RETRIEVING THE DESIGN METHOD OF

THE ISLAMIC DECAGONAL GIRIH PATTERNS

 

Dr. Hossam Aboulfotouh[1]   &   Dr. Gamal Abdelhameid

Department of Architecture, Faculty of Fine Arts, Minia University, Egypt

The 3ed International Conference of the Faculty of Fine Arts, Alexandria University, at Bibliotheca Alexandrina, 2007, Egypt

 

 

Abstract:

 

Peter J. Lu, et al, in their recent report in Science magazine, suggest that the decagonal Girih patterns on the Darb-i Imam shrine (Isfahan, Iran, 1453 C.E.) are quasi-periodic and were constructed by tessellation, using a set of five tile types, which they called girih tiles (Peter J. Lu et al 2007) and accordingly they devaluated the ability of the Medieval/Muslim architects. Contrary to what they have suggested, here we show that architects did use, and can use, a manual but swift technique to design and implement the decagonal patterns based on three types of overlapped but hidden grids with the same interval (two of them are tilted on both sides by 72° and 36°), using only the T-square and two triangle: 18°/72° and 54°/36°, without using the slow-compass. Besides, we show the simple and fast way that the students of the 1^st and 2^nd year architecture could follow in order to draw any decagonal Islamic pattern found in Iran (Peter J. Lu, et al 2007), Egypt and Morocco (Castéra, 1999 & Yves 1997); to create new patterns, based on thematic and repeated modular-units; and to retrieve the authentic Islamic motifs and visual-identity in the new architecture of the Muslim world.

 

Key words: Architecture Identity, Architecture Education, Islamic Motifs, Decagonal Patterns

                       

1-Introduction:

 

This paper discusses a technical point on the micro architectonic level, i.e., the decagonal Islamic patterns, but it primarily links it with a cultural issue on the global level, which aims to retrieve the authentic architectural identity in the cities of the current Muslim world.  It links the competition of arts in the architectonic realm with the competition of ideologies in the era of globalization. Today, Islamic ideology and identity has become in the forefront of political and scientific debates alike. In one hand, the scientific communities of the western countries dedicate the current weakness of the Muslim nations in most areas of scientific, technological, architectural and cultural aspects to, what they have called the Islamic regimes. The international journal "Nature" had published a series of biased articles on this subject (Nature news at www.nature.com), which enraged many Islamic intellectuals from around the world. Some of them have responded with comparing between the developmental conditions/statuses of the Western and Muslim countries in the medieval ears, i.e., the golden eras of the Islamic nations. But without discussing what had been changed, e.g., what are the causes of the developmental boom of the capitalist nations and the sharp decline of the Islamist nations? On the contrary, there are researchers and intellectuals who are none biased to any illogical notion. They compare the results of the nowadays discoveries, as well as the architectural and cultural development with what have been done in the Muslim counties during the Medieval eras. The paper of Peter J. Lu et al, on the Islamic Girih patterns is such an example (Peter J. Lu et al, 2007), however, it devaluate the design-patents of the Muslim architects.  Despite, they do not declare clearly that the advancement of those days had been done by the Muslims themselves, as some have stated, in the fields of astronomy and architecture (Castéra 1999, Nallino 1911), it shows at least that those creative ideas were the outcomes of, or have been inspired by, the Islamic thoughts and ideology, which had never happen before then. Besides, that situation may imply that despite the current anti-Muslim campaign, the nonbiased western researchers respect the achievements of the Muslim civilization.

 

However, enforcing the new trend of the globalization on the Muslim world threaten the continuity of interpreting the Muslims' identities in their architecture, i.e., the physical character of cities in the Muslim countries.  That issue is primarily concerning the transformation of the architectonic-urban dialogues, which are usually supported by the political regimes.  In Egypt, at the beginning of the 19^th century, Mohamed Ali started such movement in order to change the visual image of Cairo into the European character (El-Rafey, 1930 & El-Hamshary et al 2006).  In the 20^th century the transformation was influenced by the establishment of a systematic architectural study in Cairo University, which introduced the internationally new architectural trends into the local architectural education. Accordingly, that consolidated the industrial sector of the western countries more than retrieving and emphasizing the authentic architectural trends (Aboulfotouh 2005b) that Mohamed Ali's regime had started to abolish hundred years earlier, and was accented by Khedive Ismaeil at the late 19^th century. Now, at the beginning of the 21 century, the transformation is combined with replacing the basic static and abstract shapes, of the 20^th century architecture, by the dynamic shapes, entering into the realm of applying the topological sciences (Borisovich, et al 1985) on the architectural transformations, by moving, rotating, compressing or stretching the static shapes and forms

