For some time, the activity of
visualize cities with mathematical eyes, that is, discovering the guidelines that architects, designers, town halls, decorators, in general all those who participate in urban furniture, the planning of streets and monuments, have been used to shape the environment according to different purposes and concepts. Many of these guidelines (most of them I would say) have geometric, mathematical roots, since from them they at least ensure properties such as symmetry, order … and perhaps beauty, although this concept is totally subjective. That activity, which surely began with exercise proposals for groups of students in events such as Mathematical Olympics , has gone spreading and generalizing all over the world, not because of fashion or whim of some enthusiasts, but because it has really been proven that they are one more aspect to understand the philosophy behind works of art and objects in general.
In past articles, we reviewed the math behind tiling citing works carried out in many cities such as, for example, that 'Journey through the geometry of Valladolid', published by the city council of that city (book by the way that was sold out in a short time, and had to be republished). The city of Granada also had a study, 'Mathematical Walks through Granada', published in 2017 and coordinated by Professor Álvaro Martínez Sevilla, whose expanded reissue is soon to appear.
Four years later, we have the logical extension to a wider audience than a book that, a priori, is aimed at the most directly interested parties can suppose: an exhibition. Coordinated by the Discover Foundation, and the co-financing of the Spanish Foundation for Science and Technology (FECYT) of the Ministry of Science and Innovation and the Ministry of Economic Transformation, Industry, Knowledge and Universities of the Junta de Andalucía, the sample consists of 24 landscape panels (following the Cordoba proportion), with three leading cities:
Granada (11 panels), Córdoba (6 panels) and Seville (5 panels). To these are added a presentation panel and another for the exhibition credits, with a large and brilliant team of specialists led by Álvaro Martínez Sevilla. In addition, there is a monitor with the video 'Journey to the Andalusian domes', and exhibitors with various didactic tools (measurement of proportions, measurement of arches, anthropometric rulers), models (arched domes of the Mosque of Córdoba, dome of mucárabes de la Puerta del Lagarto in Seville), and other building materials (Andalusian coffered ceilings and mosaics from the Alhambra), similar to those used by secondary school students and teachers who attend the workshop sessions offered in parallel to the exhibition .
As the presentation panel indicates, the organizers have tried to show
the mathematics present in Andalusian art through an interdisciplinary journey: historical, artistic, scientific, mathematical. For this they have also relied on technologies not previously used (or at least not very commonly) in this type of event, such as the generation of 3D models, mosaic simulation software, photogrammetry, augmented reality, interactive tours and of course classical, detailed mathematical explanations. In many of the photographs, analyzes are carried out (mainly geometric) that explain the mathematics present in the design of the architectural elements (arches, floor plans of the rooms, tessellations, proportions, among others). The work and effort made have been truly remarkable.
For example, in the image we observe in detail the generation of the arches of the Mihrab of the Mosque of Córdoba (the arches, together with the domes, are two of the outstanding Cordovan innovations). The Visigothic horseshoe arches are now framed in an alfiz, and the arches of the intrados (inner curved surface of an arch or of a vault by its concave face), and of the extrados (outer curve of an arch or vault), They have a different center (the red dots that we see on the photo).
Thanks to mathematical software such as Geogebra, we can perfectly draw lines, circumferences, make measurements of lengths, angles, etc., in a simple (of course, after obtaining a sufficiently precise photograph, and without perspective that can distort the measurements). Thus we can verify that the two arcs extend exactly the same angle, marked by two equilateral triangles. Precisely the center of gravity of these triangles is the center of the circumferences provided by both arcs, respectively. It can also be seen how the cant of the arch is greater than ½ and its thread and eccentricity are quite high.
The mathematical topics that are Hidden behind Nasrid art are many and diverse: movements in the plane (turns and symmetries, with and without displacement, homothecies, etc.), tessellations, geodesics, modulation, polygons and polyhedra, star polygons, even constructions that can remind us of the fractals, among others. However, the architects of this exhibition have not exhaustively expressed in each panel everything that can be found in each location in order not to overwhelm the reader / visitor, selecting from each commented space only one of its most relevant characteristics.
Thanks to mathematics, aspects are discovered such as that each of the architectural decisions taken in the construction of each monument has a meaning, not always the purely ornamental, contemplative or aesthetic, as is usually mentioned in art treatises, but also deep and thoughtful resolutions for various purposes (military, defensive, water; generally practical and pragmatic). In the well thought out explanations of the exhibition panels we discover many of them. It should be noted, however, that it is not an exhibition to go through in half an hour reading each paragraph and looking at the images, without meditating on the content, as is often the case in tourist tours in which 'having the chrome and the photo' of have been there, even for half a minute. The true benefit, and above all, the true dimension to these Andalusian builders, engineers and mathematicians is discovered and recognized with meticulousness, with reflection, just as they thought. Of course, it is not an attitude to be shown exclusively in Hispano-Muslim art, but it appears in each and every one of the corners that we can visit in cities, buildings and monuments. And it is a pity that these types of ideas go completely unnoticed, hence the relevance of exhibitions like the one at hand.
