Panelists: Rachel Fletcher, William D. Sapp, Paul Calter, Mark Reynolds
PART II: WHAT IS THE NATURE OF MATHEMATICS?
PART III: HOW DO WE DIFFER IN OUR APPROACHES TO THE WHOLE RELATIONSHIPS OF MATHEMATICS AND ARCHITECTURE BASED ON OUR DISCIPLINES?
PART I: HOW IMPORTANT IS ACCURACY?
The Nexus 2000 round table discussion on methodology in architecture and mathematics took place on Tuesday 6 June during the course of the Nexus 2000 conference in Ferrara, Italy. Moderated by Carol Martin Watts (far left in the photo), the panelists were (from left to right) Rachel Fletcher, Paul Calter, William D. (Bill) Sapp and Mark Reynolds. The following is a transcript of the audio tapes made during the discussion.
Carol Martin Watts: Kim
asked me if I would moderate this round table, partly I think
Before we get into the specific discussion I have just a few general introductory comments and then I'll be asking questions of the panel and then giving you [the audience] the chance to ask them follow-up questions or to make your own comments. The majority of the papers presented at this Nexus and previous ones have concerned themselves with analyses of past works of architecture; just a few papers have presented contemporary designs or proposals for design methods using mathematics. So because of this preponderance of historical analysis, I thought that that would be our focus today. It seems to me that, before we talk about methodology, we need to talk about our research questions. What are we trying to find out? What are we asking? We may not all be asking the same questions in each study but there are some very general questions that are relevant. We are probably all concerned with what mathematics is evident in the particular work or the work of a particular architect, whether it is geometry or proportion or whatever. We have to establish if it is present, what it is, or how do we describe it. We are also concerned with how mathematical systems were used in design and construction. What design methods produced the end result?' could be a question that some people are asking. There is also the important question of why - why such mathematical systems were used; what, if anything, was their meaning or significance. So we have the what, the how and the why. Any one study might deal with perhaps one of these more than others, but they're certainly all relevant to get a complete picture.
In terms of the general methods of research, these questions are looking at historical examples, whether it's ancient or early twentieth century, or even more recent, typically using an after-the-fact method. We cannot conduct scientific experiments, with control groups. Except in the case of very recent works, we are not able to interview or conduct surveys of architects, builders and clients, although that may be one of the methods useful for relatively recent designs. As you go far back into time, to the Renaissance or Romans, we don't have that as an option. Essentially what we are having to do is look at observations of physical traces: what remains and is present of the work in question; and different kinds of archival research, study of records, drawings, texts, perhaps about the particular example or perhaps more general about the culture and the architect's work at that time.
So we have these two essential sources for figuring out after-the-fact what happened and how something came about: observation of physical traces and archival. The physical traces may be rather limited, or in ruins, or perhaps they have changed over time, and in other cases are well-preserved. We have to establish whether we are dealing with something that has been changed over time or has been restored. The condition of archival information varies greatly; in some cases there is a lot of information, in other cases it is limited, or it's nonexistent or is very general. These are some of the variables depending on what place, what time period, what culture we are looking at. It may be the fact that we are using these after-the-fact types of analysis that we can't really prove whether our hypotheses are true. Now, although scientific experiments might not necessarily prove anything either, the general public has the sense that experiments are science, and science is accurate, and the methods that we are using are perhaps seen as not as scientific, but that is something we can talk about. I agree with Kim that this discussion is particularly important because there is a tendency to not take seriously geometrical and other mathematical studies, certainly within my field of architectural history, and most likely others as well. So we need to think about ways in which you can make as strong a case as possible.
For today's discussion I have decided to focus essentially on two questions that are of interest to the panel here, who were all suggested by Kim to provide a broad diverse group in terms of the disciplines they are coming from and the types of things they study. Everyone [on the panel] has an interest in measurement, so I thought we would start with that. I would also like to take advantage of the variety of disciplines to talk about disciplinary differences of approach. As I mentioned before, I'll start with a question and give everyone [on the panel] a chance to talk if they want to and then we can open it up to the rest of you for follow-up questions or your own comments or debate or argument. As moderator, I will claim the power to decide when to stop the debate on something, move on to something else, or to try to direct things back on track. I have a list of follow-up questions myself but I am hoping that I won't even get to those, that you will be coming up with the questions.