 

The pace of that kind of movement seems faster than the process of understanding the basic scientific knowledge behind creating the medieval but authentic architectural Islamic images. From the point of view of the young architects, following the new architectural trends may look easier than investigating on the process of creating the architectural designs during the golden Islamic eras. Besides, today one can hardly finds books in the Arabic library that might were written by the architects of the medieval eras and that show precisely their design philosophy and the process of creating the architectural Islamic motifs. It seems that the architects of those days (and perhaps the case is the same in other cultures too) had never written these kinds of books like those we read in the other social sciences. All what we have now are such theories and postulations by the nowadays researchers, and many of them are none Muslims, and their approaches are highly doubtful, particularly their approach on enterprising the Islamic ideology into the geometrical patterns, but they produce professional books though. These books (e.g., Castéra 1999 & Yves 1997) benefit the students of architecture in the western counties, because they can read the foreign texts, and many of the Arabic students can hardly read them, and thus they read only their pictures.  Concerning the Islamic patterns, the Muslim scholars on the contrary, search only on analyzing the outputs of these patterns without showing the process of designing and developing them. Al-Nahass's book is such an example (Al-Nahass 2007); despite it includes many patterns, it lacks the explanation.  That condition shows one of the arrays of weaknesses in the realm of authenticating the architectural education in the Muslim counties.

In the Egyptian realm, it is simply the lack of the authentic references for educating our architectural students, and re-continuing what Mohamed Ali had abolished, nearly two hundred years ago.  The lack of architectural books from the golden Islamic eras implies that the architectural education in those periods was handed from generation to generation on verbal and practice bases, without any writings. Despite, their progress and creativity in architecture designs, we can hardly find any sign to prove that the systematic architectural education did existed in those golden eras. Accordingly, in this context, the point in time when the political regime started to support any new architectural trend is the same moment of stopping the continuity of handing the verbal architectural courses from generation to generation, i.e., that process of verbal but practical education did not include studying the philosophy and the history of architecture, that started with the establishment of Cairo University. By the early decades of the 20^th century, the intruded foreign architectural trends had dominated the Egyptian cities and weakened the competitive ability of the authentic architectural dialogues of the Islamic trends. The works of the local but foreign architects did support that discontinuity. And even within the context of the systematic architectural studies, the authentic architectural trends were never given hands and feet on legitimate, constituent and communal bases, due to the lack of the political well (Aboulfotouh 2005b).

 

The Egyptian case is nearly similar to the cases of other Muslim countries.  In these countries, the current transformations of their architectonic-urban dialogues are indirectly dominated by the new trends of the western countries, leaving no chance for those of the Muslim world to conserve their architectural identities. Many young architects from the Muslim counties imitate the designs of others without understanding the basic sciences of either the most authentic images or the most innovative images. As an example, concerning the new topological trends, very few architects understand what topology is, due to that topology (Borisovich, et al 1985) is not taught as basic architectonic subject in the first year of our architectural education.

 

In this paper we show the development of such architectonic thought till it reached its climax of application and before its decline due to the above mentioned reasons that abolished the authentic architectural image in our cities. The hidden message of this paper is to show that the architects of these days can easily master the architectonic dialogue of our cities as it was in the medieval times, and making it able to compete with the western architectural trends, using the application of the architectural Islamic patterns as an example. However, its core objective lies within the domain of the architectural education, by retrieving the easiest process for designing and creating the architectural patterns that not only speaks on behalf of the Islam, but also fits the current needs,  resources and human skills. We focus here on the decagonal girih patterns that found in Egypt and elsewhere.

 

 

2- The Numbers and basic shapes:

 

