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)
> I love how artistic taste can be partially derived from math [...] the math itself isn't sufficient to develop artistic taste
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.
Designing in an era where calculus exists, using chains and weights strikes me as gratuitous or onanistic.
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.
Calculus exists, but analytic solutions generally don't. Gaudi's chains and weights serve as an incredibly elegant mechanical computer that were only surpassed in the last few decades by CAD. Designers used mechanical splines until the advent of CAD in the 70's/80's.
He is solving differential equations but with an analogue computer.
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.
No doubt. I call them empirical geometric methods, you call it an analog computer, same thing. He didn’t invent anything though. The method of hanging a chain and adding custom weights to find the ideal shape for a complex masonry structure was invented by Giovanni Poleni in 1743 to fix the dome of St. Peter’s basilica. Poleni himself was extending Robert Hooke's 1675 inverted chain concept for optimal arches. The techniques Gaudi used had already been in use for hundreds of years.
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.
The Gaudí models themselves were actually destroyed during the Franco years, and so what we see today exists thanks to the successor architects doing reconstructive work on Gaudí’s lost mockups.
"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
I had to put this one through Claude, but it boils down to:
> 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.
It seems somewhat important to me to know if something was done because it looked pretty, was random or because there was an intent to reflect maths, science, planetary alignment etc.
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.
Whatever one might say about method, epistemically speaking, the aesthetic is prior to the mathematical. The mathematical is found in the analysis of the beautiful.
Gaudi stuff is cool. I used to live in Barcelona, I know all about it. But numerology is not the same as mathematics- nothing in the article comes close to sniffing at the math of classical architecture. And while many have said I have a well-developed architectural taste, and I certainly understand the artistic fascination and amusement with Gaudi’s works, I never understood the magnitude of architectural devotion that they have attracted.
> 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.
The earlier Spanish measurement system that changed to meters in Gaudi's lifetime was, like the English system, more 12-centric than 10-centric.
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?
>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.
Vaguely related, there is also the unfinished Cathedral of Justo [1] near Madrid built by a solo developer since 1961 and essentially by hand and more or less from scrap and whatever people donated. Justo Gallego Martínez died in 2021 at age 96 and donated the building to some foundation for completion.
TFA does not quite explain the logic of this and somewhat suggests it as unique when it was the norm. If you are building a church, what better metric is there than the church you are building? There is no reason to measure, you can just subdivide with very basic tools. We want the entrance to be in the center, that is 1/2 church instead of 8.385 meters and that door is 1/6 church wide and 3/4 church tall. We want windows down the sides of the church, those windows are spaced 1/4 church and are 1/6th church wide and 3/4 church high. Windows have panes and doors have panels so lets subdivide those, panels/panes 1/3 window/door.
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.
A lot of words but I agree with you with all my heart. Proportions best relate to the whole, not some “2 inches from the edge” or whatever arbitrary standard.
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.
Wow that first photo almost looks like a sci-fi contraption!
(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.
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)
https://news.ycombinator.com/item?id=48860150
> I love how artistic taste can be partially derived from math [...] the math itself isn't sufficient to develop artistic taste
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.
Designing in an era where calculus exists, using chains and weights strikes me as gratuitous or onanistic.
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.
Calculus exists, but analytic solutions generally don't. Gaudi's chains and weights serve as an incredibly elegant mechanical computer that were only surpassed in the last few decades by CAD. Designers used mechanical splines until the advent of CAD in the 70's/80's.
He is solving differential equations but with an analogue computer.
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.
No doubt. I call them empirical geometric methods, you call it an analog computer, same thing. He didn’t invent anything though. The method of hanging a chain and adding custom weights to find the ideal shape for a complex masonry structure was invented by Giovanni Poleni in 1743 to fix the dome of St. Peter’s basilica. Poleni himself was extending Robert Hooke's 1675 inverted chain concept for optimal arches. The techniques Gaudi used had already been in use for hundreds of years.
Ahem https://en.wikipedia.org/wiki/Archimedes_Palimpsest
> he did not use hyperbolic cosine to calculate the dimensions of the catenary curves, he traced the outline of hanging chains.
In a way that's like doing the math, but using real-world physics as your 'calculator'. No doubt Gaudi was a smart dude.
> 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.
He was smart, but not street smart.
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.
Be sure to visit the basement museum to see the upside-down string + tiny sandbag weights model that was used to simulate the structure. Pure genius.
The Gaudí models themselves were actually destroyed during the Franco years, and so what we see today exists thanks to the successor architects doing reconstructive work on Gaudí’s lost mockups.
"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
I had to put this one through Claude, but it boils down to:
> 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.
It seems somewhat important to me to know if something was done because it looked pretty, was random or because there was an intent to reflect maths, science, planetary alignment etc.
Mathematics and Art - https://en.wikipedia.org/wiki/Mathematics_and_art
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.
Whatever one might say about method, epistemically speaking, the aesthetic is prior to the mathematical. The mathematical is found in the analysis of the beautiful.
Truth, beauty and goodness are known as the “transcendentals” for a reason. They are linked. They don’t come one after the other.
Gaudi stuff is cool. I used to live in Barcelona, I know all about it. But numerology is not the same as mathematics- nothing in the article comes close to sniffing at the math of classical architecture. And while many have said I have a well-developed architectural taste, and I certainly understand the artistic fascination and amusement with Gaudi’s works, I never understood the magnitude of architectural devotion that they have attracted.
"The temple’s dimensions are based on the number 12 and a module of 7.5m."
12 / 7.5 = 1.6 ~= Golden ratio
Call me grumpy, but numerology does not make architecture more beautiful.
That's not grumpy but ignorant. Aspect ratios are not numerology (per se)
FTA:
> 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.
When I saw the emphasis was on 12, I wondered why it was combined with 7.5 so much rather than say, a more 12-centric nearby number like 7.2.
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!
The earlier Spanish measurement system that changed to meters in Gaudi's lifetime was, like the English system, more 12-centric than 10-centric.
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?
>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.
Vaguely related, there is also the unfinished Cathedral of Justo [1] near Madrid built by a solo developer since 1961 and essentially by hand and more or less from scrap and whatever people donated. Justo Gallego Martínez died in 2021 at age 96 and donated the building to some foundation for completion.
[1] https://en.wikipedia.org/wiki/Cathedral_of_Justo
TFA does not quite explain the logic of this and somewhat suggests it as unique when it was the norm. If you are building a church, what better metric is there than the church you are building? There is no reason to measure, you can just subdivide with very basic tools. We want the entrance to be in the center, that is 1/2 church instead of 8.385 meters and that door is 1/6 church wide and 3/4 church tall. We want windows down the sides of the church, those windows are spaced 1/4 church and are 1/6th church wide and 3/4 church high. Windows have panes and doors have panels so lets subdivide those, panels/panes 1/3 window/door.
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.
A lot of words but I agree with you with all my heart. Proportions best relate to the whole, not some “2 inches from the edge” or whatever arbitrary standard.
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.
Wow that first photo almost looks like a sci-fi contraption!
(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. :)
Reminds me of diatom pictures. The soft edges in nature comfort me.
Such a spectacular building. I could spend all day watching those color gradients move across the walls and floor.
I went went once it's very beautiful inside! And the view from the top is also wow! Thanks for sharing this
Who funds these?
https://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia#Funding_a...
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.