A mathematical walk through the Alhambra: when art is based on numbers.

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In the eyes of someone who has not mastered mathematics, the arc leading these lines can convey great harmony, but it would be difficult to explain why. The cause of this harmony lies in the mathematics used for its design, for an extraordinarily calculated composition.

This particular arch is the arch of the Cordovan Islamic mihrab of the Granada palace of Madraza, a place that indicates in which direction one should pray according to the Muslim religion. If you look, **interior and exterior arcs are not concentric**that is, they do not share the same axis. The inner arch is raised R / 2 above the fascia axis, and the outer arch has raised a R / 5.

The lines that determine the segments between the arcs are all cut at the same point, which turns out to be the middle of the fascia line. This type of bow is generally used in particularly important places.

All this is explained to us **Joaquin Valderrama** and **Francisco Fernandez**, two math teachers who have already retired but have not given up teaching. At least in their own way: they took guided tours of the Alhambra in their native Granada – also in other places that could be accompanied by mathematical explanations – for children and adults, including teachers.

They make the pedagogy on the face perhaps less obvious of the art that we find in the palatial city of the Nasrid capital: during our visit, during a trip that lasts an entire morning, we do not Keep stopping practically at every corner to explain what is not obvious, but which gives meaning to the whole place: formulas, algorithms, patterns and proportions. And also, help us to find out for ourselves.

## The rectangle: one of the geometric shapes most used in Islamic art and in the Alhambra

The rectangle is one of the most used proportions in Muslim decoration, and therefore in Islam, but not just any rectangle but the one with the square root of 2 (√2). Why is it so special?

√2 is an irrational number, the result of √2 is 1.41421356… In other words, a number that never ends. A priori, it does not seem manageable for performing calculations, but as Joaquín and Francisco explain to us, it is a very easy proportion to use geometrically.

The rectangle √2 is very simple: from a square on one side the diagonal is drawn, it is folded on one of the two sides and by drawing the parallels we get the rectangle √2.

An easier way to understand this is to use a standard DIN A4 sheet, which has been produced (like all sizes DIN, A3, A2, etc.) by following this report.

If we divide the long side by the short side of the A4, the result will be 1.4142135… or √2. In hand and geometrically, this can be quickly verified by recalling the Pythagorean Theorem, which tells us that in a right triangle the sum of the squares of the legs is equal to the square of the hypotenuse.

If we make a square with our A4, we can get a right triangle. If each leg is 1, the hypotenuse measures √2. And a quick way to verify this is to compare the hypotenuse with the length of the A4 rectangle itself.

Indeed, if we check the wine door, **has the proportion of a rectangle √2**. You could say that this entrance to the Alhambra has exactly the same proportions as a DIN A4.

**Such a special proportion because it is the only one which, folded in on itself, reproduces exactly the same proportion as the original**. In other words, if we fold an A4, we get two A5s of proportion √2, if we fold an A5, we get two A6s of √2, and so on.

This proportion is one that strikes an exquisite balance when it comes to applying it to architecture, and in this case the doors and windows of the Alhambra fill it.

## The first place in all of civilization where the 17 types of mosaics were used

The Alhambra is the only monument built before the discovery of group theory that has at least one example of each of the flat crystallographic groups. There are 17 types of mosaics according to this theory, which are governed by four movements in the plane: rotation, symmetry, translation and displacement symmetry.

Combining these movements, these **17 types of mosaics, and the Alhambra brings them all together**. No other place in the world has known him before.

Although we see very different mosaics, it is possible that they belong to the same crystallographic group, so Joaquín and Francisco used an algorithmic array to recognize each one.

All these variations can be made according to these four plane movements.

Depending on how the minimum initial pattern is formed, the bends made and the types of plane movements executed, we can create different types of mosaics, like the one in Puerta del Vino.

The mosaic is an element that appears very commonly in the Alhambra not only for aesthetic reasons, but also because it represents an idea of Islam: since it does not in any way allow you to graphically represent your god, ideas , concepts are used or epigraphs.

A mosaic is an image that repeats itself endlessly, a motif that repeats itself until it occupies the entire space. **It represents the idea of a unique god who is everywhere, something unique that covers everything**, a concept that connects to Islamic theory.

## The frieze: another recurring mathematical element of the Alhambra

Friezes work very similar to mosaics, only the difference is that they are translated in one or the other parallel direction to fill the whole plane by translating the minimum pattern, instead of doing it in all directions like it happens with mosaics.

The friezes were used as a decorative motif in prehistoric dolmens, Egyptian decorations, Greek temples… And also in the Alhambra, where they were widely used in their decoration.

In geometry, there are seven types of friezes classified according to the Rose-Stafford algorithm, and as for mosaics, **all seven are in the Alhambra**.

## An oratory mathematically oriented towards Mecca

By mid-2019, Muslims can use an app on their smartphone that easily shows where they should go to say their prayers while looking towards Mecca. We can know thanks to the technology that the right orientation direction of Granada must be 100.4º SE, which direction in the Oratorio del Mexuar, which is part of the Alhambra?

If we check this using Google planimetry or GeoGebra software, we can see that an azimuth of 108.89º is obtained (« azimuth » refers to an orientation angle on the surface of a real or virtual sphere) .

If the correct mathematical orientation is 100.4º SE and the oratory is oriented 108.62º SE, we understand that it has a deviation of 8.22º SE, is the Mexuar incorrectly oriented? Not really. It is more, **is one of the best oriented of Al-Ándalus**.

To get into the context, it should be known that Abderramán I (year 786) began to build the oratory of the Mosque of Cordoba oriented with an angle of 152º SE, that is to say with an error of 51.6º SE. The point is that many subsequent mosques were built with reference to the Mosque of Cordoba, **therefore, most mosques in Al-Ándalus are not optimally oriented**.

However, later more precise astronomical and trigonometric methods began to be used to guide mosques, such as the mayor of Medina Azahara, built during the time of Abderramán III in the 10th century. Advised by his astronomers, Abderramán III built the medina of Azahara with an angle of about 108.33 °, which probably influenced the Mexuar to have virtually the same orientation.

It should be noted that the best oriented of all Al-Ándalus is the private oratory of the Sultan in the Palace of Comares, in the Alhambra, with 101º, undoubtedly a combination of exact calculations by its astronomers and exceptionally geographic coordinates precise.

If you are still curious about more mathematical secrets of the Alhambra, Joaquín and Francisco, as well as his companion Antonio Fernández, continue to explain every day all the secrets of the Alhambra, both on their walks and on the blog, The Mathematical Alhambra.

Source: Engadget