Mathematics and descriptive geometry are coherent subjects in the architectonic filed. They are as old as the architectonic profession, which goes back to the days of the pyramids builders, nearly 5000 year ago. In those days the architect was not only the designer and the producer of the architectonic enclosures; he was more like a philosopher who understands the metaphysical laws of nature (Aboulfotouh 2007b). Thus, his designs were not only to create the most suitable forms and spaces that enclose the human functions, and achieving the safety, flexibility, privacy, territoriality and protection, but also to create the image that accent the identity of his society in the three dimensional representations. The design language he used draws its letters from the field of mathematics. In this field the architect studies the numbers and its spatial representation. Al-Maqrizi said in his famous book Al-Ketat that the priests of the ancient Egypt did understand the relation between the basic geometrical shapes (circle, square, triangle, etc) and the motions of the planets and the moon (Al-Maqrizi 1849). The Physagorasean were observing each of theses shapes as it indicates a number, e.g., the triangle is three, the square is 4, the pentagon is 5, the hexagon is 6, the decagon is 10 and so on until the shape becomes close to a circle that indicates the numbers from 360 till infinity; besides the dote is one and the line is 2 (McLeish 1992). These basic shapes (or basic systems in the 2 dimensional plane) are the primary elements for constructing any space, starting from the three dimensional man-made architectonic spaces on earth, up to the god-made eleven dimensional spaces of the celestial systems, i.e., the complex systems (Ahoulfotouh 2004).

 

 

In this paper we focus on the use of the basic shapes as the thematic modules for creating the architectonic patterns, either on straight or curved planes, but not on the stretched or compressed topological planes. Since the corners of all basic but concentric shapes are distributed on equal arc distances on the perimeter of a circle, we propose two conditions for defining the basic architectonic shape: each of them is constructed from triangles, and the total sum of its inner angles should equal 360°.  Despite that the triangle of equal sides and each of its three angles equals 60°, is the first basic shape, the second condition discards it out of the proposed definition, but it makes the right angle triangle the basic repeating unit (or the module of constructing any basic architectonic shape). 

 

Accordingly, the square is the first basic architectonic shape that qualifies the two conditions of the above definition. Without using the compass, we can not draw a square using only the T-square and a scale, we need other drawing tool. The ancient architectonicians had thought about that and thus they invented the triangular tool of the right angle that could be used to draw any basic shape. For the square, they had divided 360° by 4 and got 90°, which is not below 90°, thus they divided it again by 2 and they got 45°. Accordingly, they invented the triangle 45°/45°, in order to construct a grid of squares, without using the compass, and using the scale to measure only one direction. Similarly, for constricting the hexagon and the hexagonal grid, they invented the triangle 30°/60°, by dividing 360°/6=60°, and then dividing 60°/2=30°.  For the pentagon, we divide 360°/5=72, and then divide 72° by 2 and get 36°. Thus, for the pentagon we use two triangles: 18°/72° and 36°/54°. Today one can hardly find these triangles in the stationery shop, but he can find the triangle of various angles instead, and that we can adjust it to any tilting angle.

 

Similarly we can get the values of the two angles of the triangle that we may use to draw any basic shape and construct its grid. Taking into consideration that the needed triangle(s) for drawing a square is the same as for drawing the octagon and that needed for the pentagon is same as the decagon. Using this very simple rule, one can identify the types of triangles that are needed in order to draw the grids for constructing any architectonic pattern that is based on the repeated concentric shapes.

 

Moreover the repetition of basic shapes on a plane surface in order to construct a grid is conditional to the order of symmetry of that shape. Symmetry could be defined as the ability to divide any shape into mirrored parts around an axis. Only the basic shapes have this self architectonic characteristic, and the symmetrical order of each depends on how many symmetrical axes the basic shape has. The square has 4 axes, the pentagon has 5 axes, and the hexagon has 6 axes. However, this self-characteristic do not imply that the basic shape could be used alone as a uniformly repeated modular unit in order to construct a grid on a plane surface. In addition to the triangle of equal sides, the only two basic shapes that have this second characteristic are the square and the hexagon, and that might was the reason why the triangles 45°/45° and 30°/60° were and still are the most important triangular-tools in the architectonic designs. If we draw the triangular and hexagonal grids with the aid of the triangle 30/60, as shown in fig-1 (left-side), we notice that these two grids repeats vertically with equal intervals (like x) and with 1.15 x on the horizontal direction, creating a repeating modular unite of a square shape (30x * 30x). It implies that the grid of squares is the master grid of their basic shapes.  Not only this, due to that the rest of the basic shapes do not have the characteristic of uniformly repeating them on the plane surface, the grid of squares is their master grid too, being the base for constructing the repeated module of all basic shapes.

 

      

 

Figure-1: The grids of patterns of different basic shapes: the triangle and the hexagon (left-side), the decagon/pentagon (middle), and the square and the octagon (right-side) 

 

 

3- Interpreting the ideology into geometrical patterns.

