The Islamic culture is one of the oldest cultures and also one of the greatest contributors to the sciences and mathematics fields. In particular, Islamic cultures have vastly contributed to the field of mathematics, including the modern use of the Arab numerical system amongst other contributions in geometry and cryptography. Furthermore, Islamic artists have been repeatedly known for mastering the use of geometrical implementation in works of art including parietal carvings and paintings. Recently, more research has emerged by senior historians and renowned scholars in the field of Islamic Art that suggests contrarian perspectives regarding the use of geometrical figures in Islamic art, especially in the medieval castle of the Alhambra in Spain. The current and increasingly accepted theory regarding the meaning behind the figures of Islamic art makes a wider use of mathematical principles that present a more objective interpretation which then gives scholars the freedom to anchor an explanation.[1] This paper seeks to provide evidence of the evolving understanding regarding the symbols in Islamic Art using the 14th century castle of the Alhambra, since the information and analysis are abundant for the Alhambra castle. The complex Islamic patterns perplex the mind through their intricate and divergent patterns, yet current scholars argue that Islamic artists used, on the contrary, simple geometrical figures; indeed, through various decorating phases they were able to form complex patterns. As a result, they elicited a deep interest amongst other architects through the complex façade of their works, and even a more profound amusement through their simplistic methods of accomplishing such. Such examples will assist my argument that Islamic artists extensively used geometrical patterns and mathematical principles in medieval art as political, religious, aesthetic, and possibly as laboratories of mathematical implementation in future works of art.
According to alhambradegranada.org[1], the official website of the Alhambra, the first historical documents of the Alhambra date to the 9th century. As Muslim Caliphates propagated their dominance over the Iberian Peninsula, they occupied the Alhambra and on the 13th century it was established as the royal residence by the Nasrid Dynasty and the emperor Mohammed ben Al-Hamar. Given that the Alhambra was located in a strategic valley that oversaw the whole city, it was of great importance to Muslim emperors during the time. Some of the best known patios and palaces of the Alhambra were not constructed until Yusuf I (1333-1353) took power. During his reign, and that of the proceeding Mohammed V, the edification of the Patio of the Lions, Justice Gate, Bath, Comares Hall and the Hall of the Boat were ordered for construction. An overview of the Alhambra can also show the annexation of new palaces and corridors as it changed management, see Figure 1.
In the bottom left section one can see a group of complexes. These were constructed during Islamic rule. Other buildings were edified when the Christians regained control of the castle, some of which include Charles V palace, which bears the name of the Catholic Monarch.[2] However, the main focus of this paper is on the main Islamic buildings, which can be seen on the
lower left of the complex. It is important to note the location and arrangement of the complexes since it will be helpful in clarifying some arguments that will be posed in coming paragraphs. More importantly, it is important to note the location and arrangement of the Palace of the Lions and the Comares Palace.
The Comares Palace
The most important structure of The Alhambra is the Comares Hall, which was the residence of the king and which also surrounded various other structures such as the Court of the Myrtles and the Hall of the Ambassadors. Being of such importance, if not the most important and highly decorated structure, it is evident that we would expect to find a multitude of geometrical figures and Islamic writings. Indeed, the Comares Hall is full of text and shapes that elicit one’s imagination while deeply perplexing it with their infinitesimal patterns. Even more, the walls of The Comares Hall are replete with ancient poems and Islamic verses. According to Valerie Gonzalez, “the texts, including Qur’anic quotations, poems and pious expressions, present an imagery full of pictures and poetic metaphors that create a highly animated textual world coexisting with the visual world.”[3] The contents of these texts ranged from religious texts describing the predominant theme of God to texts that glorified the king and elevated him to celestial heights; these poems repeatedly emphasized that the king was the closest one can be to God. Such poems included:
For how many are the gorgeous robes in which thou,
O Sultan, has attired it, which surpass, in brilliancy
of colour, the vaunted robes of Yemen.
You conquered Algeciras by the sword and opened a gate which had [until then] been denied to our victory...
Praise be to God!
The first and second poems demonstrate the political means that these poems were used for. They attempted to immortalize the emperor by making his victories forever remembered and his clothing of a celestial comparison. Furthermore, for any visitor or dignitary visiting the Alhambra during medieval times would be impressive and perhaps intimidating to see a leader being glorified to such extent. The last poem demonstrates a reoccurring theme not only throughout the Alhambra but also across all of Islamic Art, that is, the repeated use of religious metaphors. This is, of course, of no surprise; however, the importance of religious text is important given the location of the Alhambra, that is, in Southern Spain. Through these texts, we can safely infer that the Muslim kings included religious texts not only to express their non-secular background, but also to emphasize their firm and overarching power in the Christian-European continent. As a result, the poems and texts that can be found on the walls of the Comares Palace served as adornments through the melody-like style of Arabic writing. Furthermore, they also conveyed deeper meanings that were well-planned and considered. Poems that lionized the king and the Muslim God had ambiguous meanings, that is, they were meant to glorify and reaffirm the Dynasty’s power.
