Carol Watts: Let me, at this point, ask a general question, which is what I think we are getting at here: How do we differ in our approaches to the whole relationship of mathematics and architecture based on our disciplines? You know, we've already talked a bit about just how our bias of our own culture obviously has an impact on what we're looking for, how we see, what kind of methodologies we use, but also it's clear that the more we discuss, the more even definitions of terms, such as "is geometry a subset of mathematics?" or "what is geometry?" or "what is mathematics?"; we could debate these endlessly. But maybe it would be useful to give a sense of some of the basic differences between our disciplines in terms of how we approach similar kinds of analyses, because it seems to me that that's something that can stand in the way of communication; not that there's one approach that's right, but that there are clearly values to differing sorts of people from different backgrounds asking similar questions or different questions about the same thing. But we need to be able to communicate with each other as best as possible and that's where there could be a possible danger.

Judy Moran: And we need to be able to build on what others are doing...

Carol Watts: Right, that's true too, we need to be able to understand that no one discipline can necessarily study a particular question completely, just because of inherent biases. Now, interdisciplinary teams, a mathematician and an architect together, can perhaps overcome some of that. Maybe that's one of the things that can come out of Nexus, is we know people who have similar interest but perhaps different backgrounds that might be able to do some joint work in the future. So I guess there are a couple of questions here. One is "How do you see your discipline as different from the other disciplines?". The other would be, "Do you think there are ways in which you could take advantage of approaches that other disciplines have to work in a more collaborative fashion?"

Bill Sapp: I think that, getting back to one of the things that you said earlier, from my perspective [as an archaeologist], I'm searching for order in patterns, and whether mathematics was discovered or invented or whatever doesn't really matter, that's the language that I need to use in order to describe things. It doesn't matter where it comes from, that's sort of the common basis that we all have that we can communicate with. In dealing with non-Western cultures, and from what I've heard over the past two days, I think part of discovering order and patterns, and one of the difficult parts, is separating universals from cultural constructs. What things are universal?; what occur in Italian Renaissance architecture that occurs in Pre-Columbian architecture or Mayan architecture or Mesopotamian architecture; what are those particular things that had to do with their own belief system or ideology or cosmology or their own view of the universe. That to me is the most difficult part. I think collaboratively that, at least from my perspective, I know very little about mathematical analysis. I know when I see something that there are patterns, but how to pull out of those patterns real mathematical meaning so that I can now communicate those patterns to somebody else is something that I can't do without the help of other disciplines. And I would think and hope that others might feel the same way.

Kim Williams: Does anybody want to go to Peru?

ALL: Yes!

Mark Reynolds: I mentioned earlier about squaring the circle, and it could be seen also as the Yin Yang, from the great goddess Yin Yang who divided the universe into male and female. There is a positive and negative charge to almost everything in this universe and without those two different opposites, you and I wouldn't be here today. And for me, when I see a six-pointed star, or I see the Yin Yang symbol, or I see an attempt at squaring the circle, it makes me feel that working with geometry is almost a responsibility that I have to myself, to see just how much geometry exists. And it's extraordinary how much geometry exists. I approach my geometry from that point of view, the unity-duality principle. And I do admit to looking at the symbolism and looking at the religion and looking at the philosophy and looking at those touchy-feeling things that some mathematicians would yell at me about, but I do anyway. I feel at this stage of the game that the geometry works with me more than the other way around. The more I work with geometry, the easier it gets for me to just start drawing and just let geometry pieces come out of my hands, so that's why I do it.

Paul Calter: Well, as you know, I'm part of a team. Maybe I'm Yin and Kim's Yang, and Susi [Knight] is the unifying principle. I think that's a very good working model, with each bringing their own insights and expertise to a fairly complicated problem and working with their own strengths. And my role is very limited, obtaining accurate dimensions that Kim can then use to make drawings and look for relationships.

Kim Williams: When I'm not giving conferences...

Paul Calter: When you're not giving conferences.

Rachel Fletcher: It's not how I work with others; I guess it's more how I work with myself. It's a combination of work and play. I try to be as rigorous and discerning as I possibly can, and at the same time to allow room to be imaginative and playful so that maybe something that might never have been thought can come into play. I try very hard to balance rigor and imagination and try to conserve both.

Carol Martin Watts: Do we have any related comments from the audience? I think we'll be finishing up shortly, so some last words.

Judy Moran: About what Bill had said. Many mathematicians would define mathematics as a search for pattern.