 

The Ideology of any nation implies their thoughts-on, believes-in, and interpretations-of three laws in their daily live and working mechanisms: the metaphysical law of creation, the law of adoration, human relations and justice; and the law of governance and development (Ahoulfotouh 2005b).  In most ancient civilizations, their architects were given the authority to study and to interpret the metaphysical laws of nature in their architectonic designs. They were not forbidden to imitate the shapes of any living or mythological creatures, as pictures on walls and ceilings or as statues inside or outside their designed buildings. Due to the myths that were created by imposters and bad soothsayers on some of this artistic figures, that were perfectly grafted with the architecture design, the illiterate people worshiped them, and forget that  Allah (the divine cosmic-name of the great Geometer with his multiple names, in different tongues) is the sole creator of these images (Aboulfotouh, 2006). Worshiping the man-made statues and art-figures were insulted and prohibited by many profits (in Koran) and philosophers in the ancient times; they showed the difference between respecting the works and the achievements of the man in the figure or the hidden natural characteristic of such creature and that were made by the sole creator; and they emphasized on worshiping only the great Geometer. But not before the establishment of the strong Islamic governance system, in the Arabian-Peninsula and expanding its administrative jurisdiction to include other countries that formed, in latter periods, the geographic domain of the Islamic empire, any other ideological regime was not able to abolish that offensive ways of illiterate adorations completely. Only with the empowerment of the verses of the holy Koran it was finally accomplished with no return. Under the umbrella of the Islamic regime, that holy objective was accomplished in two ways, but their corresponding administrative actions differed from country to country.  First, prohibiting the imitation of the living or mythological creatures in any kind of art or grafting it with any architecture.  Second, the existing artifacts that include these kinds of things but not illiterately worshiped may remain on conditional bases, based on the opinion of the Islamic governor "Wally", at that time. In Egypt Amr Ibn Al-Ass, did not demolish the Ancient Egyptian temples and statues, but in latter period Ibn Tulun demolished some statues (Al-Maqrizi 1849).

 

Concerning the first condition, in those days the architects of the Islamic empire, whether they were Muslims or none Muslim, were enforced to find other alternatives to beautify their new buildings and that do not violate the verses of the holy Koran. Their applications differed from place to place depending on their local but inherited architectonic experiences. All of them did find the solution in using the basic shape to interpret the metaphysical laws of the great geometer on the walls, the facades, the domes and/or the ceilings of their buildings (Hossam Aboulfotouh 2006). Besides, the Islam did not prohibit the studies of the astronomical and astrological subjects, but on the contrary, these subjects were greatly advanced during the early Islamic eras by many scholars (Nallino 1911 & Pedersen 1993), continuing  the researches  of the ancient Egyptians priests and  the Greek philosophers, and that are supported by studying mathematics and geometry.

 

The scientific heritage in these two fields contains many theories that were used in architecture designs. Interpreting the dynamic motions of the celestial bodies, that occur in multiple frames and multiple dimensions into two-dimensional geometrical forms on a plane surface were initiated by the ancient Egyptians, in the so-called the Ankh diagram, the symbol of cosmic life (Aboulfotouh 2007b).  When observing the motion of any planet, i.e., the number of times it changes its shape, position or appearance, one can interpret that number in a basic form. For example, during the lunar month, one can observe the eight different shapes of the moon, from the first crescent till it becomes full moon and then reduces till it disappears, and starts a new lunar month again. The ancients thus, had linked the octagonal shape with the moon and the lunar year. Also, they linked the square with the Sun for its four seasons. Similarly, they linked the triangle with Venus, the square with Mars, and the hexagon with Saturn. In short, they linked the basic shapes with the cosmic motion it indicates.

 

All ancient civilization were respecting number 10 because it’s the sum of the first four numbers 1, 2, 3 and 4, and it’s the divider of the most ancient and basic cosmic measurement units: meter, kilometer or above that (Aboulfotouh 2005a). Physagoras and Euclid had talked about the characteristics of the perfect numbers like 6 & 28, and gave a definition to them without any cosmic reference, but Aboulfotouh have established the basics of a new scientific field that called "the Architectonic Mathematics" which shows and investigates on the cosmic and physical characteristics of any physical-number (Aboulfotouh 2004). Using the characteristics of the numbers and what it indicates from the different ideological points of views and that were changed from place to place and from civilization to civilization, the architects did use in their designs the numbers and their corresponding basic shapes that symbolize the ideologies of their societies. In the early ears of Islam in Egypt, the Coptic designer of Ibn Tulun's Mosque in Cairo did use different hexagonal patterns in more than hundred windows in that mosque (Al-Maqurizi 1849). The decagonal patterns are closely linked with the Shiaa Doctrine (el-Ashria or the ten Imams), thus we find these patterns on most Shiaa buildings, like the Mehrab of Al-Azhar mosque in Egypt and Darb-i Imam Shrine in Asfahan and in Moroco too, as shown in fig-2. Also, the Egyptian Christians respected the number ten, and used the decagonal shapes in their early churches in Egypt of the 7^th century, e.g.., in the Hanging church in Cairo. The Sonnies' Muslims did use the octagonal patterns that are related to the moon and lunar years (Higry). Similarly, one could use the symbolic meaning of any number in order to create any architectural pattern that speaks on behalf of the society that respect or understand that geometrical symbol.