However, the direct and lively words of poems, like the ones mentioned, led a great majority of historians and visitors to the Alhambra with the perception that their sole intent was to communicate the braveness of the emperor and holiness of God while ignoring the endemic geometrical shapes that surrounded them. The answer is no, not all of the symbols and mathematical topography that surround poems and verses necessarily represented their meaning.[4] “The more closely one studies the functions and the iconography of its various parts… the more mysterious the buildings and their inhabitants seem.”[5] By this, Robert Irwin referred to the inherent complexity of the art and architecture of the Alhambra and the Comares Palace. However, from a mathematical standpoint, mathematics has continuously taught us that things can be made simpler, but not more simple, as Einstein once said. According to Lievovitch and Scheurle, who studied the complexity of fractals and Chaos Theory, never-ending patterns that seem too difficult to analyze or explained, can, most of the times, be made simpler through simple mathematical applications such as continuous functions. [6] In a similar manner, the confusing and foreign text and geometry of the Comares Palace can be made simple to understand as will shortly be analyzed.
There has been a strong agreement that the ceiling of the Comares Hall is a representation of the seven Islamic heavens, which is a false and rather complex assumption based on the mathematical analysis of the ceiling.[7] First of all, this analysis was done by historians who most likely did not have the string background to make the necessary analysis. It is understandable that other historians have sought to attach a meaning to the geometrical figures when there is none, since it is in their nature to provide the most feasible explanation given the evidence and their experience. The belief has been that the six rows of stars and the central cupola represent the seven heavens indicated by a scripture on a wall below the ceiling, see Figure 2. First, the argument falls apart since the stars and divergent patterns appear in other areas of the Alhambra, including ceilings, walls, floors, etc., and thus giving a less unique and important characteristic to the ceiling, which is meant to represent the seven heavens of Islam[8]. This would seem unconventional since a religious theme would deserve a more distinctive place and portrayal. Furthermore, the rows that gradually compress to surround the central cupola do not fully convey any evidence that they may be a representation of the celestial theme that has been widely accepted. Even if one looks for further patterns, which in this case it may seem highly probable due to the countless figures and patterns, one may not directly find any suggestion of the number seven. “In order that... a sentence should assert a certain fact there must, however the language be constructed, be something in common between the structure of the sentence and the structure of the fact.” [9] In other words, while there seems to be a link between the seven parallel lines that converge at the center, this is poor mathematical analysis which allowed historians to safely assume they represented the seven heavens. Even the star-like patterns found all around the ceiling may not represent celestial objects that would give an indication that the ceiling is a representation of heaven(s).
The Topkapi Scroll, see Figure 3, which was the blueprint artists followed to create the geometrical patterns for the Alhambra and many other medieval castles[10], can further explain that the ceiling of the Comares Hall was not intended to be a representation of the seven heavens. The geometrical patterns seen in the Appendix Figure is a page from the original Topkapi Scroll. These were constructed beginning in the lower left corner with a few simple shapes such as triangles and squares, and then continuing the pattern based on the previous shapes in a regression sort of fashion. As it can be seen, the purpose of this pattern is to create star-like shapes by continuing to fit other shapes around, without regard to how accurate they may look. Furthermore, the polygons are related to each other by means of the intersection of the circumferences of circles or by the Golden Mean .[11] When the geometric network has been constructed, the artist with a thorough understanding of geometry, or perhaps it could have been drawn by a skilled mathematician, had the liberty to fill the spaces as desired to account for any misalignments or incongruences. Similarly, the shapes of the ceiling of the Comares Hall can be better explained as a pattern that was meant to create the distinguishable star-like figures by continuing to surround these, quite randomly, with circles and other figures. Even though this explanation is simple and perhaps unsophisticated, it fits the context of and the ancient manual Muslim sculptors used to created architecture and art during the middle ages.
In other words, the geometrical figures were intended to kindle the mind and provide a large degree of freedom for individuals to create their own interpretations, one which cannot be proven nor refuted. According to James Turrel and Valerie Gonzales, the strong emotional power of the inscribed verses stimulate the mental associative skills to give meaning to the mathematical structures[12]. Thus, an important note to make is that many historians are not well equipped to do mathematical analysis, as it is the case with the ceiling of the Comares Hall, and were primarily guided by their own mental associative faculties and the resemblance of stars and constellations of the ceiling. “Associative constructions of the mind or iconographic metaphors created by analogy, these pcitures nevertheless cannot be identified with the didactic, material images that characterize representation.”[13] In the words of the renowned finance professor at Columbia Business School, “you’ll meet a bad fate if you extrapolate.” [14] In the chapter where Mauboussin mentions this, he explains the mistake many data analysts make when analyzing large compiled data. They seek to find patterns by mining information, and Mauboussin contends that they most often than not do find a pattern. The problem is that most of the times there is no pattern, but instead our brains are wired to make connections. In connection to the analysis of the ceiling of the Comares Hall, the analysis previously done may not have included the opinions of people with some mathematical understanding. Instead, much of the interpretations were guided by a historical context rather than by more objective guidelines.
