Guillermo León de la Barra Alvarez: I have a more elementary doubt. I apologize for my English, but I hoped (I think maybe I am wrong) in this "Round table: Methodology for mathematical analysis in architecture" to discuss a methodology that can be used in architecture. There are two kinds of general models: qualitative models and quantitative models. I won't change the focus of the discussion, but at this moment I suppose we are discussing the quantitative model. I know that all these precisions are based in Euclidean geometry, considering proportion , curves, the Golden Section, etc. But that is one step of the analysis, or one way to look at the architectural model, I suppose (I am not an architect). But there is another kind of possible step to take in mathematics, a qualitative step, that using topological and dynamical structures, and it could be another way to begin to study the architectural models. Dynamical and topological structures are more complex, considering not only metric relations and proportions. If you consider, for instance, a model of a Raphael painting, Notre Dame cathedral, Strasbourg Cathedral, etc., and you take up the model, a static model, and consider a set of points, lines and graph then you can compare them. But this is one level of comparison. Maybe they look different in that level, but as you are studying a second level, what then? Adding a dynamical or topological structure, maybe it could happen that we have less or more models. You might end up considering that the model corresponding to Notre Dame Cathedral as the same type of a Raphael painting. But then it could be possible and convenient to consider this another point of view to study an architectural model, not only from the point of view of a static model, but also a more dynamical and topological one.
On the other hand, I think that it is really important to look at the past, but also to project the future, and now we have the technology, the computer, it is easier to put a dynamic in any structure and follows its space-time development. At the same time by considering an appropriate mathematical model, you can discover that, for example, 300 lines or points and several curves, can be reduced to a few classes of objects, and it is easier to study 6 or 7 objects than 300.
I think that a quantitative method, using essentially Euclidean geometry and trigonometry, is a good way to study and compare architectural structures and artistic creations. And you don't need to use high level mathematics (homotopy theory, algebraic geometry, etc.) to be rigorous. But now, I suppose, a possible next step for a methodology for mathematical analysis in architecture could be to consider more dynamical and topological models. I suppose I changed the focus a little, but I wish and need to know more about this.

Carol Watts: Does anyone want to respond to that? I'm not sure I followed all of what you were saying, but I think you were certainly making the point that there is a variety of types of analysis, that measurement, which was our focus here to start with, is perhaps not relevant to all types of analysis.

Kim Williams: I think it comes also to the question of what your goals are. I think in this case what your goals are very directly connected to what kind of analysis you choose.

Rachel Fletcher: I want to comment on that. I've moved on from my research on Palladio and am now working on some of the architecture of Thomas Jefferson. We know a lot about Thomas Jefferson, but we don't know about his work in geometry. He doesn't provide any geometric analyses of his drawings; he doesn't provide us with that information. So I am doing what to many people would be a very speculative kind of inquiry, analyzing the drawings that he himself studied. He worked with the Leoni edition of Palladio's drawings of the Pantheon, and I thought if I studied those geometrically, and looked at the proportions came out of them, even though the measures of the buildings are totally different, would there be the same proportional techniques as in the Rotunda? And I am finding a lot of similarities. So what you are saying about trying to distill from a multiple of things is happening. To me that is a worthy pursuit, but there is no proof, you know, because it's very speculative, and yet it has been very rewarding. And I think that all of us who are in this area of inquiry, I mean, it's all speculation. I mean, we're trying as twentieth, twenty-first century people who were raised to see the world in a particular way, to see the world in another way. And all we can do is try to see, I mean, we all have to understand that we are exploring on faith, but if we don't explore those possibilities then we will only stay where we are now. We have to understand that they are only possibilities, but we have to give room to explore them.

Kim Williams: I think that though that we can explore a building and learn a great many lessons without having to put words into the mouths of the architects. Personally I've been criticized a lot for speculating, and I think the way around that is, I mean, there are certainly characteristics of the building that are objectively there, and even if you can't enter into the architect and you may not know if he meant to, but they are still there, and at that point we are very free I think to abstract those and make them known.

Bill Sapp: I think that for people dealing with historic architecture, you've got other lines of evidence that you are able to follow so that, as was brought up the other day, you could deal with the theological concepts of the time and try to understand architecture or the measurements you've taken or the perspective or whatever; Dr. Speiser was talking about relative to the religious concepts we can follow as well. I think the big danger as you point out is confusing our analysis with the intent of the architect, the intent of the builder, and that's a particular danger for archaeologists or people dealing with prehistoric societies where we don't have that sort of information available to us, so that we have to be very explicit about what our analysis includes, why we are doing it and how we are going to interpret it and present that interpretation. And to realize that it's our interpretation and that the context is our western mentality. I don't want to be too postmodernist about it, but there are certainly culture contexts that govern the kind of interpretations that we have that may have nothing to do with what the architect did.