 

 

 

Figure- 2: Examples of using the decagonal patterns in Shiaa buildings.

 

 

4- Drawing and designing the decagonal patterns:

 

Figure-3 shows the postulation of Peter J. Lu et al, in their reports; as they suggest that the decagonal girih patterns on the Darb-i Imam shrine are quasi-periodic and were constructed by tessellation, using a set of five tile types. (Peter J. Lu et al, 2007a).

 

See figure-3 (as Fig-1 & 2) at this link: http://www.sciencemag.org/cgi/content/figsonly/315/5815/1106

 

Figure-3:  Peter J Lu's proposed Girih tiles (left-side), and on Gunbad-I Kabud tomb tower, in Maragha-Iran (right-side); it devaluate the creative abilities of the architectural designers in the Muslim countries, during the Medieval eras, and supports the industrial sector of the western countries when producing it as tiles (Peter J. Lu et al, 2007a).

 

Responding to that wrong postulation, Aboulfotouh's e-letter to Science magazine showed the hidden grid approach for designing and drawing the decagonal girih patterns (Aboulfotouh 2007a), which will be explained hereunder in details. See also the response of Peter J. Lu et al to Aboulfotouh Science magazine; he thought wrongly that the inner-level of details in an intermediate part of the decagonal patterns, that includes only the hexagons, which were used on the Gunbad-I Kabud tomb tower, in Maragha-Iran, include curves, as shown in fig-2  (Peter J. Lu et al 2007b)

 

Contrary to the approaches of the mathematicians towards understanding the process of designing and implementing these patterns, as we could architects design them manually; using only a scale, T-square, and various types of triangles, and produce their working drawings in 1:1 scale for the artisans. Since these patterns don't include circular curves, we can draw them swiftly without using the slow-compass. For the decagonal stars, we can use two triangles instead: 18°/72° and 36°/54°, which are not produced today, but the triangle of various angles is an alternative.

 

        

 

Figure-4: The process of drawing a decagonal stare without using the compass.

 

To draw a decagonal-star, as shown fig-4, if x is the radius of its inner circle that the ten sides of the inner decagon are its tangents, we draw a square that its side equals 4x, and draw the perpendicular grid 4x*4x inside it. Using the triangle 18°/72°, and starting from the "center of the square", we draw the second but inclined grid every x, and its tilt equals 72° on both sides. Similarly, using the triangle 36°/54°, we draw the third but inclined grid, every x, and its tilt equals 36° on both sides. Then, we can observe the perimeters of the decagonal-star and its surroundings, i.e., one of the multiple design outputs of the three hidden-grids.  

 

      

 

Figure-5: the decagonal patterns that flow the rhythm 1.4x-2x (Left-side) and the rhythm 2x, 1.5x (right-side) with the repeating thematic-unit of 4x*5.5x (in red)

 

If we repeated the 4x*4x unit vertically, we would produce a vertical strip of repeated stars, every 4x. On the horizontal direction there are various options. While the horizontal grid repeats every 4x, the vertical grid may follow various continuous rhythms, e.g., 2x, 1.5x, 2x or 0.7x, 2x, 0.7x. The rhythm 2x, 1.5x generates a fifth decagonal-star in the middle of each four stars, creating the stagger-shape, as shown in fig-5 (right-side), within the repeating thematic-unit of 4x*5.5x. The rhythm 1.4x-2x generates the grid-shape, as fig-5 (lift-side) and the repeating thematic-unit of 4x*3.4x.

 

If one analyzed the girih patterns that were designed by the architects in different places and during different Islamic eras, he may observe that some of them are small portions of repeated thematic-units that its vertical but hidden grid-sides were rotated, inclined, and/or located outside the domains of the design-motifs, and in other cases like that on the Drab-i Imam Shrine in Asfahan, the inner-grids were subdivided in order to design the second and third level of inner-details.

