and the circle is divided for the star layout, but there are many points which need to be defined to lay out the star. -What determines the width of the arms of the star? -What determines the angles at the end of the star? -The sides of the arms are parallel. How is that constructed? -If tapered arms are desired, how is the layout changed? -Why do the lines defining the arms intersect where they do? -The small five point stars are not perfectly symmetric. How can they be made them as pleasing as possible? There are only two decisions to make to answer these questions. The infinite variety of Islamic geometric art is mostly submission to the elegance of geometry. Relatively few decisions are required from the artist.

The desired pattern is overlaid on the basic divided circle above. The fit inside the tiling hexagon is required by the rules of tiling. For this simple case, the first decision made was that the stars should tile arm tip to arm tip. For six of the arms, the tip of the arm meets the tiling hexagon side, at the center of each face. Certain characteristics of historic patterns are common enough to be called rules. The cardinal rule is to maximize symmetry. There are many elements in the pattern which are effected by design choices. The common, almost universal, choice is to choose the design with the maximum symmetry in all elements. One additional “general rule” will be introduced later which is determined by tiling and commonly followed in historic patterns. It is perhaps surprising that one decision, to create the best possible symmetry for all elements, will determine all of the remaining questions laid out above.

The most prominent minor element in this pattern is the small five point star created by the tiling of the 12 point star. The symmetry and balance of this star plays a major part in creating a pleasing infinite pattern. Minor elements like these stars are the places where historic artists generally tried to create the best possible symmetry. What does that mean and how is it defined in the layout? The enlarged figure below shows the elements which control the symmetry of the minor five point star. It is created by the tiling, and not independently constructed. Tiling determines that it cannot be a perfect five point star. A perfect star would have an angle equal to 72° instead of 75°. The objective is to make it as symmetric as possible. What does “as symmetric as possible” mean? -The star is divided in half by the tiling hexagon; this means that the center point lies on edge of the tiling hexagon. -A star with the best possible symmetry will have arms of equal length; the points of the star lie on a circle: the red minor layout circle.

A circle which satisfies all of these conditions is centered at point (o’) in the figure above; (o’) lies on an interradius. The minor layout radius is set by the point (a), the intersection of a radius with the edge of the tiling hexagon at the tip of the star arm. This minor radius is (o’ - a). For the best symmetry, the lengths of lines L1 and L2 should be equal. The only remaining decision is the length of these two lines. The angle at the end of the arm and the taper of the arm, or parallel arm geometry, is not yet fixed. This is an independent decision, the only truly independent decision for this layout if we agree that maximum symmetry is to be achieved. A large variety of tapered and parallel arm stars are common in historical patterns but their geometric definition and construction is absolutely identical.

The dodecagon defined by the radii and the layout circle is drawn in in light blue above. For many historic patterns, particularly the parallel arm star, the ends of the star arm follow this dodecagon. The end of the star arm follows the dodecagon side from point (a). A new point is named, point (g) where the dodecagon intersects the bisector of the indicated angle, line (o’- k). The use of this bisector to define the length (a g) is again determined by symmetry requirements. If the point (g) lies on the bisector, lines (a g) and (g b) are of equal length, as required by best symmetry for the five point star. [see appendix II for details. Direct Link to II] If the line (a g) follows the light blue dodecagon, the sides of the 12 point star arms will be exactly parallel. [Lines (a o) and (g c) are parallel. This is exact and easy to prove.]

The points (b) and (c) need to be transferred to all of the inter-radii and radii, as well as to other repeating layouts. The circles (ob) and (oc) are transferred to all repeating layouts and the layout of the star is complete. This construction is quite simple and fast. The divided circle requires two circles and six arcs to construct. All radii and inter-radii are drawn in. One minor circle and a bisector are then required to define all of the remaining parameters of the 12 point star. For repeat patterns, the minor layout circle and its bisector do not need to be constructed. All points are geometrically defined; nothing is measured. No definition depends on a tangent, which is difficult to use precisely. An exact parallel arm 12 point Islamic star with the best possible symmetry for the minor five point star element will result.

The actual star pattern is constructed by connecting points on circle (oc), on the radii, and (ob) on the inter-radii. These lines are extended outward to intersect the light blue dodecagon (center figure), and inward until they intersect (right figure).