Carol Watts: I wanted to ask another, related question. We talked about accuracy of measurement. What about, getting away a bit from the measurement, what about consistency of the use of a particular mathematical system. Let's say we found the Golden Section, or ad quadratum, or whatever.

Rachel Fletcher: Really important. When somebody shows me that there is an absolutely precise, spot on Golden Mean Rectangle on the facade of a building, I do not care, because it's just a ratio, just the difference between one side and another. That has nothing to do with the real value of proportion, which is the relationships between those differences. If that Golden Mean rectangle then divides and subdivides and subdivides again, continuing to highlight the important parts of the building, so that there is a logic behind the use of it, that carries through from the general to the specific, then I think that there is justification, even if that analysis is 2-3% off. I'd rather see that kind of approximation if it's been carried through so that you can almost hear it as a musical composition, then I think it's interesting.

Mark Reynolds: I confirm that from an artist's point of view, and I always wondered why the half-diagonal of a square, when you use the half-diagonal of a square it always connects you to your umbilical chord where you were connected to your mother. It's like, "How come it works like that?" Why is that? I just thought I'd say that. When you get hungry, that's the very first place that you recognize the Golden Section area of your body, "Oh yeah, my Golden Section!"

Carol Watts: Any other comments on that?

David Speiser: I would like to make a discussion a little concrete, that is, I give a concrete task to the people in this case who are up to measuring Michelangelo.

Paul Calter: We're on the spot!

David Speiser: Which they do in wonderful way, and precisely in the Sagrestia Nuovo, where they are in fact working. You all know that the four famous figures placed around the space sit on these perfectly curved sarcophagi, and incidentally, these are the same curves which you see in Raphael's Sposalizio, but inverted. So, what are they? First of all, I would like to point out that you can measure them as exactly as you wish, you can of course measure the height and put it into a table; I doubt that this brings you closer to the solution. In fact, a very precise and accurate photograph informs you, brings you as much and more information as a table of numbers. So it's not the numbers. Now, what can the answer possibly be? The answer will in any case be a kind of theory. And I have to verify that theory. Which is not what a mathematician usually does; a mathematician can classify the curves. I will just need the proportion, which you can very easily falsify...

Paul Calter: I'd be happy to...

David Speiser: ...which would make Sir [Karl] Popper extremely happy, who said, "So it's a theory, then it can be falsified." And I think my proposal would be the following, though I'm sure you'll have a better one, and the mathematicians would have another one, I think Michelangelo and Raphael were thinking of steel blades, which they curved this way and were inspired by the natural tensions and forces which are in these steel blades. Now you can make this experiment. Then you can see does it comes close in any way to what Raphael had painted in the Sposalizio or what Michelangelo had put in his sculpture, and then it is up to you to say this is as close or as far from the truth. Of course, you'll have to change the different steel bands...

Carol Watts: So you can actually do some scientific experiments.

Paul Calter: This is quite do-able by the methods we use with the theodolite, measuring points along the arc and then using curve-fitting techniques you could decide what kind of a curve it is, and then you could do the same thing with a steel blade and compare the two and see how they compare. They won't of course compare exactly, but you could decide if it is close enough to verify this hypothesis or not. And if you'd be willing to write a small grant, we'll be happy to go and do this.

Carol Watts: While we are waiting for the tapes to be working again, what I'd like to do is ask if there are other ideas to share about how do you get started? How do you know what to investigate?

Kathleen (Kathy) Reynolds: Carol, I think this is the creative part of the scientific method, where you are researching, proving a thesis, recording, analyzing on one topic, and during that process you will bump into something that you may save as an idea; in the back of your mind you are saying, "When I finish this, I would like to look at this." It will give you the stimulus, your success in one area will stimulate you to then investigate another area. I've seen this in Mark's projects, where he'll say "Well, this is another thing I want to look at one day". I think that's a creative aspect of being scientific, doing a lot of research also.

Carol Watts: One thing leads to another.

Mark Reynolds: There's an aesthetic, a desire to want to know how something is structured, its beauty and so on. When you begin to realize that you can find so much geometry in nature, and when you go back as far as you can to the caves and you see stick figures whose legs are triangular shaped and you see branches of trees that remind you of things geometric, I think you are just naturally inclined to want to know something about geometry because it's what builds what our world is, partly at least. And I think all of us go to those things that attract us of their beauty, or because are amazed or stunned by them; we're dumbfounded by how could somebody even do that.