Additionally, the inside of the Comares Hall can be credulously described as a masterpiece of geometry, see Figure 4. With its extensive and divergent patterns, it elicits the imagination to give meaning to their respective mathematical topographies. One such meaning is that the walls represent a division between earth and heaven. With its division between lower polychromatic tiles and upper revetment of sculpted shapes, the two opposites become metaphors for heaven and earth. [15] The upper shapes and patterns give a sense of movement, thus implying the chaotic and yet organized movement of the skies. As it can be seen, it is divided into four main horizontal platforms each carved by its own unique pattern. This reaffirms the belief that the upper division constitutes the heavens since it is quite hectic and yet there is a level of organization to it. Even more, the light that enters through the glassy windows is an indictment of the celestial nature of the upper division of the wall. The bottom tiles, on the other hand, with their regular and monotonous shapes, give the impression of ground which is generally conceived to be immovable. The different colors may represent different types of ground of different regions of the earth. Moreover, the vertical rectangles that surround the entrances are in perpendicular position with the horizontal amalgamation of shapes, as if directing the eye to continue looking from the ground and up. [16] According to Valerie Gonzalez, it is as if the eye moves in a defined direction, in contrast to wandering indifferently guided by the fractal nature of the carvings. [17]
Finally, the geometrical figures in the walls of the Comares Hall served a different purpose as those of their close neighbors in the ceiling. They served to direct the gaze of the viewer from top to bottom without getting lost in their infinitesimal motion. In contrast to the ceiling of the Hall, the shapes in the walls allowed for a greater degree of opinion in regards to their meaning. One of these included their embodiment of heaven and earth by the upper and lower division. Nonetheless, the geometrical patterns of the Comares Hall served for aesthetic and viewer perspective purposes, in addition to objective mathematical geometry whose sole intent was to continue a sequence of shapes. This characteristic is one that resembles the Comares Hall, most likely due to its importance and central viewing itinerary.
Palace of the Lions
After Muhammed V replaced his father Yusuf I, he finished the building projects his father had started in addition to building the beautiful Palace of the Lions (Palacio de los Leones). [18] It is this palace that deviates from the traditional and excessive use of geometrical and abstract decorations and where we see the stronger resemblance to Christian decorations and architecture, primarily guided by Muhammed’s close relationship with the Christian king Pedro the Cruel. [19]The Nasrid Palace obtained its name from the fountain in the courtyard, which includes sculptures of lions positioned in a circle and supporting a fountain.
Ø They have raised him to be like their king so that he became
a lion-whelp among them, and they became like oxen. [20]
In other words, the Court of the Lions resembles Solomon’s temple fountain of brass with the difference that the Court of the Lions is being supported by lions instead of oxen, see Figure 5.[21] This indicates the superiority that was meant to convey by replacing a prey beast with the superior characteristics of a hunter animal such as the lion. According to Bargebuhr, the ferocity and strength of the lions is a good representation of the characteristics a prince should embody, while the lions’ subjugation portrayed by supporting the basin fountain implies humility and consent, features that he believed a just king should also bear.[22] This is important in explaining the remodelations that Muhammed V continued after his father’s death and the implicit message Yusuf might have wanted to convey to his son regarding the characteristics of a fair prince.
Moreover, The Palace of the Lions is comprised of a central patio surrounded by various halls, such as the Hall of the Abencerrajes, where an elegant stucco ceiling can be found resting on an octagonal cylinder with eight windows, and with intricate stucco decorations. [23] As it was already noted, the Palace of the Lions carries a different essence than the rest of the structures in the Alhambra. The first difference is its perpendicular alignment in respect to the Court of the Myrtles and most of the other halls and palace, see Figure 6. It is this positioning that makes the Palace of the Lions stand out at first glance and provides a clue to the usage of the space and the semantics of the symbols.
While in the Comares Palace there was a strong emphasis on its celestial interpretation and its predominant use of geometrical figures to embellish the walls and ceilings and sometimes convey a powerful meaning, the Palace of the Lions carries a slightly different purpose. This Palace has caused confoundment and incited curiosity regarding its use. It is the design of the complex that supports a more widely accepted idea that the Palace of the Lions was used as a place to educate and inspire, or more commonly referred to as a madrasa which comes from the Arabic and refers to any type of educational institution, secular or religious. Indeed, it is argued that Muhammed V’s exile into North Africa inspired him to mimic the educational style of major Arab institutions in North Africa after returning to power. [24] Furthermore, the architectural style of the Palace of the Lions is indicative of this idea. As has already been mentioned, the Palace is compose of a “central patio... occupied by a quadripartite, sunken garden, emphasizing the four cardinal directions, or paved with white marble, calling to mind through invocations of the Qu’ranic parable concerning the Queen of Sheba and King Solomon (Qu’ran 27:44), its likeness to a shimmering sea.”[25] (see Figure 6). As it can be seen, the open structure of the Palace of the Lions has allowed for different interpretations; from a “pleasure palace” to a “new-throne room” and to a place dedicated towards the elite of academia. [26] Even further, the open structure of the pillars and the slightly cold temperatures of this region might suggest that the Palace of the Lions might not have been used as a “pleasure palace,” or even less likely as a “new-throne room.” This further supports the idea that the Palace of the Lions was a place dedicated for intellectuals and academia.
The stucco and text in the walls of the Palace of the Lions serves a different purpose as that of the Comares Palace. As it has been noted, the text and writings of the Comares Palace served a detached purpose from the geometry that surrounded it. It allowed for expressing the royalty of the king and the greatness of God. On the other hand, the geometry and text of the Palace of the Lions served the purpose of inspiring the pupils and intellectuals that attended the complex. The columns that surrounded the central fountain were decorated by marvelous stucco of intricate and complex nature, see Figure 8. According to Robinson, the carvings and texts engraved on the pillars were primarily inspired by poets such as al-Qartajanni who strongly emphasized letting paintings and engravings speak for themselves. That is, to allow text and verses to explicitly express their meaning and references in a direct and poetic manner.
