The mathematical secrets of Barcelona's Sagrada Familia
mappingignorance.org
mappingignorance.org
Strictly speaking, it isn't "math" as math is the science of quantity and structure, both of which are objective features of reality. We all perceive structure and quantity as it is instantiated in concrete things and ensembles of concrete things and so on. We all respond to and reason about quantifiable and structural properties of reality at varying depths all the time. All math does is pursue them intentionally and methodically. It isn't surprising, then, that a competent artist should intuit various mathematical truths. Indeed, quantity and structure as essential to art. The artist is therefore closer to a domain-specific application where such properties are understood in relation to the subject matter. This introduces a domain-specific aesthetic dimension that is not present in abstracted properties, though one can certainly make aesthetic judgements about abstracted properties.
The Ancient Greeks and Romans also used the same or similar empirical geometric methods to generate ellipses, parabolas, and hyperbolas in their architecture. The difference is, they were still 1000-2000 years away from having formalized calculus.
Doing it faster and with less doubts over fidelity and existence of a solution too.
Solving partial differential equations numerically and vetting the solution so obtained is not a trivial matters. Many things can go wrong in non obvious ways.
Analogue computers are a worthy alternative when applicable.
Oh wait I misread, it was Robert Hooke as you said, but Poleni used and developed it.
I find the study of funicular shapes very gratifying.
In a way that's like doing the math, but using real-world physics as your 'calculator'. No doubt Gaudi was a smart dude.
He also died crossing the tram track, presumable not looking both ways before crossing. To be clear, I've also nearly been hit by the Barcelona tram too, so I don't blame him, but "smart" is always relative.
Also, I do belive the story was that he had the appearance of a homeless man and thus received sub-standard care. When someone finally recognized him, it was already too late.
> A culture's felt sense of proportion, ratio, and spatial order manifest directly through the hands of masons and sculptors, without necessarily needing the mathematical formalism of proofs, axioms, and treatises.
Not sure how I feel about this, as the Familia was absolutely built in a context of formalised mathematical sciences.
Wikipedia on "Sagrada Família" - https://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia (see "Geometric Details" section).
Gaudí used hyperboloid structures in later designs for Sagrada Família (more obviously after 1914). However, there are a few places on the nativity façade—a design not equated with Gaudí's ruled-surface design—where the hyperboloid appears.
12 / 7.5 = 1.6 ~= Golden ratio
> the number 12 features heavily in the Bible: Jacob’s 12 sons, the 12 tribes of Israel, the 12 apostles and the crown of 12 stars in the Book of Revelation are just a few examples.
And about "aspect ratios" (the article doesn't mention aspect ratios):
> 12 is a number particularly well-suited to establishing proportions, as it has many divisors
> the apse, at 75m (7.5 × 10), followed by the vault of the transept at 60 metres (7.5 × 8). The nave vault is 45m high (7.5 × 6), the side aisle 30m (7.5 × 4), and the choir 15m (7.5 × 2).
neither 10 nor 8 are divisors of 12, and 10 has a factor of 5. TBH it seems like 15m is the base length here, not 7.5, then you get multiples 1, 2, 3, 4, and 5, but I'm guessing these sequential integers are not considered "beautiful" numbers by the author so they picked 7.5 to pigeonhole some divisors of 12 in there.
7.2 = 12 * 12 / 10 / 2
But having integers for ones' for lengths (e.g 90m rather than 84.4) does seem a bit nicer in a way.
I think you make a good point about 15 vs 7.5 and symmetries around 1/2/3/4/5.
It undercuts the 12-centricity though; 15-centricity is just so much more man-centered, not fitting for someone about to be beatified as a saint!
In a way, base-10-centric measurements are the victors of a three thousand year battle between group-by-10s beancounter arithmatical people in Egypt/Mesopotamia and the also ancient 12-centric make-things-easily-divisible architect/construction geometers.
True, 12 doesn't match the number of fingers or toes... but have you ever considered it does match the number of spaces around our fingers and toes?
We used to learn to count on our fingers instead of counting our fingers, our four fingers have twelve knuckles and our thumb is the pointer. There are quite a few schemes for counting and doing math on our fingers like this, some include the tips of our fingers as well or tips and two knuckles, or the segment between the joints or all of the above, and both hands for complex math and different fingers for different things like place so you could count higher and do math on larger numbers. Someone saying you counted on your fingers was once a compliment, only the unlearned would count in their head, too easy to lose place because of distractions.
It is more complex than that and that example assumes the church is a cube but that is the basics of it and generally there is a square in there somewhere and that square is the basic unit. We want a church with no posts in the main space and our trees produce timbers so long, that length becomes 1 square, the church is 1 square wide and 1 square high to eaves, 2 square long, roof peak is 1 square above the eaves. The wisdom of the wood (or what ever material was being used) often played a role.
Personal style in this proportional sense came down to how they used the divisions and their subdivisions and their subdivisions and what random numbers they interjected into the easy divisions, which might be suggested by the divisions and subdivisions or come from just about anything. I am building a cabinet to hold my pots and pans, I have 7 pots and pans so I will figure out a way to get 7 in there so everyone knows it is meant to hold 7 pots and pans, or at least that one random person who got obsessed with that one peculiarity of my cabinet will figure out that 7 had some sort of importance too me. Easter eggs have been with us for a long time.
People often reduce the imperial system/base 12 to the foot and its inches, but it really is this sense and use of proportion that it is built on. If you are building a cabinet for your home, there is no reason to measure, it needs to be as wide as the space it will fit in so use that as your metric, make that one square and subdivide out the rest of the dimensions. Dividers, sector and straight edge are all you need, no need to deal with the fact that the space is 4.738 whatever wide and a sense of proportion to the whole will just happen for free. Admittedly, the sense of proportion might not be the best when you are first starting out but it will have a sense of proportion that could easily not be there if you got fixated on 4.738 and rounding errors in your cad software, and you never have to measure those odd lengths.
What I would add also is the human scale. Along with the dimensions of the space you are filling, the scale of the humans using the cabinet are important as well. Height, eye level, arm length, waist height, etc.
Standard table height is stand up straight, let your arm hang straight down, make a fist, your knuckles are at table height. Works perfectly with chairs measured by elbow to finger tip and perfect for working at while standing when doing things like kneading bread or pushing a hand plane, puts the surface low enough that you can put your upper body into it and not just your arms.
A great many of things we interact with on a daily basis developed the standards we have today centuries before we had standards for measurement, back when calibrated measurement was academic and legal, impractical and abstract.
(insert suggestions here)
And wonderful such a building that embodies in stone all kinds of mathematical relations, religious references etc. Let's hope it'll stand at least as long as it took to build. :)
Construction on Sagrada Familia is not supported by any government or official church sources. Private patrons funded the initial stages. Money from tickets purchased by tourists is now used to pay for the work, and private donations are accepted.