islamic geometric ornament: the 12 point islamic star. vi: al azhar panels; complexity. 12 plus 8...

7/16/2019 Islamic Geometric Ornament: The 12 Point Islamic Star. VI: Al Azhar Panels; Complexity. 12 plus 8 Point Star Tiling

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Part VII: Al Azhar Panels in Stone; Complexity.

Twelve Point plus Eight Point Star Tiling

Islamic Geometric Ornament:

Construction of the Twelve Point Islamic Star

These two rather remarkable panel compositions are found on the facade of the Mamluk Al Azhar mosque of

Cairo (in monochrome stone).9a They make very different visual impressions and it takes a moment’s reflection

to realize that these are exactly the same pattern.

The fourth order Islamic star on the right is very common in the surviving fine quality work of Mamluk Cairo.

Various versions appear in wood, stone and marble in many of the famous mosques and complexes. The eight

point star of linked almond shaped kites on the left is also common in other Mamluk work and later Ghurid

work.There are several remarkable facts about this pattern. The central star is a 12 point star, the corner star in the

square tiling an eight point star. Yet the pattern is strewn with what appear to be regular pentagons. These

pentagons are formed by by elements of both the 12 and eight point star.

--How is the pentagonal symmetry element produced in a twelve point symmetry layout?

A mixed tiling of eight and 12 point stars was already constructed in Part VI.

--How closely related are these two figures with identical symmetries and tiling?

Alan D Adams, Holland, New York, 7 June 2013. License: Creative Commons -Attribution 3.0 Unported (CC BY 3.0) Text, photos and drawings.



The fourth order Islamic star was already constructed in part V, with the example from the Blue mosque of

Aqsunqur. The examples constructed there were parallel arm stars. As a result, the external angles of the star

are those of the sixfold symmetry polygons, 120°.

As always, moving the intersection of the arm end along the bisector of the minor layout circle introduces taper

into the star arms. There are an infinite number of tapered stars.

How is a taper selected to fit with the external angles of pentagonal symmetry? The taper here was selected at

random. It is close to the desired pentagonal symmetry, but noticeably inaccurate on close inspection. Several

more guesses would arrive at a solution, but the method is not very efficient. There also arises the problem of

how to reproduce and scale the figure. Choosing at random has some problems.

http://www.scribd.com/doc/138956805/Islamic-Geometric-Ornament-The-12-Point-Islamic-Star-5-Blue-Mosque-of-Aqsunqur





A pentagon could be constructed and the twelve point star built around its roof angles, but this is also not very

practical. A deeper look at possible divided circles provides the answer. The circle is almost always initially

divided with staggered hexagons due to the simplicity of the layout. There are other ways to lay out 12 point

divisions. The fourth order star base layout circle always needs to be moved inward from the tiling polygon.

This was accomplished with hexagons in the figures above. It can also be accomplished by spacing squares.

Only one square is needed to define the dark blue basic layout circle. At least two are needed for the next step.

The figure is examined for a defined intersection which will yield the desired angle.

The possibilities are simply surveyed, and the desired angle is found to be accessible. It is not uncommon to

find that the desired angle, or a close approximation, can be constructed easily.



The angle marked as 54° is not exact. The calculated angle is 53.894°. The 0.106° error, roughly two parts in a

thousand, is certainly not exact but is only a problem for the more extreme precision requirements. In a

compass and straight edge construction, the 57° angle would have been a clearly inferior choice. The 55° angle

might or might not have been noticed as inexact.

This is the first example of empirical measurement in these constructions, but it is important to note that it is not

used to construct the layout. Measurement testing of the possible layout intersections, which is done most

easily with a carefully cut template, is used to define what intersection will be used in the layout work. Layoutsare still defined by intersections. Clearly this approximation will yield hexagonal figures with excellent

precision by visual standards. There is probably more than one such empirical solution, but a similar solution

was almost certainly used for the historic figures. No one wants to construct 12 individual pentagons!

As another useful curiosity, the approximate pentagons fit exactly in the basic layout circle. This layout will be

easy to use and scale. These observations construct about 80% of the chapter head figure with a trivial

adaptation of the usual method. The next part is not so obvious.

