"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 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.
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.
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.
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.
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. :)
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.
20 comments
[ 3.7 ms ] story [ 45.9 ms ] threadStrictly 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.
In a way that's like doing the math, but using real-world physics as your 'calculator'. No doubt Gaudi was a smart dude.
[1] https://en.wikipedia.org/wiki/Cathedral_of_Justo
12 / 7.5 = 1.6 ~= Golden ratio
(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. :)
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.