Paul Calter: Of course you know my involvement in this whole process very, very narrow. I couldn't even begin to address all the questions on your list, but maybe we can talk about accuracy. What is appropriate accuracy? I think it depends on two things: what it is you are measuring, and the purpose of your measurements. First, what it is that you're measuring. You wouldn't expect to use the same degree of precision when measuring, say, the Medici Chapel with nice, clean, stone walls as you would measuring a crumbling medieval tower or some foundations in Peru, for example. It just wouldn't be appropriate to use the same precision in both cases. In the other direction, measuring a small precise pavement like Kim does, you would think that an even greater degree of accuracy is indicated there; you don't just take a measurement once, you measure it several times to get a result, something you wouldn't consider doing in all cases. So that's part of it, what it is you are measuring, choosing a method that is appropriate. The other thing is what you plan to do with the measurements. In the case of the Laurentian Library doorway, we were primarily doing a survey that we were going to publish for the use of other scholars, so there we tried to strive for the best accuracy we could without using extraordinary measures, just simply taping and triangulation. If you were using the results of the survey for your own purposes, say proportional analysis, you might want to settle for a less degree of accuracy.
William D. (Bill) Sapp: I would like to say that in archaeology, when we excavate sites, we are often only going to be at that site once, which means that we are going to uncover walls and map the site and the results of our excavations are going to be what other people are going to use for anaylsis. And so because of that I think that it's important to be as accurate as we can be. I've seen overlays of maps done of particular sites in Peru in the early part of the twentieth century and in the middle part of the twentieth century and later done with a total station and a theodolite, and the differences are quite substantial, so that regardless of what kinds of questions that, let's say, I have as an archaeologist and I want to answer with this information, I have to realize that other people are going to have to use the information that I get for other kinds of analysis, so I think it really behooves us as archaeologists to get the most accurate kinds of measurements that we can get. Given that, there are still a lot of problems in terms of accuracy and what we define as accuracy, but I still think that taping gives us one degree of accuracy, using a theodolite gives us a much higher degree of accuracy, and if we can use better methods, then that's what we have to do, realizing that we may be the last person that is at that site. Also, archaeology is a very destructive science. When we dig something up, we may decide to destroy a wall because we have to excavate below the wall to see what kinds of prior constructions existed, so that means that no one is EVER going to get a chance to remeasure these kinds of things in the future. So I think that in that sense, while we may be dealing with things that certainly aren't as finely constructed as medieval and Renaissance buildings may be, we still really need to be as accurate as we possibly can.
Rachel Fletcher: And also I think at this level of taking measurements it is very useful for people coming after who are going to be using them to know what the degree of accuracy is. If you can say what that is, it's really helpful to the next generation so they can know what they're building on.
Bill Sapp: Right.
Carol Watts: So you see accurate measurement and the use of scientific technique now, sort of the most up-to-date techniques to measure, as a way of helping to convince people that there is validity to such things as geometry?
Mark Reynolds: Well, I think there is a problem for all of us, we talk about what happens over the centuries and how things shift and change, earthquakes and floods and so on, and so I don't know whether we are having accurate measurements or not, so that's problematic.
Paul Calter: I'd like to go back to the phrase that Bill used "accurate as you can possibly be". What does that mean? Using extraordinary measures? Taking each measure three times? Using better instuments? You can always make a more accurate measurement, but is it really appropriate when you are measuring, say, crumbling walls or a foundation? Would you try to measure it to a millimeter or a tenth of a millimeter; it is possible but does it make any sense in a structure of that sort?
Bill Sapp: I see what you are saying. We're talking about taping, and we're talking about taping and using a compass for example, to lay out the plan of a building, again referring back to these overlays of plans that I have seen done with compass and tape versus what's been done with a theodolite. The differences can be substantial, and as you said, with smaller things you might want to be more accurate, and yet, I see with larger things you might want to be more accurate. If I'm taping a temple that's a hundred meters on a side and has rooms on top I have a greater chance of being wrong if I'm using a tape and compass than if I use a theodolite. In terms of what degree do you want to be accurate, I think we're all bound by the fact that we have a certain amount of money to spend on a project...
Carol Watts: ...and time...