Paul Calter: Yes, yes.

Judy Moran: So, there's not an unbreachable gulf, okay?

Carol Martin Watts: So perhaps there are more similarities in our disciplinary approaches than we are aware of, maybe the language we use or the preconceptions we have of what other disciplines are like may be getting in the way of the commonalities.

Paul Calter: This is a good point to get in a plug for Nexus. That's why conferences of this sort are so important for bringing together people from different backgrounds.

Mark Reynolds: I think your point about patterns is well taken. It's one of the few things that we find all over the world from the earliest of time. The human brain itself is a pattern. And from that point forward, we are all looking for patterns that exist. I totally agree with you.

Carol Martin Watts: Okay, your last words, David.

David Speiser: As the moderator said, the last word. Now, when I think over these things, I like to think of concrete examples. Now, if we think of the architects here, we must of course think of the very greatest of them, of which, in this country, there has always been a great lot. Now what do we admire in them? We admire of course their craft, we admire their views, but most of all I think, and for me that would be the last word, we admire in them their creative imagination. Now, what how does mathematics come in? If you would ask me "what is mathematics?", the old, of course, the elementary rules and the Pythagorean theorem etc.. But really, why does it come in and can claim to have contact with the arts, which are creative and imaginative. Mathematics, ultimately, I would say, its greatness and significance is that it is imagination made methodical and systematic. That's what mathematics is, and this is why the two must come together, and why we are grateful about the Nexus. Of course, the word imagination is easily said, but a great imagination, a creative one, is very rare. But, the products, in this country, can be admired of all the corners of all the streets.

Paul Rosin: I want to say one word since I didn't get around to saying anything before when you were talking about verification of the data, but in many scientific disciplines what's also very important is verification of the solution you are proposing. So we have for instance that a lot of the questions start off with asking, we have several possible models. We have Dr. Speiser suggest that maybe we could have a model for that certain curve as a certain tension in some steel and maybe we could have some other models. Well then you have to do is verify which of these models fits best and maybe that's the stage in which a mathematical statistician comes in. But I don't think in any of the talks we've heard in the last couple of days we've heard very much verification of the solutions suggested. If we have a number of 1.6 that appears, is that the Golden Ratio, or is it 1-2/3, or neither? I think we need to have some more rigorous analysis of that.

Steve Wassell: Like Judy, I was glad to hear Bill say that he is very interested in finding order in pattern. I agree that that's what mathematics is, on some level at least, and I agree with David that mathematics is also a very creative process. I also think though that we are a very selected group here, and I think that the people in this room really probably do have a lot more similarities than differences. But we have to realize that, if you want to say, the mainstream architects and mainstream mathematicians might not feel the same ways that we do. You have to keep in mind that we have a wider audience out there that is very scrutinizing and we have to make sure that we present ourselves in the best way possible to that wider audience as well.

Mark Reynolds: Is it possible to convince them ever, do you think?

Steve Wassell: Well, you won't be able to convince everyone...

Mark Reynolds: Well, the majority...

Bill Sapp: There's certainly a great debate in anthropology about patterns and whether they should even be looked for. There are certain people, like myself, that think there are patterns of social evolution, and we're looking for those kinds of patterns. What make early empires in China similar to early Mesopotamian and Mayan and Inka empires?; what makes this cultural evolutionary process the same the world over? And there's another group of people who says, in doing so, what we do is take humanity out of the process and we make it as if the process predetermined what people do, and we dehumanize, in a sense, the cultures that we're taking a look at. And these people are saying, therefore, that you have to look at the individual and it's individual decisions that are being made that drive cultural evolution. So even within anthropology and archaeology there's a great debate whether it's worthwhile to look for patterns, let alone whether there are patterns at all. So I in particular believe there are patterns, that there are patterns in social evolution just as there are patterns in architecture, and when I was referring to the math, I felt that I was really speaking in terms of the architecture that I see in non-Western cultures, that there really are patterns. But I believe there are patterns in virtually everything because I think there is some sort of psychic unity in mankind that really says that, worldwide, people react to certain situations in certain ways because, frankly, there are a limited number of responses that are available to humans in most situations and that therefore we tend to make certain decisions worldwide, when confronted with the same sorts of problems.

Carol Watts: Does anyone think that this kind of round table discussion should be continued at the next Nexus, and if so, what would be a good focus? I knew that this would only be a starting point. There are way too many issues for us to cover all of them. Perhaps we could have had a more descriptive title in terms of just what was to be covered; but that was decided once we knew who the group was. So any suggestions? First of all, should we do it again?

