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!