Bill Sapp: ...and time, and we have to make the evaluations of how much time and money we're going to spend on that particular aspect of our project. Given that we have those restraints, I think that one is obligated to do the best job possible. I'm back to this "best job" again, but that means using the most accurate technology within the given time and monetary constraints. Taping is great, but if I've got a crew of 15 people that are working out there and some people don't stretch the tape quite as tight and we're not using surveyor's tapes and we're not using 20-lb pulls and strings to get accurate taping, I could be meters off if I'm using a 50- or a 100-meter cloth tape; I could be degrees off in terms of weighing on these things and there are certain people who would want to take my information and understand why a building was laid out in a certain direction and how it may compare to other buildings. So I may not really care whether the corner is 2 or 3 millimeters one way or another, but I want to know whether the wall is at 32 degrees off north or 42 degrees off north; I want to know if that wall is 108 meters long or 112 meters long.
Paul Calter: I hear what you're saying now. The thing that threw me that was "to be as accurate as you possibly can" implies to me that you are measuring with greater precision, say down to millimeters, in a case where the physical structure that you are measuring isn't that precise, because if you measure at the top of a wall you get one dimension, but then a few inches down [away] you get another one. That degree of precision is not appropriate.
Bill Sapp: ...right...
Paul Calter: I think what you are after is a reliable set of data that is pretty close, but not highly precise.
Bill Sapp: I think that's right.
Mark Reynolds: I think you're missing Kim's emphasis on percent of deviation as a way of verifying just how close or far off we are and by explaining the kinds of tools we've used and giving that percent of deviation we give future scholars something to go on. We better not use this old wooden ruler because it's going to be a couple of meters off versus a theodolite with a laser attachment to it, so I think Kim's emphasis is appreciated, (although I had to call her a couple of times to understand what percentage deviation meant!), I finally do understand what it means now and it's a very, very valuable tool, and an important tool for any scholar in the future to be able know just how just how far off they are. Both of you were talking about something that percentage of deviation could help.
Rachel Fletcher: One of the things you were mentioning, that a wall was one thing here and another over here, especially if you are taking very accurate measures and you can't take very many points, the question is what points do you take in order to really reveal an accurate portrayal, whether it's a door or whatever. In a certain sense I like the idea of someone who is taking measurements to go about it purely objectively without anyone telling them what points to take so that there isn't that bias. On the other hand I really appreciate having a team like you and Kim working together to say "There is a particular point that I feel is going to reveal something important", and yet that carries with it a certain tendency to read something in a particular way. So I think choosing what to measure is as important as how accurate the measurements are.
Mark Reynolds: Kim, the points were verifiability' and repeatability'?
Kim Williams: I said scientific method requires verifiability and repeatability. A result has to be verifiable and it has repeatable, and that is what we are looking for, I think.
Paul Calter: I think it is important especially if you publish your results.
Kim Williams: That's true. If you measure a building, someone should be able to go back and measure the same way and find those same numbers.
Carol Watts: Which is why it's very important that you always include in publications or any archives of your measurements how you did it and a description of the process.
David Speiser: May I humbly point out to our directress, that one of the preferred subjects in Nexus are the symmetries. Now whether the symmetry is hexagonal or octagonal, you can count it on your fingers, and you don't have an accuracy that is very difficult to achieve; I mean, if you can't count, you can use your fingers. Second, as I was once lucky to observe, the accuracy even then depends [on what you measure]; the accuracy you need for measuring an angle is not as great as for measuring the length, it's very different. So it depends very much on the subject about which one is talking, and of course the subject comes always up when what one is up to is verifying a theory. Then what you said is perfectly accurate. And I give as a dubious example the series of discussions that we have had at all Nexus conferences about the Golden Section, and there I was always sitting with our friend Volpi Ghirardini, who demanded an enormous exactness, because there you are in trouble, and you never know what you are talking about, if you are talking about a Fibonacci series, which is very different from the Golden Section, as far as the accuracy is concerned, so it really is a question that depends on what you are talking about.
Kim Williams: Well, in some ways. Let me make a comment, though. It is one thing to count the sides and say, "The Baptistery of Florence is an octagon", but I think the next step is "how regular an octagon is it?". It depends on what kind of a theory you are trying to prove, but actually for Brunelleschi, with the cupola of the Duomo of Florence, who had the job of building the world's largest cupola at that time and spanning that octagon, it did make a great deal of difference to him that that octagon wasn't regular, because it caused all kinds of problems. And so frankly the issue of how many sides it has versus does it have that many equal sides, I think is crucial.