Kathy Reynolds: Rachel, did your study of Palladio grow out of something else?

Rachel Fletcher: Yes, that was an unusual one...

Kathy Reynolds: But that was my thought when you you said that you were doing Jefferson.

Rachel Fletcher: Yes, I deliberately moved on to Jefferson after Palladio, because for me it's been kind of a journey around the world--trying to come home. I started with ancient Greek and Roman theatres, then gradually moved on to Palladio. Now I see that all this time I've been trying to get home, and right now, it's Jefferson. But as I say I think probably the most important thing I ever learned from my geometry teachers was from Lucie Lamy, the stepdaughter of Schwaller de Lubicz. I was very lucky to be invited to prepare some of her drawings when, in her later years, her eyesight began to fail. I don't have a sophisticated background in geometry at all, and at first she was kind of stunned that I didn't have the mathematical training. But she went away, and then she came back and she said, "Oh, no that's the best because you don't have to unlearn everything, you can start from scratch." And through that process the most important thing that I learned was not to impose what I wanted things to be, just to listen, to realize that whoever I am brings to the work certain limitations, and that the best I can do is get out of the way and let the work speak to me. And I try as hard as I can to do that, so I try not to say, "Gee, I have a theory to prove; I'm going to stick it on here." I rather try to meet the project at least on neutral terms.

Mark Reynolds: You know, I wonder. When I taught elementary school student teaching back in my old college days in 1880, I wondered why kindergarteners and first through third grade, when you would give them tempera paint and an 18 x 24 sheet of paper, why they would always use the circle, square and triangle to make everything. It was their family, it was their brothers and sisters, it was their neighborhood, it was their mother's car, it was just everything in the world. I wondered from the beginning, you know, we don't talk about the collective unconscious, but where did this geometry come from anyway? I mean, we didn't invent this geometry; this geometry was here before all of us were. Before any human being came to earth in any way, shape or form, geometry existed. So was geometry there at the very, very beginning of the Big Bang (if you believe in the Big Bang)? Was it responsible for the Big Bang? Was it the result of the Big Bang? Was it there before the Big Bang? Or, if it happened in between the Big Bang and the present time, who or what invented geometry? Or, does geometry have its own living consciousness throughout the universe? I don't know the answer to it.

Carol Watts: When I published my article in Scientific American on the Roman houses at Ostia, it received a very wide interdisciplinary audience because of the publication. I received a number of letters, including several from neurosurgeons and scientists studying brain structure who were fascinated by it and saw a relationship to their work of the structure of the brain. It was from Eastern Europe and the Soviet Union and places like this, from all over the world I was getting some of these letters. I didn't understand the details of what they were trying to tell me but it clearly resonated as geometry (and this was fairly simple, the Sacred Cut), that geometry was somehow innate in the structure of the brain.

Mark Reynolds: Did you see the January issue [of Scientific American magazine]? The human skin is covered with small equilateral triangles that conduct electricity. Well, they did a study and said that our skin is very similar to the way Buckminster Fuller did his work. I don't know if that's true or not, I'm just telling you that that's what was reported in the Scientific American, for whatever it's worth. We may even have equilateral triangles all over our bodies and not know it!

Rachel Fletcher: I have to disagree. I do think that there is probably a relationship between geometry and the structure of our brains as we understand the structure of our brains. I'm on the side of thinking that mathematics is probably a human construct, but that doesn't make it any less of a miracle. The wonder of geometry is not how it was created, but the magnificence of the complexity and the richness of it.

Paul Calter: Can Judy get a word in? We have a real mathematician here...

Judy Moran: Because all of our talk has only of course been about one geometry. And when you were talking, you were just talking about Kant, right? and Kant's theory of knowledge.

Mark Reynolds: Exactly.

Judy Moran: My field is actually hyperbolic geometry, partly. So here is Kant who believes that the brain is hard-wired to perceive Euclidean geometry and that that's what we have to have, and then the nineteenth century comes along and we discover that we can have any geometry we want. And so we can have curvy triangles, we can have hyperbolic geometry, we can have all kinds of non-Euclidean geometry, which is in fact the basis of relativity theory. So we can pick and choose our geometries, I think.

Mark Reynolds: What's that that Descartes was taking a nap one afternoon and he was looking at the ceiling and he said, "Gee if there was a fly in each of the four corners and each one of them saw each other out of their right eye, and started to walk towards one another, what would the resulting pattern be?" So did that come from the construct of the brain, or was that just something creative that he came up with? There's no answer I guess to that, but...

