Pedimental Beasts

Toledo calls itself the city of three cultures. This is most visible in the monastery of San Juan de los Reyes. I walked through the Jewish quarter, and past the synagogue, into this large monastery built in the Mudejar style by Islamic architects and artisans, and found a stunning building.

Toledo: San Juan de los Reyes, detail 3

There is a wonderful mix of Christian spaces with Islamic decorations in the tiles on the floor and the woodwork in the ceilings along the corridors. But my eyes were caught by the exuberance of the details on the carvings on pillars and pediments.

Toledo: San Juan de los Reyes, detail 3 Toledo: San Juan de los Reyes, detail 4

I have seldom seen this kind of naturalistic detail outside of India. I walked slowly along the verandah bordering the central courtyard, admiring each piece of sculpture and taking a few photos. I’m sure each of these has been fitted into a symbolic belief structure, and if I were well-versed in medieval Christian symbolism I would see other layers of meaning under them.

Toledo: San Juan de los Reyes, detail 5

I just took pleasure in what the simple artists saw: a dog, a pig and a duck. Satisfied with my slow circuit around the central garden, I waddled out.

A Magic Square

I wandered through the Sagrada Familia in a daze until I came to the western facade and its Magic Square (see the grid of numbers in the featured photo). The facade is called the Passion Facade because it depicts the crucifixion and resurrection of Christ: the events are called the Passion of Christ in Christian dogma.

Work on this facade began in 1954, after Gaudi’s death, but followed his plans. The sculptures were made by Josep Maria Subirachs starting in 1984. I thought that the sculptures looked unlike the rest of the church, and found later that there is indeed a bit of controversy related to the look of the facade. When Subirachs took the commission, he insisted that he should be allowed artistic freedom, and not be forced to follow Gaudi’s designs slavishly. So, I wonder whether the magic square behind the sculpture of the Kiss of Judas is Subirachs’ or Gaudi’s.

A magic square is a square filled with numbers such that each row, each column, and the two diagonals all sum to the same number. In this square the sum is 33, which is the age that Christ was supposed to be during the events portrayed in this facade.

A magic square may (or may not) have other characteristics. One that people often insist on is that the numbers used be consecutive, starting from 1, and that no number be used more than once. This convention is clearly violated by the Passion Square, since 12 and 16 are missing and 10 and 14 are repeated.

Note also that this magic square has other magic properties. The square can be divided into four 2×2 squares, by a horizontal and a vertical line bisecting each side (the upper-left square has the numbers 1, 14, 11, and 7). The sum of the four numbers in each of these squares is also 33.

The 2×2 square at the centre contains the numbers 7, 6, 10, and 10. These also add up to 33. If you take the two numbers above this central square and the two below it, then they also sum to 33. The two numbers to the left of the central square and the two below it also give the same sum.

There is an easier way to think of these two disjoint blocks of numbers. Imagine a large floor tiled with copies of the Passion Square. I’ve tried to show this in the image here: the thin black line marks each copy of the magic square. Mark out on the floor all copies of the central square with the numbers 7, 6, 10, and 10 (I’ve coloured them red in the image on the left). Then you will see that the two numbers above and below the central square also become a 2×2 square, repeated on the tiling (I’ve left them white). Similarly, the disjoint set of 2 pairs of numbers, two to the left and two to the right of the central square also become a single 2×2 square (also white).

When you think in terms of the tiling, then you discover that the corner squares become a single 2×2 square on the tile (the blocks of blue tiles). And, of course, the numbers on the corners of the Passion Square also sum to 33!

The diagonals join blue and red squares. The four numbers along the forward diagonal sum up to 33, as do the backward diagonals. On the tiled floor you discover other diagonals. Of these, only the diagonals removed from the circled ones by two spaces are associated with the same arithmetic magic, ie, they sum to 33. So there is a set of magic diagonals in the coloured squares, and another in the white squares. This is easier to see if you bleach the colours, as I’ve done in the image below. In the resulting chessboard pattern, there is a set of diagonals on coloured squares, and another in the white squares. Both kinds of diagonals are magic. Interestingly, magic squares of this kind are called pandiagonal squares, or, more interestingly, diabolical magic squares.