The stucco and carvings were meant to represent light and leafy branches in water; even more, the verses and texts on the columns explicitly state that the column is a column and the wall a wall. [27]The overlapping stucco of one of the columns is meant to represent “leafy branches,” and it is quite clear by making a comparison to average branches. However, the stucco takes this idea further by its highly embellished patterns and Arabic writings which incite curiosity and perhaps kindle inspiration. The simple nature of the engravings is a metaphor for the complex purpose the Palace of the Lions was meant to represent. It is recurring ideas like these that have inspired the greatest philosophers, from Aristotle to Ptolemy. Indeed, Irwin has repeatedly suggested that this palace was built and ornamented by scholars for scholars in the most simplistic way possible and yet with the most complex intentions possible. [28]
Ø Oh, what rainment of embroidered stuff have you thrown about it! It makes one forget the tulle of Yemen!...
Her columns are so beautiful in every aspect that word
Of their fame has reached far and wide!
Her smooth, diaphanous marble brighten the farthest
Corners darkened by shadow...
It is clear from this verse that the engraving was meant to convey a clear meaning, mostly how beautiful the garden is and the level of contemplation that it deserves. [29]
In summary, the Alhambra has proven to be a rich place for research and analysis. Its interesting and ancient history make it more mysterious; the mathematical implementations on walls and architecture throughout the complex make it a valuable academic asset apart from its historical importance. The geometrical shapes and recurring patterns have perplexed many visitors and historians. However, it has been shown that their interpretation was easier than previously conceived. Furthermore, the Comares Palace also embodied texts and verses with a wide a range of themes and meanings. From political to religious, these poems had ambiguous meanings that were meant to convey authority in a Christian land while maintaining a strong religious influence. The Palace of the Lions also further showed to be a building with a recurring symbols and texts. Through their direct and motivational verses, the Palace of the Lions was meant to inspire academics through simple methods. Indeed, the idea of simplicity appeared very often throughout the discussion of the Alhambra. Given that only two complexes of the castle were analyzed, we can extrapolate and infer that the other buildings are characterized by simple geometrical figures and direct poetic verses. We can even infer that the sculptors of the artwork might have had knowledge of the Golden Ratio and used it in their carvings. It has been shown that the Greeks repeatedly used this measure, which means that this mathematical tool was available during the middle ages. This could be potential research in the future. Finally, we leave with a deep appreciation for a medieval castle that captivated our attention with its mathematical simplicity and stunning architecture.
[1] alhambradegranada.org, Area 25,Granada, 2000, November 12, 2014.
[2] alhambradegranada.org, November 12, 2014.
[3] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[4] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[5] Robert Irwin. “The Alhambra.” Harvard University Press. Cambridge, 2004.
[6] Larry Liebovitch, Daniella Scheurle. “Two Lessons from Fractals and Chaos Theory.” Complexity, Vol.5, 2000, 34-43. http://www.ccsfau.edu/~liebovitch/complexity~20.html.
[7] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[8] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[9] Ludwig Wittgenstein; Bertrand Russell. Tractatus Logico-Philosophicus. Oxford University Press. Vol. 32, No. 128.
[10] Gülru Necipoǧlu; Mohammad al-Asad. The Topkapi Scroll: Geometry and Ornament in Islamic Architecture. Speculum, Vol. 73, No. 2 (Apr., 1998), pp. 566-568
[11] Ettinghousen, Grabar, & Jenkins. “Islamic Art and Architecture 650-1250.” Penguin Books Limited, 2001
[12] James Turrel, Valerie Gonzalez. The Comares Hall in the Alhambra and Space That Sees: A comparison of Aesthetic Phenomenology. BRILL. Vol. 20 (2003), pp. 253-278.
[13] Turrel, Gonzalez. The Comares Hall in the Alhambra and Space That Sees: A comparison of Aesthetic Phenomenology.
[14] Michael Mauboussin. More Than You Know. Columbia University Press. 2007
[15] Turrel, Gonzalez. “The Comares Hall in the Alhambra and James Turrel’s Space That Sees: A Comparison of Aesthetic Phenomenology.” Muqarnas: Brill. 2003.
[16] Turrel, Gonzalez. “The Comares Hall in the Alhambra and James Turrel’s Space That Sees: A Comparison of Aesthetic Phenomenology.” Muqarnas: Brill. 2003.
[17] Ibid, pg 261.
[18] Area 25, alhambradegranada.org, Granada, 2000.
[19] Fairchild Ruggles. “The Alcazar of Seville and Mudejar Architecture” : The University of Chicago Press. 2004
[20] Frederick Bargebuhr. The Alhambra Palace of the Eleveth Century. Journal of the Warbaulg and Cartauld Institutes. 1956.
[21] Bargenbuhr. 1956.
[22] Bargebuhr. 1956.
[23] Fred Kleiner. Art Through the Ages. Wadsworth, Cengage Learning. 2013.
[24] Fairchild Ruggles. “The Alcazar of Seville and Mudejar Architecture” : The University of Chicago Press. 2004
[25] Cynthia Robinson, “Marginal Ornament: Poetics, Mimesis, and Devotion in the Palace of the Lions”. Frontiers of Islamic Art and Architecture: Essays in Celebration of Oleg Grabar's Eightieth Birthday. (2008). BRILL
[26]Ibid, 2008.