The eight point stars, in the center of the left side panel or in the corner of the right side panel, look very

different from the stars constructed in earlier discussions. This eight point star of linked almond shaped kites is



common and it is frequently not as different from the normal stars as it looks. To give the best visual harmony,

the eight point stars of the pattern are constructed based on the same enforced external angle as the large 12

point star. The left side fourth order star below would respond exactly to the fourth order 12 point star. This

rather odd reversed taper star is sometimes seen, but it is rare. The star could also be completed as third order

normal tapered arm eight point Islamic star in the center or the interlaced kite star on the right, a completely

different lacing of the same layout. All of these stars would mate to and respond to the larger 12 point star.

The forced choice of this external angle directly results in the unusual layouts with a reversed or a very strong

taper in the two left stars. Neither is a particularly appealing star. The alternate lacing to the interlinked kite

star is often found in cases like this where the taper in the arm of the stars would be unusual. There are other

solutions to this problem, but they can result in out of scale distorted stars.

One problem remains with this figure remains before it can be laid out. How are the stars spaced? The base

layout of a 12 plus an eight point star was already developed in the first part VI.

The first thing to try is the obvious choice. Will the same layout ratio work? The first attempt is the normal

layout circles for both 12 and eight point stars. A square is bisected corner to corner. The 45° angle in one

corner is bisected and then the 90° angle is trisected.[This is a special case; a 90° angle can be exactly trisected.]

http://www.scribd.com/doc/146376472/Islamic-Geometric-Ornament-The-12-Point-Islamic-Star-VI-8-Plus-12-Point-star

http://www.scribd.com/doc/146376472/Islamic-Geometric-Ornament-The-12-Point-Islamic-Star-VI-8-Plus-12-Point-star



This is just 1/4 of the staggered hexagon layout. The intersection of the bisector and the trisector defines the

ratio of the 12 point and eight point layout circles. Both layout circles pass through the point indicated in red.

The 12 point layout circle is divided with staggered hexagons as shown and radii and inter-radii are drawn.

This circle transfers the eight fold division around the layout defined by the two indicated intersections. In this

case, staggered squares are used to space the star layout inward. The base layout circle for the fourth order star

is inscribed.

The intersection marked in red above is used to set the angle of the 12 point star arm. The intersection of that

arm end with the minor layout circle bisector defines the arm layout and taper as usual. [See App.II] Drawing

the three required layout circles and connecting them yields the star polygon.

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Completing the star by extending the layout inward to intersect and outward to the layout circle results in a very

pleasing tapered 12 point star. It is not yet clear how this will interlace with the corner eight fold stars. They do

not overlap. More layout elements are obviously required.

The ring of pentagons around the star, the interlace elements, are provided by extending the layout using the

external angle of the star.

These peripheral pentagons are constructed

with a circle, centered on the vertices of the

spacing squares with the radius set to the sameintersection originally used to set the arm end

layout, the “roof” of the pentagon.

This new circle defines four new points in

yellow where it intersects the star points and

the original base layout circle.

This pentagon is very close to a perfect

pentagon.

Extending the sides inward will define the

interlacing kites in the larger figure.

Layout circles are used to transfer the

intersections with the star points and all of the

new sides are drawn in. This construction

creates the first link between the star layouts.



If the eight point star is to respond to the 12 point star and complete the pentagonal elements, it should have the

same external angles as the 12 point star. The next task is to create that star to respond to the 12 point layout.

The dark blue line is the boundary between the 12 point and eight point layouts. The question is how to transfer

the structure and angles to the complimentary layout in the eight point space.

The structure of the eight point layout should reflect the angles and spacing of the 12 point star to preserve the

minor pentagonal elements. This is done by regarding the dark blue boundary line as a tiling plane andrecreating the structures and angles by reflection across the boundary. Simple cases of this problem have been

dealt with above and in previous examples. Reflection across a boundary is easily done by locating two points

equidistant from the reflection plane. These are almost always located with a circle.

The problem requires some preparation. Structures need to be extended to the reflection plane; the reflection

points and a convenient circles to give a cleanly defined intersection need to be identified.



The key point first used to define the external

angle of the 12 point star, in red, is

transferred by drawing a circle from the point

marked in blue, on the reflection plane, on a

layout radius.

The angle of the lines is determined byextending the arm ends of the 12 point star to

intersect the reflection plane. Connecting

these intersections with the point mapped by

the circle creates a reflected triangle.

The resulting similar triangle in the eight

point layout space is identical to the triangle

in the 12 point layout.

Similar strategies can be used to transfer any

point across the reflection plane.

The intersection defining the pentagonal

elements are transferred similarly.

The blue points on the reflection plane define

one end of a line; a circle from that point to

the intersection of that line at the triangle

sides defines the reflection of the other end.