ALL: Yes.

Carol Watts: Okay. And what would be suggestions for topics to focus it down a little bit so we would have a starting point for a discussion?

Rachel Fletcher: I love Kim's idea of taking one building and different scholars...

Carol Watts: ...and different approaches to the same building.

Kim Williams: That was felt by some to be too specific. People felt that if we focused on one building, it wouldn't generate enough interest. But I think we could use it maybe as a springboard...

Carol Watts: I would agree that as an introductory springboard ... Part of our problem is that we are talking in very general terms, and our own research is so varied in terms of the culture and what it is we are looking at, but if we could control one variable maybe we could talk more about different approaches.

Rachel Fletcher: We just need a moderator like Carol who can extract the principles so that it's not just about the building.

Carol Watts: The idea would be a beginning point for discussion, not just a series of talks on the same building.

Mark Reynolds: Carol, suppose we took a number of buildings, wrote them down on a sheet of paper, folded them up, put them in a bag, so that it was totally random and each one of us had to do whatever we got. I mean, that way there would be no prejudice.

Kim Williams: Yeah, but what would they have to do?

Carol Watts: You'd have to do the analysis.

Mark Reynolds: You'd go and do your analysis...

Kim Williams:...and then ten years later...no, what you need is people who have already approached a building from some point of view.

Carol Watts: So we'd have to pick a building that a lot of people have analyzed.

Kim Williams: You'd have to pick something like the Pantheon that everyone is familiar with...

Carol Watts: ...and that a number of people have studied ...

Kim Williams: Like Rachel said, you'd have to have someone who was able to extract from individual comments the thematic that would structure the discussion.

Paul Calter: You said you were going to have a ‘call for papers' for the next conference. Maybe you could see what comes in and find some that group around some theme that could be the subject of a round table discussion.

Kim Williams: You could maybe do it around a period, like the Renaissance, and have people who have looked at different buildings but they're still looking within the same social context.

Carol Watts: I think the suggestion of if you had papers with some similar themes, maybe there would be several on the same building, or by the same architect.

Paul Calter: It may just happen that a logical grouping would appear in the papers that were presented.

Carol Watts: And then that could be the basis for a discussion that would be specifically comparing the methods used by those people.

Paul Calter: And you'd be working with what you had already had at hand, rather than making something up.

Rachel Fletcher: I have just one comment. You were saying something about picking just the best papers for the next conference. I would rather see papers that have a relationship to one another.

Kim Williams: Well, I agree with you; that's what I've always done. But if you issued a call for papers, and you mentioned specific topics, you be likely to get more than one paper about a certain subject and then you could pick the strongest. In every conference there are always a couple of weak papers also, so hopefully what you'd be doing with a ‘call for papers' is generally raising the level.

Kathy Reynolds: And was your focus going to be on Asia?

Kim Williams: No, I mentioned that strictly as one of the areas that hasn't been gone into yet that I think needs to be. I don't want Nexus to be a European, Renaissance, or 15th century conference, I really don't want that to happen. The other thing that has been suggested for Nexus, is that every conference have a theme, so we should have a conference just on Leon Battista Alberti, then we should have a conference just on Roman architecture, which is kind of interesting but exclusive, and we don't want to be exclusive; we want to be inclusive, at this point. When we reach the point where we've had 10 Nexus conferences and we've had a wide range of material, then at that point it might be time to focus, because we are focusing from many different view points, but not yet.

Mark Reynolds: So we just don't have anything from the Far East; we have the Middle East, and we have Europe...

Kim Williams: Well, we just barely have the Middle East. We don't have Russia, we don't have the Far East at all. Your contribution on Egypt in the Nexus Network Journal was the first, we don't have anything in the Nexus conferences about Egypt, of all things...

Rachel Fletcher: ...and vernacular...

Kim Williams: ...and vernacular. We don't have any native American architecture. Really what we haven't covered is vast. This is an on-going process. I really want to make a big archive, and I'd also like to create an environment where even if there's one poor soul in China who is making studies, I want him to have a place to publish.

Carol Watts: I think that the Nexus Network Journal with its presence on the web is going to be the most important for giving a more global participation.

Kim Williams: It helps a lot. The response has really been international and not just from a native English speaking group.

Carol Watts: Okay, I think it's exactly 6:00.

ALL: Applause!

Copyright ©2000 Kim Williams