Finally, the first two numbers in the second row and the first two in the third row also sum to 33, as do the last two numbers in these rows. Shifting down by one row, the first two numbers in the second and fourth rows also add up to 33, as do the last two in these rows. If you try the same thing column-wise, the magic goes away.

Of course I could not have been the first person to discover these marvellous tilings hidden in plain sight. A quick search led me to a paper by two mathematicians from the University of Las Palmas, Jose Pacheco and Isabel Fernandez, who examined these symmetries some time ago. They also refer back to the first (paywalled) description of this square in the mathematical literature in 2001 by Pieter Maritz of the University of Stellenbosch. The symmetries of magic squares was first discussed in the early 20th century.

The beautiful hidden symmetries of the Passion Square open up through the tiling into a beautiful doubled lattice. This can be represented in many ways, and I show one in the image above. This harks back to the Mudejar style of the Spanish churches of the middle ages. Is this Subirachs’ hidden contribution to the Sagrada Familia?

I left feeling quite satisfied with the discovery of this mystery.

Mudejar towers

The Merriam-Webster dictionary defines Mudejar as a Muslim living under Christian kings especially during the 8th to 11th centuries. It also says that the first known use of the word is from 1829. Soon afterwards it was applied to a style of Spanish architecture in the sense that it still carries. The style is seen in all the photos here.

The main characteristics of the style are brickwork, decoration with tiles, intricate carpentry, and the use of geometrical motifs in decoration. The featured photo is of the grand mosque in Granada, and shows all these features. The mosque was closed because I went there in the afternoon of the last day of Ramzan, so I did not manage to see the interior. Still, the exterior gave a good indication of what the interior would have been like.

In Granada, near one end of Plaza Nueva is the church of San Gil and Santa Ana (photo above), an exemplary piece of Mudejar architecture. Look at the beautiful azulejos (tiles) at the top, above the bells, the woodwork just below it, the beautiful tiles above each of the romanesque arches, and the ornate pillar just below the bells. Again, because I was always there at the wrong time, I could not see the interior.

Notice the beautiful woodwork in these three photos. The one on the left is the exterior of the church of San Nicholas. I climbed up to it on a really hot day, and was too dehydrated to walk in. I sat in the shade outside and drank a lot of water. The other two photos above show the church of San Pedro and San Pablo. Typical Mudejar elements are visible outside. Inside, the ceiling is covered in incredibly beautiful woodwork. A friend told me about his perception of the difference between Islamic and Christian architecture in the Alhambra: that they embody different philosophies of what make something grand. These churches combine these two world views: the high interiors are grand in the manner of churches you see all over Europe, and the beautiful intricate workmanship brings to it the Asian love of detail. One wonders about the missed paths of history; if only the Jews had not been expelled in 1492, and the Muslims had been allowed to stay on after the 25 more years that they did, what kind of new architectural styles would have evolved as engineering improved.

In my mind the Giralda of Seville (photo above) is the grandest of these towers. Technically it does not belong to the Mudejar style, because it was built before the Christian reconquest, and was only converted from a minaret to a bell tower afterwards. However, it has all the elements of this style: brickwork to build up a really high tower and incredibly detailed external ornamentation.

Finally, a moment of duh-ness or serendipity, whatever you may call it. I looked back at the photos I took a year ago inside the Royal Palace of Sintra and realized that I had seen the Mudejar style before. The chapel inside the palace, which you see in the photo above, is an example of this style. And a particularly fine example, one must say.

The first thing we saw in Seville

We arrived at night in Seville and thought it looked charming. It still looked wonderful in the morning when we walked through the centre and joined a long queue of people waiting to buy tickets for a visit to the Alcazar. This is an old palace which was built about eleven centuries ago, and remodelled many times later.

I will write about it in more detail later. The part which you see above was built by the Moors, and remodelled later by Moorish craftsmen who had not converted to Christianity. These craftsmen remodelled the palace, and gave their name, Mudejar, to the style of architecture which you see here.

Such an ornate palace has to be a location where movies or serials are shot. Sure enough, the room you see in the photo above is supposed to have featured in The Game of Thrones. I don’t follow the serial, but if you do, then maybe you can recognize it.