Judith (Judy) Moran: I'm speaking now as a mathematician, and I know in Kim's project for this discussion she's talking about, not believability...
Kim Williams: Verifiability.
Judy Moran: No, the discipline itself...
Paul Calter: Credibility
Judy Moran: Thank you. And I really wanted to reinforce a lot of the points that Paul made because I know how some members of the mathematical community would view this, that it's not just how accurately you can take a particular measurement, it's the mathematical operations that you perform on these measurements and not understanding the kinds of precision you lose. Paul was talking about that and I thought that some contribution on that might be important because I don't know if people know a lot about that. If you're just doing the measurement, that's one thing, but if you're ever making a claim about the proportions, you have to really know how numbers work, and really understand it in order to ever prove the Golden Section, because you can't really measure it because of course it's irrational, so there needs to be a sophistication and an acknowledgment of that I think in publications for the credibility that you're looking for, that when you start operating with numbers there are other constraints.
Mark Reynolds: Can I respond to that? I found after 30 years of working in geometry that a lot of times the geometry will give me the mathematics rather than the other way around. In other words, that I can make a harmonic ratio in a rectangle or a square by construction, I don't need to know any mathematics at all to be able to do it and there are sometimes there are no mathematical procedures needed in order to do a particular paving, for example. Perhaps three-dimensionally or structurally speaking you may need to know the mathematics so the thing doesn't fall down, but if you're doing a paving, a lot of times you don't need to know any mathematics whatsoever, the geometry has the math in it already.
Judy Moran: I think we are talking a little at cross purposes. I'm not talking about the mathematics that's required to construct the structures...
Mark Reynolds: Okay...
Judy Moran: I'm saying, some people here are looking at measurements that they're taking, that's the original question, and then talking about precision, there's the precision of the individual measurements, and I'm not talking about the architect used to construct the building, but then when you perform numerical operations on your numbers to get proportions, you need to be aware of some of the things that Paul spoke of in terms of how accurate, degrees of accuracy, decimal points. I know this sounds really boring, but the results of what you do with your numbers may not be as precise as some of the individual measurements and you need to know that and refer to it in the final publication.
Rachel Fletcher: I don't crunch numbers, I lay something out geometrically with a compass because I aspire to get as close as I can to the intention of the proportion, but I am also getting as close as I can to how a carpenter or a craftsperson would lay it out. I don't understand why it is always the Golden Mean that seems to require an extra perfection in laying things out, while other measurements seem not to, no one seems to care very much. But a Golden Mean isn't a Golden Mean unless it is so precise that you can't even draw it, you can't even put your pencil on the paper because the thickness of the line compromises the accuracy. You can never get as perfect a Golden Mean as can be conceived in your mind. Architecture is meant to be experienced, it is meant to be lived in, it is meant to be perceived, it has all of those experiential qualities which I think don't require the super perfection that only the mind can hold.
Judy Moran: That's not what I was talking about. All I was talking about, and this is a huge issue, is the very rudimentary numerical sense that of those people who take measurements, and some people at this conference were spinning out numbers in the decimal places, and when you do that and then you take one and combine it with another one to get ratios and things like that, then you need to be aware that, in terms of credibility, you need to be aware of how precision works under plain old arithmetic operations, that's all, so that when you talk about your results you're don't claim an accuracy of proportion that is greater than you can get from the input.
Paul Calter: So that wouldn't apply to a purely geometrical analysis, but to a numerical analysis.
Carol Watts: So if you're adding up a string of numbers for example, even if each one is very accurate, the total may not be.
Rachel Fletcher: For example, I was interested in Mark's derivation of the equilateral triangle and the Golden Mean, because mathematically speaking you cannot reconcile two incommensurables; that is not possible, and everybody knows it. But the fact that there is such an intersection suggests a potential in the mind to bring them together. The great geometric exercise that all of us go through, that we have all been searching for, is the squaring the circle, reconcile the perimeter of a finite form with an infinite form. Now we attempt it not because we feel we can accomplish it, but because in the quest there are all kinds of values and qualities that come about through the exercise of it. That's a part of the experience. I think that's what we are looking for.
Mark Reynolds: I agree with you and I think that sometimes the mathematician and the artist hold hands and everything goes really well, but sometimes we part company when mathematician says, "A line has no width." Well, an artist is going to say, "I'm sorry, but in my drawing if it has no width, you're not going to see my drawing." So sometimes we agree and and sometimes we don't and I think if we could come back together as artists and mathematicians together, that would really help to solve this issue that has been brought up.