Drawing in the ends of the arm along the dodecagon completes the star inside its tiling hexagon. One additional rule or common practice is introduced here, and it arises from tiling practice. Where the 12 point star arm ends do not meet the tiling hexagon, a rather large empty space would result when tiled. In most cases, the lines forming the ends of the arms are extended to meet the tiling hexagon, at point (f). The result is a more balanced pattern. The triangular figure to the right is the minimum definition of this pattern. It contains all information required to construct this star and it sometimes appears in historic pattern books or modern discussions.2d This can be used for some very interesting and complex tiling designs, but that discussion is deferred until later. It does illustrate one common and important design element of tiling figures. Where a pattern line meets a tiling edge, along the bottom of the triangle, it can terminate, or it can re-enter the pattern at the same angle it exits. A correct tiling figure will always result. This tiling rule will be used in some of the more complex tilings studied later. This construction method is old. The first clear description and use I have found is by James William Wild in the 1840s. 1f It was also published with a good explanation in 1987 by A.J. Lee.1a Daud Sutton uses some examples of this method in his book. 1g This layout is probably closely related to the method used by historic

geometers based on two pieces of evidence. The most persuasive is that this construction reproduces proportions of historic patterns remarkably successfully. It is versatile and creates large families of patterns without modification of the method. The second piece of evidence is the presence of construction and layout lines on ancient design scrolls. Many of them have layouts which correspond quite closely to the method which is used here. This evidence is not conclusive, and it is ultimately unimportant. What matters is that this construction method is very successful at reproducing a large number of historic patterns. The method is general for 6, 7, 8, 9, 10, 12 and higher symmetry stars. The best symmetry star is always produced by this layout. There is a short cut for even number, parallel arm stars. It is attached here as an appendix, but it teaches less about why the star is constructed as it is. It is useful since it can be more accurate for smaller layouts. It does not work for odd numbers or tapered stars. After completing the construction of this parallel arm star, a construction where (g) is moved toward the center of the star along the bisector will be used to construct a tapered star. A very interesting and very common historic pattern is generated when (a- g -c) becomes a straight line. As a preview, a very small change in the position of point (g) produces this tapered pattern.

Tiling and decorating the 12 Point Islamic Star. The Islamic star or rosette is seldom seen in isolation. It usually appears in larger repeating patterns. Several classic versions are described in later chapters. These patterns are found carved in plaster, stone or metal, molded from clay, cut from tile, assembled from carved wood and ivory or drawn in inks and gold leaf. The examples are almost endless. Most of these applications are elaborated beyond the simple incised or drawn line of the pattern. One of the most common elaborations is the interlace. Most historic patterns are “alternate” patterns. They can be interlaced by an infinite pattern of over and under alternating crossings of the pattern lines. This gave one of the

best known western reference works on Islamic pattern its name, "Les éléments de l’art arabe: le trait des entrelacs," Traits of Interlacing, by Jules Bourgoin. Patterns like this one can be “laced” in a continuous over under alternation in any size pattern if all nodes, where lines meet or change direction, have two or four lines meeting at the node. If three lines meet at a node, it cannot be alternating laced. The interlace to the right repeats infinitely. Constructing the large numbers of parallel lines of an interlace is one of the most tedious parts of pattern drawing, but one of the most impressive in final result. There are many ways to lay out the parallel lacing, but only the one requiring additional layout on the drawing will be shown here. This is one of the more accurate methods, but also quite tedious. For all terminal points where the lacing changes direction, a small circle, of a diameter the same as the desired lacing width, is laid out. These must be drawn very precisely. For this purpose, any line which continues across the pattern only needs two layout circles. Lacing this star requires 30 circles, almost doubling the layout work for this pattern. Some circles, 6 in this pattern, are shared between repeating patterns but this requires a substantial amount of very careful layout work. For each pattern line, two parallel lines are drawn tangent to the lacing layout circles. Layout using tangent lines is tedious, but the alternative is simply too much work. With very careful work, this layout will produce very good results. It requires a very sharp pencil and a very steady hand. For small patterns, the very small size of the layout circle becomes impractical. Drawing very precise, very small circles is quite difficult. I almost never use this method. For most work, one of several mechanical methods is more practical. One of several versions of a “parallel rule” can be used. One rule lies on the pattern layout line and the parallel lacing lines are drawn against a second attached parallel rule. Good parallel rules are quite expensive. The most practical method uses the virtues of modern plastic drafting tools. They are transparent. A dedicated tool is made with a line inscribed into the back of a drafting triangle. Any method

works to inscribe the line so long as it is exactly parallel to the edge.Note 1 This reference line is placed along the layout line and parallels line can be drawn with impressive precision to define the interlace.