The exhaustive work carried out to prepare the exhibition has made it possible to correct some commonly transmitted errors, such as wrong proportions, or the confirmation, no matter how much it hurts, of the non-presence in the Alhambra of each and every one of the 17 groups of symmetry in the plane (the different ways of creating mosaics; one is missing). Little-studied avenues of work and research are also suggested, such as the search for the symbolic meaning of mosaics. Some are known, but not most. In some of the images of the panels, instead of photographing the real elements, they have been recreated using algorithms implemented in computers that have captured them with total accuracy. Using applications such as Google Earth Pro and Google Maps, the precise and exact orientations of buildings, streets, and oratories have also been determined, surprising the miniscule errors that Arab mathematicians of the 14th century made, taking into account the media of the time. In the following image, we observe with total clarity, the stacking of six friezes of four different types in the Patio de los Arrayanes of the Alhambra in Granada.
The panels are made up in a clear and simple way: a title of the place being analyzed, a phrase that tries to summarize the content to be highlighted, an explanation above all historical and artistic, and a wide set of images in which the mathematical concepts that can be found in the place are perfectly visualized. Thus, the tour of Granada (focused mainly on the Alhambra) has the following panels (in parentheses, in italics, the phrase that summarizes its content): Puerta de la Justicia (The art of defensive construction), Puerta del Vino (Los Horseshoe arches set the tone for the space in a transit door), The Comares Façade (An enigmatic portal where duality sprouts), Patio de los Arrayanes (The splendor of a symmetrical and proportionate palace like Solomon's), Throne Room (A majestic qubba for the sultan's throne: hierarchy and order), Palace of the Lions (Where paradise becomes stone and water. Modulation), Courtyard of the Harem (The unique geometry in the women's rooms), Qubbas de Abencerrajes and Dos Hermanas (The stars come to life in the octagonal domes of muqarnas), Oratorios de la Alhambra (The religious precept in a hedonistic palace: orientation), Tiling and plasterwork (Geometric art with symbolic meaning), Friezes (A infinite line to separate spaces and write calligraphy). The one in Córdoba, the city where the exhibition was previously, contemplates Arcos Califales (An exhibition of interlaced curves over a forest of columns), The domes of the Mosque (Pioneering techniques that will tour Europe), The monumental portals (Blind arches, mosaics and proportions inaugurate the “doors of honor”), A mosque that does not look at Mecca (The doctrinal schools dispute how to orient the architecture), An Islamic qubba for a Sephardic synagogue (The mathematics that define the architectural gene), The Bimah and enlightenment (A lobed arch to symbolize the Menorah). And finally, the one in Seville runs through La Giralda (An imposing minaret enhanced with geometry), La Puerta del Lagarto (The first muqarnas vaults in Al-Andalus. How do you build such a roof?), El patio del Yeso (A Geometric lattice portico for a solar architecture), The Palace of Pedro I (The Mudejar conquers the victors) and La Torre del Oro (Polygonal prisms for a defensive architecture). The following image shows us the heptalobulated arch of the Cordoba synagogue. It is perfectly visible that the lobe centers (indicated by a yellow dot) lie on the curved sides of a Reuleaux triangle. The symbolic meaning of the number seven to Jews is well known, appearing in many of his objects of worship, such as the arms of his ritual candelabrum, the menorah. His inspiration is clearly caliphal, with similar vestiges (with a different number of lobes) in the nearby Mosque.
Like everything in this life , the exhibition also has some (few) aspects that may be uncomfortable for the visitor. The most obvious (but that depends on the training of the visitor himself) is that the panels contain terms, both artistic, historical, and mathematical, with which you must be familiar to usefully follow the description. It is not possible to explain everything in great detail because then twice as much text would be needed, which would make the whole more objectionable. For this reason, these types of exhibitions get the most benefit when attending a guided tour. But this is a drawback not attributable to mathematics, but it appears in any type of subject. In this case, there is a wide range of free complementary activities to the exhibition (workshops for students and guided tours). Another aspect that would improve it would be the possibility of having replicas of the materials in the display cases to be able to manipulate, but unfortunately this is not advisable due to the pandemic situation we are going through. Materials of this type are those that allow us to better understand the concepts, much more than all the images that we want to visualize, because in this way, building it with our own hands, we perfectly understand how domes and mosaics were devised, for example, and we come across the problems to be solved and the solutions to be adopted. That is why workshops are so recommended for students and the general public. This is how they understand it, given the high attendance it is having.
The place chosen to host the exhibition could not have been more successful: the Royal Room of Santo Domingo, Granada. It is a 13th-century Muslim palace of which only one tower remains, with really remarkable plaster ornamentation and tiling thanks to a meticulous restoration, extensive gardens, a pond and excavations of various areas of the original building. Its interest lies in being one of the few and oldest remains of a residential building, and a model for some of the most famous rooms in the Nasrid palaces of the Alhambra, among others. The exhibition is located on the top floor of the modern building (see image), and will remain there until next January 9 to later move to the Exhibition Hall of the Andalusian Institute of Historical Heritage (IAPH) in Seville. Then you will go to Algeciras, Malaga, Almería and even Tunisia, in the Arabic version.
Finally, if you do not plan to be in Granada these days, or in the successive locations that the exhibition will visit, you can enjoy a magnificent alternative virtual tour at the Virtual Tours link, a real joy in which we can enjoy 360º panoramic views of the places described. And, also, if we are lazy, or we are tired, with full audio of all the explanations.
Alfonso Jesús Poblacion Sáez is a professor at the University of Valladolid and a member of the Disclosure Commission of the Royal Spanish Mathematical Society (RSME)
The ABCdario of Mathematics is a section that arises from the collaboration with the RSME Disclosure Commission.