Paul Calter: I think we are getting far afield here. All Judy is saying is that if you do use numbers, measured quantities, and carry them through a computation as some of us do, you should know how the accuracies or inaccuracies propagate through the computation and treat the numbers properly.
Judy Moran: That's all!
Carol Watts: Let me say something as moderator. I think this is a good example of the differences between the disciplines, even in terms of communicating, right? A mathematician, an artist, an architect, an archaeologist think in very different ways, communicate in very different ways, and that's what's good about a conference like this is that we are coming together, but it can be easy to have misunderstandings.
Paul Calter: There's a difference between a numerical approach and a geometrical approach...
Carol Watts: or a visual approach...
Mark Reynolds: Can't those numbers change, though, over time? I mean, we've all seen like something shrinks over years, or there's an earthquake and it bends. How do we know that the numbers themselves are telling us the truth?
Rachel Fletcher: Well, that's the question that I have with buildings. I mean, we're all talking about taking simple, accurate measurements of buildings which have settled over time. Are they closer to what was intended than the original working drawing? I mean, what kind of tolerance do you have in the case of settling?
Paul Calter: I think we're not really taking super-accurate measurements to begin with. We're using very basic surveying techniques, taping, usually just once, without repeating the taping; we're not making corrections for temperature or sag or for pull, just doing the measurements...
Rachel Fletcher:... the best you can do...
Paul Calter: ...the best you can do, and I think in most cases it's more than enough, because of the things you've said, because of irregularities in the buildings, and of alterations that may have occurred, because of earthquakes and floods and bombings. I think taping is fast and simple, it's not an extraordinary means, and it gives you accuracy that's fine for most cases.
Mark Reynolds: Do you think that most mathematicians feel the way you do?
Paul Calter: I think they do, sure, sure.
Carol Watts: I think Leonard had a comment earlier.
Leonard K. Eaton: Yes, I'd like to see a little more effort at thinking back to the design tools that were used by the people laying out architectural objects. Now, I am not a historian of technology, but I will bet that Michelangelo did not have a theodolite when he laid out your doorway. I visualize the old boy as having something like a yardstick and maybe a compass and perhaps a triangle, and those instruments survived until well into the nineteenth century. So I just would like to see, when this whole issue of measurement comes up, a little more emphasis on what the architect at a given time had to do design and layout with.
Paul Calter: I think you're interested in what kinds of precision, what kinds of tolerance were acceptable during the periods...
Leonard Eaton: Sure...
Paul Calter: and I think that's important..
Carol Watts: And that's the kind of information that architectural historians can find in some cases. There may be correspondence, there may be instruments that have survived
Paul Calter: But when we go now to measure an existing building that may have been put up with crude tools, I don't think you are suggesting that we should use similarly crude tools to measure what's there now.
Leonard Eaton: No, but I am talking about Wright's windows tomorrow and I had a substantial difference with a scholar at Wisconsin who came up with some results for his measurements of a window that I think were a great deal finer than any that were available with the instruments that would have been used in Wright's office in 1904.
Paul Calter: Hmm-mm. I think he's padding the data.
Leonard Eaton: No, I don't want anybody to go back to using the tools of 1600 or 1700 or 1800, but I want us to think back a little as to what was available when things were laid out, designed and set up for us to measure centuries later.
Mark Reynolds: What a wonderful thing it would be for us to find Michelangelo's set squares. You know we could get up to the door and could just actually take his tools and use them to see how they work in that doorway and forget all the modern stuff that exists.
Kim Williams: I think I could use Michelangelo's tools and I still wouldn't come up with what Michelangelo did.
Carol Watts: Because he didn't build it himself, right?
Paul Calter: The problem is that the superintendent of these places won't let you do that; they won't let you go up the walls and measures with tapes or anything, or go anywhere near these things, so you need an indirect method like a theodolite or a transit just because you can't get permission to go and measure directly.
Leonard Eaton: That's one reason I stick to windows.
PART III: HOW DO WE DIFFER IN OUR APPROACHES TO THE WHOLE RELATIONSHIPS OF MATHEMATICS AND ARCHITECTURE BASED ON OUR DISCIPLINES?
Copyright ©2000 Kim Williams