Kim Williams: I want to say something about this as an architect and not as a mathematician. I think it's is possible for us to have whatever geometry we want, but I think, as an architect and maybe a person just walking on the earth, we want it to be Euclidean geometry, and we don't want our builders to know that parallel lines ever meet. In a very practical sense, I think our first experience of the world is as Euclidean geometry, and that is, we are vertical beings on a horizontal surface. I think we need horizontal surfaces in order to move because we could have hyperbolic floors with hyperbolic tilings that would be really beautiful, but we'd trip a lot.

Mark Reynolds: Escher, when he did his circle limit series, was using that type of geometry.

Judy Moran: Oh, absolutely...

Mark Reynolds: All of the points were outside the circle but when you look at the pieces they are geometric as can be. So maybe the architect could use some Escher.

Stephen R. (Steve) Wassell: I just want to say one thing, that mathematics is discovered, not invented.

David Speiser: Say it again?

Steve Wassell: Mathematics is discovered, not invented.

David Speiser: No, it's both actually.

Judy Moran: Mathematicians don't agree on that. That's a big debate in the math community.

Kim Williams: Who said that's not true?

José Francisco Rodrigues: I said that mathematics is a human creation. It's not that something that exists independent of man. Of course, when man created mathematics, he creates because he lives in this world and because he wants to master the world he lives in. So I do not think that geometry has anything to do with something outside us. Of course...

Mark Reynolds: You are saying mathematics and Steve is saying geometry...

Steve Wassell: No, geometry is math......

José Francisco Rodrigues: Well, what is geometry? Geometry is what? A chapter of mathematics? What is the difference? Geometry is a small subset of all mathematics.

Mark Reynolds: Because I can make a circle and not know anything about math. I wouldn't have to know pi to make a circle, I could just do it.

Rachel Fletcher: You should read the Meno dialogue.

José Francisco Rodrigues: But are we sure you are making a circle? You are making a physical circle, but when you think about a circle as a mathematical object, that is a different thing; that is a human creation. When you make a circle...

Mark Reynolds: But, see, I'm in trouble because I separate geometry from mathematics, so I'm already in trouble; I separate the two.

José Francisco Rodrigues: Why? What is the difference? Why do you separate them?

Mark Reynolds: I'm not really sure...

José Francisco Rodrigues: Is geometry so important for you, and mathematics of so little importance?

Mark Reynolds: No, no, it's not that at all, it's just that it is easier, I suppose. The number and the math are present in the geometry. You can find a lot of mathematical formulae by doing analyses of the geometry that is already there. I mean, the spheres in the universe have been there...

José Francisco Rodrigues: They are not spheres...

Mark Reynolds: ...well, whatever you want to call a star...

Kim Williams: But a planet is really not a sphere...

Mark Reynolds: Okay, whatever you want to call a planet...

Kim Williams: But it's not a matter of what you call it...

Guillermo Leon de la Barra Alvarez: Maybe most of you have reason, and that is a great problem, when two people have reason, they can't reach a point, because they are talking from a topological point of view. What you see is not a sphere; it could be a cube, you don't know that, because it is transformed, this sphere, but this is the oldest discussion and it is not finished here...

José Francisco Rodrigues:...no...

Guillermo Leon de la Barra Alvarez: ...because one is the abstract object and the concrete object, you know? Mathematics, somebody said, is a language, and others said it is created, and another said it is discovered. But mathematics is more than all that, because it is the same as architecture. Architecture, I suppose, I have talked with some close friends who are architects, and they told me that architecture is art, technology, philosophy, sociology; it is everything. So it is multifaceted. To study architecture you must merge, in a mathematic way, in every dimension of the space, so it is very difficult, and I agree with you, that when you look at something, you look at the sky and you look at the star, you find all the polygons, all the structure of the Euclidean geometry, of the spheric geometry, hyperbolic geometry; you find everything. But let me retroscind, in mathematics you have an object, when you study homotopy, you can find something that is called the generator, and it is a mathematical object, and it is very real. It is so real that you can use your cellular phone to make a call, so it couldn't be more real than that. But you don't see that inside your telephone there is a chip, and a circuit, and the circuit is a piece of wire, but why can you take a piece of wire and use it to make a call? Because the engineer or two engineers have adapted that wire, have changed its position and put in this pole and this other pole in such a way that you can talk. And this is the difference...

Copyright ©2000 Kim Williams