A technical pen or other precision pen works best for inking patterns. Round nib pens or felt tip pens produce a rounded terminal point which is less pleasing than a sharp intersection. note 2

A simple design interlaced with the parallel line tool shown above and inked with a 0.3 mm Koh-i-Noor technical pen. 6 mm Lacing (A3)

The tiling discussed here is the simplest and most compact infinite tiling of the classic 12 point star. It is, however, only one of dozens of common tilings of this star. The final lattice pattern can be changed substantially by modifying the pattern inside the tiling polygon or changing the tiling to another polygon tiling. This simplest pattern allows few common modification of the layout without changing the tiling polygon since it contains only a single star which fills the entire tiling polygon.

The following chapters will discuss; Alternate Lacings, Tapered Arm Stars, New Tiling Polygon, Square Repeat Tilings. All of the variations will retain exactly the same basic star layout structure. The final chapters will show how these patterns can be constructed in wood.

The Twelve Point Star in quarter-sawn red oak, The 103 parts of the lattice have only three unique types. 27 x 16 inches: 69 x 41 cm. Alan D Adams, 2011.

The Twelve Point Star in American Black Walnut. This is a modified tiling to fit a square. It will be covered later. 31 inches square, 78.8 cm

References 1a) Lee, Anthony J, Islamic Star Patterns, Muqarnas, Vol. 4 (1987), pp. 182-197. 1f) James William Wild’s notebooks are held by the Victoria and Albert Museum, earlier the Kensington Museum. The layout is clearly shown and noted on a page of his research notebooks from a study in Egypt in the 1840s. The construction is a square tiling of this 12 point star. That square tiling will be discussed later in the documents. (V&A accession number 2011EP3581) See the Wikipedia article. 1g) Duad Sutton, Islamic Design; A genius for Geometry, Wooden Books, Walker & Co, New York, 2007. See page 11 top center and page 21 for specific cases of the same layout method. Sutton’s specific examples are shown for parallel sided stars only. As a result, he does not need the bisector. 2d) Abas, Syed J.; Salman, Amer S., Symmetries of Islamic Geometric Patterns, World Scientific, Singapore, London, Hong Kong, 1995. This volume sits uncomfortably between a symmetry and computer generation discussion and a pattern catalog.

Notes 1 I use one or two plexiglass (perspex) drafting triangles stacked flat to give the desired spacing as a precise reference to inscribe the lines on the parallel spacing tool triangle. A sewing needle with a 60 degree flat ground on one side makes an excellent graver for plexiglass (perspex). 2 The tea mug was custom printed with a pattern from a later chapter at zazzle.com.

Appendix to Part 1: A short cut. The precise construction of the bisector and location of point (g) in the minor layout circle can be difficult in small layouts. Even number arm stars are symmetric across the pattern. We can use that property of the patterns as a shortcut. To use this shortcut we need; even numbered arm stars, parallel arms and at least one half pattern in the layout. This shortcut does not teach clearly why the pattern is drawn as it is, so it was not used for the main text.

Construction is exactly the same up to the minor layout circle. Now a second minor layout circle is drawn on the opposite side of the layout in the same position. The red line on the left below connects the intersection of the minor layout circles with the inter-radii. These points define the circle (ob) we used to construct the star polygon. The bold blue line on the right shows how it defines the second circle (oc) which completes the definition of the star polygon.

We can now complete the layout exactly as it was completed in the main text.

It is fairly easy to prove the we have constructed the exact same bisector in the minor layout circle using the dodecagon to define the ends of the arms, as before. For practical layouts, this shortcut can be helpful. Very small constructions are usually not very accurate- and the minor layout circle can be small in some layouts. This layout uses larger separations of the points which define the star polygon and is therefore often more accurate. Bigger is usually better for compass and straight edge layouts. The layout is geometrically identical for this special case. Extra layout lines can be constructed to use this shortcut on an odd numbered arm star. It is difficult to use this to set up a tapered arm star correctly.