Hello, and welcome to those who have joined up since our last newsletter.
In this issue
- This week: Tiles
- Virtual Snacks
- Bizarre Searches
- Quotation and joke
Something special for you this time! Our pal Mike Kingdom-Hockings of New Freebooters has kindly contributed this week’s topic….
Alhambra. The Red Fortress. People associate the name of this Moorish palace and fortress complex to the south east of the Spanish city of Granada with many things.
For me, it always brings to mind Francisco Tarrega’s virtuoso composition for Spanish guitar, but I want to tell you about something else: the tiles that decorate its walls, floors and ceilings. If you’ve ever seen the remains of a Roman villa, you’ll have seen mosaics – pictures built from freehand arrangements of coloured tiles. But the artists who decorated the Alhambra were forbidden to do that kind of thing. For Muslims, depicting anything believed to have a soul was taboo.
They had to seek beauty in inanimate forms, so they competed with one another for the praise of their Sultan by seeking variation in ways of tiling a surface. Tiling is a system for covering a surface with pieces that fit exactly, leaving no gaps. Modern mathematicians such as Roger Penrose have discovered tiling systems that are not regular, but the Moors hadn’t reached that stage. The Alhambra’s surfaces are all covered with patterns that repeat.
There are only three basic shapes that you can use to tile a flat surface with a repeating pattern: a triangle, a square (or rectangle) and a hexagon (like a honeycomb). Once you have chosen your base shape, you can modify it in two ways: distort the edges in a way that still allows them to fit perfectly, and decorate them with patterns. Here’s a diagram:
Tiles and their orders of symmetry/tessellation
Without knowing anything about symmetry as defined by modern mathematicians, the Moors managed to create tiling schemes that, between them, demonstrated all 17 of the possible symmetry groups for two dimensions.
Let’s take a moment to explain what I mean. You probably think of symmetry as being the property of having one side a mirror image of the other. That’s one kind of symmetry, but mathematicians talk of symmetry in a tiling scheme as being any movement where you pick up a tile (or the whole tiled surface), turn it round or flip it over, put it back down – and then find no-one can tell that anything has changed.
Look at an equilateral triangle (one with all three sides equal in length). You can rotate it by one-third or two-thirds of a turn and it will still look the same. Mathematicians also count doing nothing as one of the moves, so they say it has three rotation symmetries. You can also flip it about a line from one corner to the middle of the opposite side. Mathematicians call this a reflection, and since there are three corners a triangle has three reflection symmetries, making a total of six symmetries.
Now look at a six-pointed star. You can rotate that one-sixth of a turn at a time, giving you a total of six rotation symmetries. It also has six reflection symmetries – three about axes through opposite points of the star, and three about axes through opposite “dents”.
A hexagon (a beehive cell) has exactly the same symmetries. It’s just a six-pointed star that’s got so fat that the dents have become straight lines.
Here comes the clever bit.
If I decorate a triangular tile with a pattern that doesn’t have the same reflection symmetry as the triangle, I destroy the tile’s reflection symmetries. It no longer looks unchanged if I reflect it. That means my decorated tile exhibits a different group of symmetries (in this case, only the three rotation symmetries) from the undecorated tile.
That’s how the Moors created their systems, each demonstrating one of the 17 possible different symmetry groups. Many of the ones that look completely different actually belong to the same symmetry group, but it takes a mathematician who understands Group Theory to prove it.
I love reading popular books about science and mathematics written by acclaimed experts. This little piece was constructed from a longer and more entertaining chapter in the book Finding Moonshine – a mathematician’s journey through symmetry, by Marcus du Sautoy. Marcus takes his 10-year-old son Tomer along with him in many chapters. A delightful read, and an introduction to a fascinating topic.
Thanks, Mike! You can follow Mike Kingdom-Hockings on his own explorations of whatever takes his fancy at:
Do you have anything to say about this topic? Or do you have some suggestions for other issues we might discuss in our weekly email? Why not comment and tell us?
Just a few suggestions if you have a little time to spare:
The Math Forum has lots of fascinating, well written articles explaining various aspects of mathematics – their article What is a Tessellation? is particularly good.
+plus magazine is an excellent resource if you’re interested in explanations of cutting-edge mathematics for the intelligent lay reader. A Knight on the Tiles is their article about Sir Roger Penrose, a leading British expert, and his explorations of tiling theory.
Some strange search terms which have led people to visit British Expat recently:
- duke of edinburgh idiot
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- mark last houses pigi
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- a joyful song of reverence relative to hollow metallic vessels which vibrate and bring forth a ringing sound when struck
- website for people who think the holy bible is a bunch of filth
- jetpacks flying duration
- granny measure
Till next time…
Kay & Dave
Editor & Deputy Editor
British Expat Magazine
“Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives’ mouths.”
– Bertrand Russell, mathematician and philosopher (1872-1970) in The Impact of Science on Society (1952)
The Flood is over and the ark has landed. Noah lets all the animals out and says, “Go forth and multiply.”
A few months later, Noah decides to take a stroll and see how the animals are doing. Everywhere he looks he finds baby animals. Everyone is doing fine except for one pair of little snakes.
“What’s the problem?” says Noah.
“Cut down some trees and let us live there,” say the snakes.
Noah follows their advice. Several more weeks pass. Noah checks on the snakes again. Lots of little snakes, everybody is happy. Noah asks, “Want to tell me how the trees helped?”
“Certainly,” say the snakes. “We’re adders, so we need logs to multiply.”