A quick extra post today: I’m in the Big Internet Math-Off, which decides who will become the World’s Most Interesting Mathematician of 2019.
My first group match is today, against @kyledevans, and I’ve done a video for it! Go over to the Ap’, have a look at the pitches, and vote for the one you like best (especially if it’s mine).
Here’s a transcript of the video, in case you prefer text - but the video is definitely better!
### Ext: beach
Famed for its endless golden beaches… an unrivalled hub of commerce, culture and sport, and home to one of the world’s most iconic buildings.
Are you sure this is Barcelona? There’s a Punch-and-Judy stand over there.
I’m here in… Barcelona to look at some of the maths behind a UNESCO World Heritage Site, one of the most famous buildings in Europe, the instantly recognisable Sagrada Familia.
I know we’re on a tight budget, but that sign quite clearly says “Holy Trinity”
[off] I gather they subtitle the whole city for tourist season.
A Magic Square
In the austere Passion facade, there’s a grid of numbers - it’s a magic square! Or nearly.
A proper 4-by-4 magic square contains each number from 1 to 16 exactly once, but this one is missing 12 and 16, and it has 10 and 14 twice each.
That means, instead of adding up to 34, the rows, columns, diagonals, squares, rectangles and corners each add up to 33 - supposedly Jesus’s age when he died.
What they’ve done here is take a regular 4x4 magic square - and it’s a nice one, it’s not just the corners and the diagonals that work, there are all sorts of patterns you can play with here - and they’ve taken one away from each of these four squares. So I suppose the question is, why those four? Which other sets of four could they adjust without breaking the main properties of a magic square? Which patterns does doing this break?
I have to say, I understood the magic square to be carved into the Montjuïc sandstone rather than Sharpie-d on a napkin, but I’m sure Señor Subirachs knew exactly what he was doing.
Another mathematical artefact of the Sagrada Familia is that the architect - Antoni Gaudi - made great use of ruled surfaces: curved surfaces made up of straight lines.
You might be thinking of something like corrugated iron - in one direction, it’s all wibbly; in the other, it’s dead flat. And while that is cool, it only drains one way.
Or, if you’re unlucky, no ways.
Gaudi asked: why must the ends be in phase? So instead of linking the bottom of one curve to the bottom of the other, he linked the bottom of one curve to the top of the other.
That way, you end up with a surface like this that drains in both directions!
And I think that’s lovely… only it’s not in the Sagrada Familia.
A shape that is in the Sagrada Familia is the hyperboloid, one of my favourite shapes.
If you draw straight lines between corresponding points on two circles, one above the other, then you get a right circular cylinder. But if instead you connect each point on the bottom with a point on the top that’s rotated some specified angle around, you get a curving effect on the surface.
And that’s a really good shape for a deep window, not because it lets more light through, but because it lets light through more evenly - if your window is a cylinder, you only get even light if it’s really shining straight along the axis.
With a hyperboloid, you’ve got much more freedom in where the light can be coming into the window to get the same lighting effect.
Throughout the basilica, there are catenaries and paraboloids and all sorts of cool mathematical shapes.
In classical architecture, you’ve got three basic types of column: you’ve got smooth, cylindrical columns like you get at the Pantheon in Rome; you’ve got striated columns with jaggedy bits all the way round - you get those at the Parthenon in Athens; and you’ve got Solomonic columns, which have a twist in the patterns as you go around, you might see those at St Peter’s in the Vatican.
And Gaudi said “I’m going to bring in elements of all three of those - so you start with a polygon cross-section, I’m going to twist it around a bit, split it, and end up with something that’s virtually a circle at the end.”
So, a relatively simple version would look like this: Start with two copies of a square in the same place at the bottom of the column. I’m going to rotate one one way and the other one the other way as you go up, just very gradually… Until in four metres’ time the points of the squares are at the vertices of a regular octagon.
Going to make a copy of that octagonal star, and twist that in different directions at the same speed. Do that over the next two metres and then you’ve got a 16-pointed star.
Split again, twist again… 32-pointed star, 64-pointed star, and it’s already pretty hard to distinguish from a circle.
[off] And you do that infinitely many times over the last metre?
Well, Gaudi was kind of smart about that - he’d generally cover up the last metre or so with something else – even he wasn’t so pernickety that he’d insist on infinitely many bifurcations on the nanometric scale.
[off] … but why?
Something to do with “being able to include motion and change,” it says here.
[off] But surely columns, by their nature, stand still?
Look, I just came here to do some maths, I didn’t expect the Spanish Inquisition.
I’m not convinced that was Lionel Messi.
* Don’t forget to go to The Aperiodical and vote!
A selection of other posts
subscribe via RSS