Granada: Things besides the Alhambra

In Granada, our airbnb host let us know that (in her opinion) the Sevillians act a little bit superior about their city, but that Granada is "far more charming".

Our apartment in Granada is situated in the only part of the city which retains the old Moorish style of architecture (called the Albaicin area).

Here's a view of the apartment, and the large attached balcony.

Our little area's winding, cobblestone streets and white plaster facades certainly were charming.

On our first day we walked uphill until we couldn't anymore and found a nice mirador overlooking the city.

A few blocks from our house we discovered a bustling narrow street lined with Moroccan restaurants, pastry shops, and street vendors.  Down a quiet side-street we found this restaurant:

 The proprietor greeted us in Spanish, but quickly switched to English, and then to French as he chatted with the couple coming in behind us.  His accent in English was impeccible (and, so far as we could tell, in French as well).  We later asked him how many languages he spoke, and he said English, Spanish, German, French, Italian, and an Arab language we didn't catch the name of.  He was very charming, and made a stream of small jokes — to a table of girls from LA, "oh, your're from Beverly Hills?  Give me a minute to raise the prices, please!", to the couple next to us "if you need any help, or need me to recommend the most expensive things on the menu, please just ask!".

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We both enjoyed really friendly service in this neighborhood, and suspect it's because of Maria's background. At one point, we bought some breakfast items at a small Indian grocery and the man ringing us up was very interested in where Maria was "really" from (and surprised that with parents from Karachi, I only spoke Gujarati). He threw in a couple of free chilis with the rest of our purchase. We also stopped in at one of the pastry shops selling all kinds of baklava variations and had a lovely interaction there with the clerk who explained that the name of one of our selected pastries was based on it resembling a bird's nest (it was delicious).

All in all, we rarely found it necessary to leave our small part of town, though that means we likely missed out on other, very different parts of Granada.

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As an addendum to this post, we are very proud to have figured out how to take the bus to the airport as it was very poorly marked and a completely different system than the other bus system in town (which had numerous, clearly marked route maps).  We're getting pretty good at this!

The Alhambra - Tile patterns

The tile mosaics in the Alhambra are not only beautiful, but mathematically interesting as well (...but I repeat myself).  One way to better see and understand the visually complex patterns is by describing their symmetries.

Group theory is a branch of math used for describing symmetries.  It not only lets you see how different patterns are constructed or related to each other, but can also tell you what different kinds of arrangements are possible.

Two interesting mathematical results related to 2d tilings:

• 17 Wallpaper Groups Theorem:  There are only 17 different types of symmetric pattern possible for a regularly repeating two-dimensional pattern.  Of course, there are more than 17 patterns.  But if you look at the symmetries of how the patterns repeat, the collection of symmetries must be one of 17 known types.  Below I will explain this in more detail and show you a way you can figure out what type of pattern you have, using the same notation used in crystallography (which studies the patterns in the atomic structure of crystalline solids).

• Undecidability result:  There is no procedure you can follow to decide whether a set of tiles is capable of creating a repeating 2d pattern that covers the plane.  (You might be able to figure this out on a case-by-case basis, but there is no general approach to this question which will work in all cases).

But I'm getting ahead of myself.  First, a selection of a few of the tile patterns.  Maria and I are hoping to create an interesting tiling in our study (details forthcoming) and if anyone wants to help think/plan/create, we'd love to collaborate!

These certainly look impressive, but I had no idea how labor intensive they are to construct!  Here's a brief video showing the process of fitting together the tiles into these kinds of mosaic.

It's no wonder the Spanish started painting patterns on square and rectangular tiles which could be mass produced and easily fitted together!

Anyhow, on to the story at hand.

Symmetry

In school, most people learn about reflection and rotational symmetry.

For reflection symmetry, if you "mirror" or fold a shape across an axis of symmetry, it remains the same.  For rotational symmetry, you rotate the shape around a point by some amount until it fits back onto itself.  In this example, you can rotate the shape around its center by 120 degrees (or 240 degrees, or 360 degrees, or any multiple of 120 degrees).

What do these two things have in common that make them both kinds of symmetry?  After all, rotating something is very different from folding it.  Why are they both symmetries?

The answer is:  They're both actions (transformations) you can perform on the shape which leave them unchanged.  And that's a good general definition for what it means to be symmetric.

"A symmetry of something is an action that leaves it unchanged"

You may have heard about "symmetry in the laws of physics" and this definition of symmetry is general enough to apply to things other than physical objects (like equations, for example).  For an interesting discussion of this, I recommend this very approachable Feynman lecture.

For now, let's use this definition to identify...

The Four Types of Planar Symmetry

Any regularly repeating two dimensional pattern (periodic tiling) is made up of copies of a single pattern.  For example, here is a picture of the plinth from the Hall of Ambassadors.

And here is the same pattern, extended a little bit further.  You can imagine it extending infinitely in all directions.