 

4- Conclusion:

 

Great architectures are the outputs of understanding the basic knowledge; this basic knowledge includes the accumulation of sciences of many scientific disciplines. Despite that the architects of the golden Islamic ears did not wrote any books on their philosophy of design, their master pieces that still standing on earth now  show and prove that they were mastering these kinds of basic knowledge and that made them able to produce their great architecture. We the architects of today should learn from their practices. The architecture education in our universities should give some emphasis on the basic sciences that supports the design abilities and capabilities of our students, not only to cope with the new trends but also to retrieve the old authentic lessons and to understand how to do it in the likely effective contemporary ways to conserve the Identity of our Arabian societies.  Concerning the minor example of the decagonal patterns that we had shown in this paper, designing and implementing these patterns without tessellation were and still are not difficult tasks, as some none-architects from the western countries like to limit our authentic architectonic abilities in this marvelous realm, that we call the architecture of man and that was inspired by the architecture of the great geometer, our God Allah.

 

 

6-References:

 

 

1- Aboulfotouh, Hossam M. K. (17 April 2007a), the Hidden Grids of the Decagonal Girih Patterns, E-Letter to Science Magazine.

http://www.sciencemag.org/cgi/eletters/315/5815/1106

 

2- Aboulfotouh, Hossam (2007b), the Relativistic Tilts of Giza Pyramids' Entrance Passages," MAA International Journal Vol. 7 No. 1, (accepted for publication).

 

3- Aboulfotouh, Hossam (2006), What Islam have introduced to Architecture?, And The Architecture of the Ethereal and Material Bodies, Portrait magazine, Vol, 3 and Vol 4, Cairo.

 

4- Aboulfotouh, Hossam (2005a) The Horizon Theory-Part-II: Internal design concept of the great pyramid,  proceedings of the first conference of the Department of Conservation, Minia University, Minia, www.fotouh.netfirms.com/great-pyramid-introduction.htm

 

5- Aboulfotouh, Hossam (2005b), The Global Ideology and the Architectural Heritage: between vulnerability and empowerment, proceedings of the UIA's IIXX World Congress of Architecture on Cities, Grand Bazaar of ArchitectureS, Istanbul, www.fotouh.netfirms.com/uia2005-Ideology-461.htm

 

6- Aboulfotouh, Hossam (2004), Determining Planetary Spin and Musical Gravitation in the Spheres of Cosmic Systems of Perfect Numbers," ã 5984-2004 Dar el Kutub, Cairo, ISBN 977-17-1386-X www.fotouh.netfirms.com/spin-gravity-introduction.htm

 

7- Al-Maqrizi (1849) Al Mawaes Wal A'atebar Bezeker Al-khetat Wal Asar (Sermons and Lessons with the Discourse on Alleys and Monuments), Vol. I, Dar Al-Tahrier, Bolaque Edition, Cairo.

8- Al-Nahass,  Osama (2007), Islamic Patterns, Cairo.

 

9- Castéra, Jean-Marc (1999), Arabesques, Decorative Art in Morcco, ACR Edition, International Courbevoie, Paris, pp. 250-256

10- Borisovich, Yu, et al (1985), Introduction to Topology, Mir Publications, Moscow.

11- El-Hamshary, Mohamed & Abdel-Hameid, Gamal (2006), The Issue of Development and local Architecture, proceedings of the fifth Conference of the Faculty of Fine Arts, Helwan University, Cairo.

12- El-Rafey, Abdel-Rahman (1930), Mohamed Ali's Era, Cairo.

 

13- Korbendaq, Yves (1997), L'Architecture Sacrée de L'Islam, ACR Edition, International Courbevoie, Paris, p.177

 

14- Lu, Peter J. & Steinhardt, Paul J. (2007a) , Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture, Science, Vol. 315 no. 5815 pp. 1106-1110  [Abstract] [Full text] [PDF]

 

15- Peter J. Lu, et al.   (17 April 2007b) Response to H. M. K. Aboulfotouh on the Hidden Grids of the Decagonal Girih Patterns, Science magazine

http://www.sciencemag.org/cgi/eletters/315/5815/1106

16- McLeish, John (1992), Number from Ancient Civilizations to the Computer, London.

 

17- Nallino, Carlo, Arabian Astronomy: Its History During the Medieval Times, Oriental papers for publications, second edition in Arabic, Beirut, 1993. (first edition, 1911)

18- Pedersen, Olaf (1993) Early Physics and Astronomy, a Historical Introduction, Cambridge University press, New York, pp. 160-163.