[27] Cynthia Robinson, “Marginal Ornament: Poetics, Mimesis, and Devotion in the Palace of the Lions”. Frontiers of Islamic Art and Architecture: Essays in Celebration of Oleg Grabar's Eightieth Birthday. (2008). BRILL
[28] Robert Irwin. “The Alhambra.” Harvard University Press. Cambridge, 2004
[29] Cynthia Robinson, “Marginal Ornament: Poetics, Mimesis, and Devotion in the Palace of the Lions”. Frontiers of Islamic Art and Architecture: Essays in Celebration of Oleg Grabar's Eightieth Birthday. (2008). BRILL
[1] Ettinghousen, Grabar, and Jenkins. Islamic Art and Architecture 650-1250. Penguin Books Limited, 2001.
According to alhambradegranada.org[1], the official website of the Alhambra, the first historical documents of the Alhambra date to the 9th century. As Muslim Caliphates propagated their dominance over the Iberian Peninsula, they occupied the Alhambra and on the 13th century it was established as the royal residence by the Nasrid Dynasty and the emperor Mohammed ben Al-Hamar. Given that the Alhambra was located in a strategic valley that oversaw the whole city, it was of great importance to Muslim emperors during the time. Some of the best known patios and palaces of the Alhambra were not constructed until Yusuf I (1333-1353) took power. During his reign, and that of the proceeding Mohammed V, the edification of the Patio of the Lions, Justice Gate, Bath, Comares Hall and the Hall of the Boat were ordered for construction. An overview of the Alhambra can also show the annexation of new palaces and corridors as it changed management, see Figure 1.
In the bottom left section one can see a group of complexes. These were constructed during Islamic rule. Other buildings were edified when the Christians regained control of the castle, some of which include Charles V palace, which bears the name of the Catholic Monarch.[2] However, the main focus of this paper is on the main Islamic buildings, which can be seen on the
lower left of the complex. It is important to note the location and arrangement of the complexes since it will be helpful in clarifying some arguments that will be posed in coming paragraphs. More importantly, it is important to note the location and arrangement of the Palace of the Lions and the Comares Palace.
The Comares Palace
The most important structure of The Alhambra is the Comares Hall, which was the residence of the king and which also surrounded various other structures such as the Court of the Myrtles and the Hall of the Ambassadors. Being of such importance, if not the most important and highly decorated structure, it is evident that we would expect to find a multitude of geometrical figures and Islamic writings. Indeed, the Comares Hall is full of text and shapes that elicit one’s imagination while deeply perplexing it with their infinitesimal patterns. Even more, the walls of The Comares Hall are replete with ancient poems and Islamic verses. According to Valerie Gonzalez, “the texts, including Qur’anic quotations, poems and pious expressions, present an imagery full of pictures and poetic metaphors that create a highly animated textual world coexisting with the visual world.”[3] The contents of these texts ranged from religious texts describing the predominant theme of God to texts that glorified the king and elevated him to celestial heights; these poems repeatedly emphasized that the king was the closest one can be to God. Such poems included:
For how many are the gorgeous robes in which thou,
O Sultan, has attired it, which surpass, in brilliancy
of colour, the vaunted robes of Yemen.
You conquered Algeciras by the sword and opened a gate which had [until then] been denied to our victory...
Praise be to God!
The first and second poems demonstrate the political means that these poems were used for. They attempted to immortalize the emperor by making his victories forever remembered and his clothing of a celestial comparison. Furthermore, for any visitor or dignitary visiting the Alhambra during medieval times would be impressive and perhaps intimidating to see a leader being glorified to such extent. The last poem demonstrates a reoccurring theme not only throughout the Alhambra but also across all of Islamic Art, that is, the repeated use of religious metaphors. This is, of course, of no surprise; however, the importance of religious text is important given the location of the Alhambra, that is, in Southern Spain. Through these texts, we can safely infer that the Muslim kings included religious texts not only to express their non-secular background, but also to emphasize their firm and overarching power in the Christian-European continent. As a result, the poems and texts that can be found on the walls of the Comares Palace served as adornments through the melody-like style of Arabic writing. Furthermore, they also conveyed deeper meanings that were well-planned and considered. Poems that lionized the king and the Muslim God had ambiguous meanings, that is, they were meant to glorify and reaffirm the Dynasty’s power.
However, the direct and lively words of poems, like the ones mentioned, led a great majority of historians and visitors to the Alhambra with the perception that their sole intent was to communicate the braveness of the emperor and holiness of God while ignoring the endemic geometrical shapes that surrounded them. The answer is no, not all of the symbols and mathematical topography that surround poems and verses necessarily represented their meaning.[4] “The more closely one studies the functions and the iconography of its various parts… the more mysterious the buildings and their inhabitants seem.”[5] By this, Robert Irwin referred to the inherent complexity of the art and architecture of the Alhambra and the Comares Palace. However, from a mathematical standpoint, mathematics has continuously taught us that things can be made simpler, but not more simple, as Einstein once said. According to Lievovitch and Scheurle, who studied the complexity of fractals and Chaos Theory, never-ending patterns that seem too difficult to analyze or explained, can, most of the times, be made simpler through simple mathematical applications such as continuous functions. [6] In a similar manner, the confusing and foreign text and geometry of the Comares Palace can be made simple to understand as will shortly be analyzed.