When you're finding symmetries of a tiling, you're looking at actions you can perform on the whole infinite pattern which will leave it unchanged.  This means that the repeating pattern might have a symmetry which the individual tiles that make it up don't have.

Every tile pattern has two directions of translation symmetry.  This means you can slide the entire pattern over until it fits over itself again.  For the pattern above, you can slide it along either yellow arrow to match it up to itself again.  These translation vectors help define the smallest area of the pattern that can be used to construct the whole (sometimes called the primitive cell).

After translation symmetry, you can look for reflection symmetry.  Here, the yellow lines identify a few of the lines of reflection symmetry in this pattern.  (Imagine the entire infinite pattern reflecting around one of these lines).

Next, identify the rotational symmetries in the pattern.  This pattern has several places where you can rotate it by 180 degrees to fit onto itself.  I've marked a few of these places with yellow dots.  You can see that the top two dots (where the arrow shapes come together end-to-end) are centers of rotation that are also on lines of reflection.  The lower (slightly larger) dot is a center of rotation which isn't on a line of refection.

The last kind of planar symmetry is called the glide reflection.  It consists of a reflection, followed by a translation.  I've tried to indicate this by a yellow reflection line, and some yellow arrows showing how to slide the pattern after you reflect it.  (In other words, the upper left corner of the black arrow first reflects across the yellow line, then slides over to match up with the lower left corner of the other black arrow).

All together, this covers the four possible kinds of planar symmetry:  translation, reflection, rotation, and glide reflection.  In three dimensions there are other possibilities (such as inversion, which are important for understanding 3d atomic structure, and which you can read a simplified account of here).

A few live examples

See if you can spot the different types of symmetry in the examples below!

Practice #1

This pattern has horizontal, vertical and diagonal lines of reflection symmetry, as well as several different centers for 90-degree rotational symmetry.

Practice #2:  Tiling from the Patio of the Lions

Ignoring the gradation of color from yellow to green, this has a 90-degree rotational symmetry, just like the last example.  

It looks like it might have reflection symmetry as well, but if you look closely the black stripes sometimes go over and sometimes under the other stripes.  If you try and reflect it, you'll see that this over/under pattern doesn't match up.  The same is true of any possible glide reflections.  So this pattern, though similar to the last, only has rotational symmetry.

Only 17 Types of 2d Symmetry?

Even though two patterns may look very different, they can have the same set of symmetries.  Take these two example.

These patterns both have 90-degree rotational symmetry, but no reflections or glide reflections.  This means they are classified as the same type of symmetry (the group named p4 using the crystallographic notation).

So, the sense in which they're the same is that the same set of actions will leave each of these patterns unchanged (even though both patterns look different from each other).  Finding deep similarities between apparently different things is one of the things that makes math so wonderful!

If you're curious about the 17 types of symmetry group or want to see visual examples of all 17, wikipedia has an excellent page on the subject.  You can also see a step-by-step classification chart that will tell you what symmetry group you're looking at if you can identify the types of symmetry in your pattern.

Conclusion

This just scrapes the tip of the iceberg.  Group theory (the type of math that describes symmetry) is a fascinating subject used extensively in chemistry and physics.  Back when we used CDs, group theory (and ideas of symmetry) were behind a method for encoding the audio information that let you play cds perfectly despite scratches on the surface.  The ideas of symmetry applied to polynomial equations helped to explain why there's a quadratic formula, but no formula that will let you find the solutions to equations with an x^5 or higher.  It's a wonderful and amazingly applicable kind of math!

The Alhambra - Historical and cultural information

During our chat over our delicious Moroccan dinner this evening, David mentioned his idea for the Alhambra posts - an all-photo post followed by a mathematical discussion of symmetries and crystalline forms. I decided immediately that I wanted to offer an alternate view, for those who are less mathematically inclined. Let's begin!

First, a brief history.

The Alhambra is a collection of monumental buildings atop the Cerro de la Sabika, or Red Hill, overlooking Granada. The Alcazaba fortress was the first of the buildings to be constructed, somewhere around the end of the 9th century. The Nasrid Palaces and Generalife gardens making up much of the rest of the Alhambra were added 100 years later by Muslim sultans. The name Alhambra likely originated from "al-Hamra", the Red, because of the red glow coming from torches used to illuminate the buildings at night. Over several centuries, as the area changed hands between Muslim and Christian rulers, buildings were added or changed to accommodate their new inhabitants.

The buildings.

The Alcazaba, as mentioned earlier, was a fortress. A small city was constructed just around the fortress to house the military. Many of the walls and buildings making up the original fortress are unfortunately no longer standing. We were able to go up some of the towers that are still intact, photos of the views over the city from those vistas don't quite do them justice. That's basically true, though, of all the photos from this visit!

