The mathematical secrets of Barcelona's Sagrada Familia

I visited this and several other Gaudi buildings about 15 years ago in Barcelona, and many of them are truly breathtaking, or at least dramatically original and unique. I went to the Gaudi museum as well and found it fascinating that the architect himself was not a professional mathematician - he did not use hyperbolic cosine to calculate the dimensions of the catenary curves, he traced the outline of hanging chains. Really interesting to hear about how he also heavily used ratios and symmetry. I love how artistic taste can be partially derived from math (but the math itself isn't sufficient to develop artistic taste)

"Gothic cathedrals and Doric temples are mathematics in stone. Doubtless Pythagoras was the first in the Classical Culture to conceive number scientifically as the principle of a world-order of comprehensible things—as standard and as magnitude—but even before him it had found expression, as a noble arraying of sensuous-material units, in the strict canon of the statue and the Doric order of columns. The great arts are, one and all, modes of interpretation by means of limits based on number (consider, for example, the problem of space-representation in oil painting). A high mathematical endowment may, without any mathematical science whatsoever, come to fruition and full self-knowledge in technical spheres." ~ Spengler, Decline of the West

Such a spectacular building. I could spend all day watching those color gradients move across the walls and floor.

I took a look at the magic square from the article.

>On the Passion façade there is a magic square in which the sum of all rows, columns and diagonals is 33.

I did the math since my caffeine load is currently ramping up.

It is simple to deduce that the rows and columns each add to 33.

The main diagonals each add to 33 (1+7+10+15) and (13+10+6+4)

Construct the matrix such that you have <rows,columns> be <x,y> as follows:

<x1,y1> = 1; <x1,y2> = 14; <x1,y3> = 14; <x1,y4> = 4

<x2,y1> = 11; <x2,y2> = 7; <x2,y3> = 6; <x2,y4> = 9

<x3,y1> = 8; <x3,y2> = 10; <x3,y3> = 10; <x3,y4> = 5

<x4,y1> = 13; <x4,y2> = 2; <x4,y3> = 3; <x4,y4> = 15

I think they also missed that the values in the corners,

<x1,y1> + <x1,y4> + <x4,y1> + <x4,y4> also add to 33 (1+4+13+15)

In addition the center square values,

<x2,y2> + <x2,y3> + <x3,y2> + <x3,y3> also add to 33 (7+6+10+10)

I think they also missed that the paired parallel short diagonals,

<x1,y2> + <x2,y1> + <x3,y4> + <x4,y3> also add to 33. (14+11+5+3)

<x1,y3> + <x2,y4> + <x3,y1> + <x4,y2> also add to 33. (14+9+8+2)

The paired parallel diagonals with three values are a tougher nut but it appears that the symmetry of the matrix allows them to be related as follows:

<x2,y1> + <x3,y2> + <x4,y3> do not add to 33. (11+10+3) adds to 24.

<x1,y2> + <x2,y3> + <x3,y4> do not add to 33. (14+6+5) adds to 25.

Neither of them gets us to the magic number until...

...we look across the matrix and add the last value (or first value) of the row as seen here:

<x2,y1> + <x3,y2> + <x4,y3> + <x2,y4> now adds to 33. (11+10+3+9).

For the other pair we see:

<x1,y2> + <x2,y3> + <x3,y4> + <x3.y1> now adds to 33. (14+6+5+8).

Looking diagonally orthogonal to this, the other paired three-value diagonals break this pattern.

<x3,y1> + <x2,y2> + <x1,y3> do not add to 33. (8+7+14) adds to 29.

<x4,y2> + <x3,y3> + <x2,y4> do not add to 33. (2+10+9) adds to 21.

When we look across as we have done for the other 3-value diagonals we don't quite get there.

<x3,y1> + <x2,y2> + <x1,y3> + <x3,y4> now adds to 34. (8+7+14+5).

<x4,y2> + <x3,y3> + <x2,y4> + <x2,y1> now adds to 32. (2+10+9+11).

Taken together their average is 33. I guess that's something.

The last thing I have for you also involves those 3-value diagonals.

If you sum the two parallels you do not get 33 nor do you get something that immediately suggests a relationship. It is only when you sum all four of the 3-value diagonals that you get to something related to 33. Let's walk through this together since I already did the math.

<x2,y1> + <x3,y2> + <x4,y3> do not add to 33. (11+10+3) adds to 24.

<x1,y2> + <x2,y3> + <x3,y4> do not add to 33. (14+6+5) adds to 25.

<x3,y1> + <x2,y2> + <x1,y3> do not add to 33. (8+7+14) adds to 29.

<x4,y2> + <x3,y3> + <x2,y4> do not add to 33. (2+10+9) adds to 21.

However, if we sum the totals of these 3-value diagonals we will find our relationship:

(24+25+29+21) = 99 = 33 * 3

That's all I have for you today.