There has been a strong agreement that the ceiling of the Comares Hall is a representation of the seven Islamic heavens, which is a false and rather complex assumption based on the mathematical analysis of the ceiling.[7] First of all, this analysis was done by historians who most likely did not have the string background to make the necessary analysis. It is understandable that other historians have sought to attach a meaning to the geometrical figures when there is none, since it is in their nature to provide the most feasible explanation given the evidence and their experience. The belief has been that the six rows of stars and the central cupola represent the seven heavens indicated by a scripture on a wall below the ceiling, see Figure 2. First, the argument falls apart since the stars and divergent patterns appear in other areas of the Alhambra, including ceilings, walls, floors, etc., and thus giving a less unique and important characteristic to the ceiling, which is meant to represent the seven heavens of Islam[8]. This would seem unconventional since a religious theme would deserve a more distinctive place and portrayal. Furthermore, the rows that gradually compress to surround the central cupola do not fully convey any evidence that they may be a representation of the celestial theme that has been widely accepted. Even if one looks for further patterns, which in this case it may seem highly probable due to the countless figures and patterns, one may not directly find any suggestion of the number seven. “In order that... a sentence should assert a certain fact there must, however the language be constructed, be something in common between the structure of the sentence and the structure of the fact.” [9] In other words, while there seems to be a link between the seven parallel lines that converge at the center, this is poor mathematical analysis which allowed historians to safely assume they represented the seven heavens. Even the star-like patterns found all around the ceiling may not represent celestial objects that would give an indication that the ceiling is a representation of heaven(s).
The Topkapi Scroll, see Figure 3, which was the blueprint artists followed to create the geometrical patterns for the Alhambra and many other medieval castles[10], can further explain that the ceiling of the Comares Hall was not intended to be a representation of the seven heavens. The geometrical patterns seen in the Appendix Figure is a page from the original Topkapi Scroll. These were constructed beginning in the lower left corner with a few simple shapes such as triangles and squares, and then continuing the pattern based on the previous shapes in a regression sort of fashion. As it can be seen, the purpose of this pattern is to create star-like shapes by continuing to fit other shapes around, without regard to how accurate they may look. Furthermore, the polygons are related to each other by means of the intersection of the circumferences of circles or by the Golden Mean .[11] When the geometric network has been constructed, the artist with a thorough understanding of geometry, or perhaps it could have been drawn by a skilled mathematician, had the liberty to fill the spaces as desired to account for any misalignments or incongruences. Similarly, the shapes of the ceiling of the Comares Hall can be better explained as a pattern that was meant to create the distinguishable star-like figures by continuing to surround these, quite randomly, with circles and other figures. Even though this explanation is simple and perhaps unsophisticated, it fits the context of and the ancient manual Muslim sculptors used to created architecture and art during the middle ages.
In other words, the geometrical figures were intended to kindle the mind and provide a large degree of freedom for individuals to create their own interpretations, one which cannot be proven nor refuted. According to James Turrel and Valerie Gonzales, the strong emotional power of the inscribed verses stimulate the mental associative skills to give meaning to the mathematical structures[12]. Thus, an important note to make is that many historians are not well equipped to do mathematical analysis, as it is the case with the ceiling of the Comares Hall, and were primarily guided by their own mental associative faculties and the resemblance of stars and constellations of the ceiling. “Associative constructions of the mind or iconographic metaphors created by analogy, these pcitures nevertheless cannot be identified with the didactic, material images that characterize representation.”[13] In the words of the renowned finance professor at Columbia Business School, “you’ll meet a bad fate if you extrapolate.” [14] In the chapter where Mauboussin mentions this, he explains the mistake many data analysts make when analyzing large compiled data. They seek to find patterns by mining information, and Mauboussin contends that they most often than not do find a pattern. The problem is that most of the times there is no pattern, but instead our brains are wired to make connections. In connection to the analysis of the ceiling of the Comares Hall, the analysis previously done may not have included the opinions of people with some mathematical understanding. Instead, much of the interpretations were guided by a historical context rather than by more objective guidelines.
Additionally, the inside of the Comares Hall can be credulously described as a masterpiece of geometry, see Figure 4. With its extensive and divergent patterns, it elicits the imagination to give meaning to their respective mathematical topographies. One such meaning is that the walls represent a division between earth and heaven. With its division between lower polychromatic tiles and upper revetment of sculpted shapes, the two opposites become metaphors for heaven and earth. [15] The upper shapes and patterns give a sense of movement, thus implying the chaotic and yet organized movement of the skies. As it can be seen, it is divided into four main horizontal platforms each carved by its own unique pattern. This reaffirms the belief that the upper division constitutes the heavens since it is quite hectic and yet there is a level of organization to it. Even more, the light that enters through the glassy windows is an indictment of the celestial nature of the upper division of the wall. The bottom tiles, on the other hand, with their regular and monotonous shapes, give the impression of ground which is generally conceived to be immovable. The different colors may represent different types of ground of different regions of the earth. Moreover, the vertical rectangles that surround the entrances are in perpendicular position with the horizontal amalgamation of shapes, as if directing the eye to continue looking from the ground and up. [16] According to Valerie Gonzalez, it is as if the eye moves in a defined direction, in contrast to wandering indifferently guided by the fractal nature of the carvings. [17]
Finally, the geometrical figures in the walls of the Comares Hall served a different purpose as those of their close neighbors in the ceiling. They served to direct the gaze of the viewer from top to bottom without getting lost in their infinitesimal motion. In contrast to the ceiling of the Hall, the shapes in the walls allowed for a greater degree of opinion in regards to their meaning. One of these included their embodiment of heaven and earth by the upper and lower division. Nonetheless, the geometrical patterns of the Comares Hall served for aesthetic and viewer perspective purposes, in addition to objective mathematical geometry whose sole intent was to continue a sequence of shapes. This characteristic is one that resembles the Comares Hall, most likely due to its importance and central viewing itinerary.