The Nasrid Palaces are where we took the majority of photos. The first, the Mexuar, has seen the most change over time. first built under the reins of Muslim sultans, it was changed to a chapel (by knocking down some walls and adding a section for a choir) after the Christian conquest. The Comares Palace is the central one, with an iconically long, still pool in its central courtyard. The purposes of such pools in Muslim architecture was to serve as a mirror, and to make the palace appear larger and as if it is floating on water. The main hall of the Comares Palace has a beautiful honeycombed ceiling built in 7 symmetrical bands, alluding to the 7 heavens in Islam. All around the wall are religious inscriptions in cufic, Mghrebie and Aladalusian characters, especially a repetition of "Allah alone is the victor." The last building, the Court of the Lions, has a courtyard filled with a dizzying array of columns, all with delicately carved arches like lace and a central fountain surrounded by marble lions (hence the name). By this point it really felt as if each subsequent palace had outdone the previous ones, with the carvings in archways, windows, walls and ceilings becoming more and more intricate. Within the Court of Lions, the Hall of Two Sisters also contains an elaborate honeycombed ceiling, this one made up of over 5000 separate pieces, and the walls are inscribed with a poem by Zamrak describing the beauty of the room itself.

The name of our final area, the Generalife, is understood to be derived from two Arabic words, "djennat" or garden, paradise, even heaven and "al-Arif" or architect, so potentially "Garden of the Architect." These areas were used by nobility living in the palaces when they needed a little "me" time. The lower gardens were designed to be long and narrow, so that sitting at one end you had an expansive view ahead of you. Long narrow reflecting pools are included here as well, surrounded by a variety of flowers and trees. Most of the original greenery has been updated since its original usage. The upper gardens have a maze-like structure and were re-created in 1931 after being bought back from private ownership. The gardens have been re-created to best represent what the gardens may have been like at the time.

TLDR, it's amazing, just go and check it out some time.

The Alhambra - Installment 1: in Photos

One of the main things we've been excited about in Granada is seeing the Alhambra, a Moorish palace with intricate and varied tile work.  This installment will be photos only, then Maria will post one about the historical and cultural aspects of the Alhambra, followed by David who will make a short post about the mathematics of the tile patterns (he's hoping to create two half-hour interactive lectures on the relationship between art and math based on things from this trip which he can give next year).

Like many of the Catholic cathedrals we've seen, the goal of many rooms of the Alhambra is to be awe inspiring, and so photos really fail to capture them.  The most ornately decorated rooms are called the Nasrid palaces, which are covered, floor to ceiling with carved stone, plaster letter forms, and geometric tile patterns.  A photo can't capture both the detail and the full scope of these rooms, but we've done our best to give a representative sample.  Enjoy the photos for now, more information to come!

A view from the outer walls (as this was once a military outpost).

Maria looks back toward the city (we're staying in one of those white buildings!)

One of the decorated hallways leading to the Nasrid palaces.



A detail of a wall and ceiling (many of which have many thousands of individual polygonal faces)



We take a self-photograph among the columns!





A view of one of the detailed honeycomb ceilings.



This is one of many Arabic inscriptions worked in among the tile patterns.



This shows the Alhambra in the foreground.  It's taken from an area called the Generalife, which was reconstructed to represent a typical Moorish garden.

Granada

(we're a little behind, but probably we'll catch up soon....maybe)

Plaza Espana & Parque de Maria Luisa

We set out with the aim of visiting Sevilla's large public park and without really realizing it was there, ran into the impressive Plaza Espania.

Sometimes we take a self-photograph...though sometimes the brightness levels are also way way off, sorry guys.

The adjoining park complex was extensive and beautiful, but also contained two wonderful surprises.  The first was that someone had put art books into the shelves of the central decorative feature of this small courtyard for anyone to read and enjoy!

The second surprise was discovered by Maria when she made a closer inspection of the tiles on these benches...

The tiles contain a sequential representation of major scenes from Don Quixote which we've been listening to on audiobook in the evenings!

Don Quixote and its characters are frequently referenced in the cities we've been to so far and seem to hold a similar cultural relationship to Shakespeare in England.

Soon we need a full accounting of tapas and other eating adventures before we leave Sevilla for Granada!

Sevilla odds and ends (vino naranja, public art)

A few odds and ends.

Our airbnb host left us his own recommendations for things to see and do around the city.  He included a "bars and restaurants" section, each entry of which encouraged us to "have a drink and some olives" there, as well as a "dining" section which instructed us to "have a drink and some olives" at each restaurant.

One notable exception was the mention of vino naranja, a wine made from oranges.  So, we decided to go try a glass together with an unknown dessert we decided to risk pointing to on the menu.  It turned out to be something like a butterscotch pudding with a sauce that included a reduction of the orange wine.  The orange wine itself was (David thought, ) delicious!  It was quite sweet, and had a similar flavor to other dessert wines, but hinting toward orange.

We also discovered an amazing public art piece in one of the plazas, near our house (strangely modern for that part of town).