With these two lines reflected into the eight

point layout, that star can be defined

completely.

One more decision needs to be made. The

star can be completed as a fourth order star,

exactly mirroring the 12 point star, a third

order star, giving a less crowded layout, or

the interlaced kite star. Here the interlacedkite is chosen. It is somewhat surprising that

the layout for that star is identical to the third

order star.

Using the transferred information, that layout

is built from the outside in.



The layout circle for the eight point star is completed, the circle is divided with two staggered squares and the

radii and inter-radii are drawn in. To construct the layout from the outside in, the end of the arm, point (a) in the

usual layout, must be identified.

With the point (a) identified, a layout

circle can be drawn through (a). One

square is drawn through any of the

eight points identified as (a) to locatethe minor layout circle. In this case,

the top of the pattern is chosen simply

to reduce crowding.

The required minor layout circle can

now be located and constructed. The

bisector is drawn and the layout

defines a complete eight point star-

constructed from the outside in.

Layout circles are drawn to transfer

the layout of the star arms and

additional layout circles are drawn to

locate the lines transferred down to

define the ring of pentagons. A total

of seven layout circles are required for

this type of star.

The star is completed as an interlaced kite but it still uses the layout of the normal eight point star for

proportions.



The star is completed by transferring the points around the layout and extending them inward, as usual, from

point (a). Completing the arms of the stars as shown on the right results in a layout which matches the Al Azhar

panels. There is some freedom to choose how the kites are defined, but this is perhaps the most common.

Completing the layout of the figure requires two simple operations. The ends of most of the pentagons will be

closed off by lines connecting points (p) and (p’), in blue. All of these lines are extended beyond (p) or (p’)

until they intersect a similar layout line.

Four lines, ending at points (q), in red, which will form interlacing kites are extended to intersect.

These final elements are relatively normal rules based closures of the pattern.

Seven layout circles are transferred to each corner to of the layout. The three more eight point quarter stars are

laid out and this rather elaborate layout is complete.

The layout is quite complex since it contains full layouts for five complete tapered arm Islamic stars. Despite

the number of operations, these five are not more complex than any standard star. Laying out the eight point

star from the outside in is unusual operation, but it is simply two steps in reversed order.

The only new and relatively complex operation introduced here involves transferring the layout angle from the

12 point star layout to the eight point layout.

The completed layout, with some of the construction lines used to transfer points across the reflection plane

omitted, is shown in full on the next page.



This figure should tile correctly as a square repeat, and it does. It is an even node figure, so it can be laced in

strict over under lacing.

Lacing the figure is straight forward but it introduces the last odd feature of this pattern. Two types of execution

of this pattern are found. The chapter head figure is drawn from this figure executed as both eight and 12 point

star centered frames in stone on the facade of the complex. The pattern is also executed beautifully in wood on

the minbar of the mosque of Sultan Qaitbay from Mamluk Cairo. The pattern is executed exactly as drawn here

on the minbar.9c

The stone panels are subtly altered.



As executed on the minbar on the left. In stone on the right.

The subtle change, which alters the appearance of the pattern noticeably, is made by moving the lacing of the

pattern to the side of the layout line, from the centerline.

All other aspects of the pattern remain unchanged in the Al Azhar panels. Small adjustments such as this are

not uncommon in complex figures. The changes are usually very small, as in this case, and very well defined.

This is avery complex figure which agrees with the historical work extremely well, but it is still based on the

simple rules set introduced in chapter one. It is a very nice demonstration of the power of A.J. Lee’s method.



The layout as drawn overlain on David Wade’s photo of the Al Azhar panel, EGY1007. Copyright to David Wade



References

9) a) Two panels on the facade of the Al Azhar Mosque, Cairo originally established 972 AH. Wade’s image

EGY1007, EGY1011 twelve star centered, EGY1005 Eight star centered.

b) Side of a minbar, al Ghuriyya Complex, Cairo established 909 AH (1504 CE). Wade’s image,

www.patterninislamicart.com EGY1721, EGY1722, EGY1723.

c) Side of minbar, mosque of Sultan QaitBay, Cairo 879 AH (1474 CE), Wade’s Image EGY1401,

EGY1402, EGY1403, EGY1406.

d) From tilingsearch.org, their reference T01472, data174/P117; Mosque of Ghanim al-Bahlawan, minbar,

Cairo, Egypt established 883 AH, (1478 CE).

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