Palace of the Lions
After Muhammed V replaced his father Yusuf I, he finished the building projects his father had started in addition to building the beautiful Palace of the Lions (Palacio de los Leones). [18] It is this palace that deviates from the traditional and excessive use of geometrical and abstract decorations and where we see the stronger resemblance to Christian decorations and architecture, primarily guided by Muhammed’s close relationship with the Christian king Pedro the Cruel. [19]The Nasrid Palace obtained its name from the fountain in the courtyard, which includes sculptures of lions positioned in a circle and supporting a fountain.
Ø They have raised him to be like their king so that he became
a lion-whelp among them, and they became like oxen. [20]
In other words, the Court of the Lions resembles Solomon’s temple fountain of brass with the difference that the Court of the Lions is being supported by lions instead of oxen, see Figure 5.[21] This indicates the superiority that was meant to convey by replacing a prey beast with the superior characteristics of a hunter animal such as the lion. According to Bargebuhr, the ferocity and strength of the lions is a good representation of the characteristics a prince should embody, while the lions’ subjugation portrayed by supporting the basin fountain implies humility and consent, features that he believed a just king should also bear.[22] This is important in explaining the remodelations that Muhammed V continued after his father’s death and the implicit message Yusuf might have wanted to convey to his son regarding the characteristics of a fair prince.
Moreover, The Palace of the Lions is comprised of a central patio surrounded by various halls, such as the Hall of the Abencerrajes, where an elegant stucco ceiling can be found resting on an octagonal cylinder with eight windows, and with intricate stucco decorations. [23] As it was already noted, the Palace of the Lions carries a different essence than the rest of the structures in the Alhambra. The first difference is its perpendicular alignment in respect to the Court of the Myrtles and most of the other halls and palace, see Figure 6. It is this positioning that makes the Palace of the Lions stand out at first glance and provides a clue to the usage of the space and the semantics of the symbols.
While in the Comares Palace there was a strong emphasis on its celestial interpretation and its predominant use of geometrical figures to embellish the walls and ceilings and sometimes convey a powerful meaning, the Palace of the Lions carries a slightly different purpose. This Palace has caused confoundment and incited curiosity regarding its use. It is the design of the complex that supports a more widely accepted idea that the Palace of the Lions was used as a place to educate and inspire, or more commonly referred to as a madrasa which comes from the Arabic and refers to any type of educational institution, secular or religious. Indeed, it is argued that Muhammed V’s exile into North Africa inspired him to mimic the educational style of major Arab institutions in North Africa after returning to power. [24] Furthermore, the architectural style of the Palace of the Lions is indicative of this idea. As has already been mentioned, the Palace is compose of a “central patio... occupied by a quadripartite, sunken garden, emphasizing the four cardinal directions, or paved with white marble, calling to mind through invocations of the Qu’ranic parable concerning the Queen of Sheba and King Solomon (Qu’ran 27:44), its likeness to a shimmering sea.”[25] (see Figure 6). As it can be seen, the open structure of the Palace of the Lions has allowed for different interpretations; from a “pleasure palace” to a “new-throne room” and to a place dedicated towards the elite of academia. [26] Even further, the open structure of the pillars and the slightly cold temperatures of this region might suggest that the Palace of the Lions might not have been used as a “pleasure palace,” or even less likely as a “new-throne room.” This further supports the idea that the Palace of the Lions was a place dedicated for intellectuals and academia.
The stucco and text in the walls of the Palace of the Lions serves a different purpose as that of the Comares Palace. As it has been noted, the text and writings of the Comares Palace served a detached purpose from the geometry that surrounded it. It allowed for expressing the royalty of the king and the greatness of God. On the other hand, the geometry and text of the Palace of the Lions served the purpose of inspiring the pupils and intellectuals that attended the complex. The columns that surrounded the central fountain were decorated by marvelous stucco of intricate and complex nature, see Figure 8. According to Robinson, the carvings and texts engraved on the pillars were primarily inspired by poets such as al-Qartajanni who strongly emphasized letting paintings and engravings speak for themselves. That is, to allow text and verses to explicitly express their meaning and references in a direct and poetic manner.
The stucco and carvings were meant to represent light and leafy branches in water; even more, the verses and texts on the columns explicitly state that the column is a column and the wall a wall. [27]The overlapping stucco of one of the columns is meant to represent “leafy branches,” and it is quite clear by making a comparison to average branches. However, the stucco takes this idea further by its highly embellished patterns and Arabic writings which incite curiosity and perhaps kindle inspiration. The simple nature of the engravings is a metaphor for the complex purpose the Palace of the Lions was meant to represent. It is recurring ideas like these that have inspired the greatest philosophers, from Aristotle to Ptolemy. Indeed, Irwin has repeatedly suggested that this palace was built and ornamented by scholars for scholars in the most simplistic way possible and yet with the most complex intentions possible. [28]
Ø Oh, what rainment of embroidered stuff have you thrown about it! It makes one forget the tulle of Yemen!...
Her columns are so beautiful in every aspect that word
Of their fame has reached far and wide!
Her smooth, diaphanous marble brighten the farthest
Corners darkened by shadow...
It is clear from this verse that the engraving was meant to convey a clear meaning, mostly how beautiful the garden is and the level of contemplation that it deserves. [29]
In summary, the Alhambra has proven to be a rich place for research and analysis. Its interesting and ancient history make it more mysterious; the mathematical implementations on walls and architecture throughout the complex make it a valuable academic asset apart from its historical importance. The geometrical shapes and recurring patterns have perplexed many visitors and historians. However, it has been shown that their interpretation was easier than previously conceived. Furthermore, the Comares Palace also embodied texts and verses with a wide a range of themes and meanings. From political to religious, these poems had ambiguous meanings that were meant to convey authority in a Christian land while maintaining a strong religious influence. The Palace of the Lions also further showed to be a building with a recurring symbols and texts. Through their direct and motivational verses, the Palace of the Lions was meant to inspire academics through simple methods. Indeed, the idea of simplicity appeared very often throughout the discussion of the Alhambra. Given that only two complexes of the castle were analyzed, we can extrapolate and infer that the other buildings are characterized by simple geometrical figures and direct poetic verses. We can even infer that the sculptors of the artwork might have had knowledge of the Golden Ratio and used it in their carvings. It has been shown that the Greeks repeatedly used this measure, which means that this mathematical tool was available during the middle ages. This could be potential research in the future. Finally, we leave with a deep appreciation for a medieval castle that captivated our attention with its mathematical simplicity and stunning architecture.
[1] alhambradegranada.org, Area 25,Granada, 2000, November 12, 2014.
[2] alhambradegranada.org, November 12, 2014.
[3] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[4] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[5] Robert Irwin. “The Alhambra.” Harvard University Press. Cambridge, 2004.
[6] Larry Liebovitch, Daniella Scheurle. “Two Lessons from Fractals and Chaos Theory.” Complexity, Vol.5, 2000, 34-43. http://www.ccsfau.edu/~liebovitch/complexity~20.html.
[7] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[8] Valerie Gonzales. Beauty and Islam: Aesthetics in Islamic Art and Architecture. London. I.B. Tauris Publishers. 2001
[9] Ludwig Wittgenstein; Bertrand Russell. Tractatus Logico-Philosophicus. Oxford University Press. Vol. 32, No. 128.
[10] Gülru Necipoǧlu; Mohammad al-Asad. The Topkapi Scroll: Geometry and Ornament in Islamic Architecture. Speculum, Vol. 73, No. 2 (Apr., 1998), pp. 566-568
[11] Ettinghousen, Grabar, & Jenkins. “Islamic Art and Architecture 650-1250.” Penguin Books Limited, 2001
[12] James Turrel, Valerie Gonzalez. The Comares Hall in the Alhambra and Space That Sees: A comparison of Aesthetic Phenomenology. BRILL. Vol. 20 (2003), pp. 253-278.
[13] Turrel, Gonzalez. The Comares Hall in the Alhambra and Space That Sees: A comparison of Aesthetic Phenomenology.
[14] Michael Mauboussin. More Than You Know. Columbia University Press. 2007
[15] Turrel, Gonzalez. “The Comares Hall in the Alhambra and James Turrel’s Space That Sees: A Comparison of Aesthetic Phenomenology.” Muqarnas: Brill. 2003.
[16] Turrel, Gonzalez. “The Comares Hall in the Alhambra and James Turrel’s Space That Sees: A Comparison of Aesthetic Phenomenology.” Muqarnas: Brill. 2003.
[17] Ibid, pg 261.
[18] Area 25, alhambradegranada.org, Granada, 2000.
[19] Fairchild Ruggles. “The Alcazar of Seville and Mudejar Architecture” : The University of Chicago Press. 2004
[20] Frederick Bargebuhr. The Alhambra Palace of the Eleveth Century. Journal of the Warbaulg and Cartauld Institutes. 1956.
[21] Bargenbuhr. 1956.
[22] Bargebuhr. 1956.
[23] Fred Kleiner. Art Through the Ages. Wadsworth, Cengage Learning. 2013.
[24] Fairchild Ruggles. “The Alcazar of Seville and Mudejar Architecture” : The University of Chicago Press. 2004
[25] Cynthia Robinson, “Marginal Ornament: Poetics, Mimesis, and Devotion in the Palace of the Lions”. Frontiers of Islamic Art and Architecture: Essays in Celebration of Oleg Grabar's Eightieth Birthday. (2008). BRILL
[26]Ibid, 2008.
[27] Cynthia Robinson, “Marginal Ornament: Poetics, Mimesis, and Devotion in the Palace of the Lions”. Frontiers of Islamic Art and Architecture: Essays in Celebration of Oleg Grabar's Eightieth Birthday. (2008). BRILL
[28] Robert Irwin. “The Alhambra.” Harvard University Press. Cambridge, 2004
[29] Cynthia Robinson, “Marginal Ornament: Poetics, Mimesis, and Devotion in the Palace of the Lions”. Frontiers of Islamic Art and Architecture: Essays in Celebration of Oleg Grabar's Eightieth Birthday. (2008). BRILL
[1] Ettinghousen, Grabar, and Jenkins. Islamic Art and Architecture 650-1250. Penguin Books Limited, 2001.