Michael LeytonDepartment of Psychology Rutgers University New Brunswick NJ 08904 USA This
is the second part of a two-part series. Click on the link to
go toArchitecture and Symmetry 1: Nested Symmetries
In this second tutorial we are
going to look at the What we wish to consider, in this paper, is the following issue: Why was classical architecture dominated by symmetry; i.e., what purpose did symmetry serve in classical architecture? Correspondingly, why is modern architecture dominated by asymmetry; i.e., what purpose does asymmetry serve in modern architecture? The answer to this question comes
from my previous book
If we define "memory" to be information about the
past, we observe that there are many forms that memory can take.
For example, a Memory is always in the form of asymmetry.Symmetry is always the absence of memory.I can give you a simple illustration of this as follows: Imagine a tank of gas on the table. Imagine that the gas is at equilibrium, at TIME 1. The gas is therefore uniform throughout the tank, in particular, symmetric—left to right in the tank. Now use some means to attract the gas into the left half of the tank at TIME 2. The gas is now asymmetric. Someone who has not previously been in the room now enters and sees the gas. The person will immediately conclude that the gas underwent a movement to the left. This means that the asymmetric state is memory of the movement. Now let the gas settle back to equilibrium, that is symmetry at TIME 3, that is, uniformity throughout the tank. Suppose another person enters now, someone who has not been in the room before. This new person would not be able to deduce that the gas had previously moved to the left and returned. The reason is that the symmetry has wiped out the memory of the previous events. The conclusion is that from symmetry, you can conclude only that the past was the same. We can summarize the rules used here, in two principles:
In mathematics, symmetry means indistinguishability under transformations. Thus, for example, a face is reflectionally symmetric because it is indistinguishable from its reflected version, and a circle is rotationally symmetric because it is indistinguishable from any of its rotated versions. Now, what we will see over and over again in this paper is that the way to use the above two rules is as follows: You first partition the present situation into its asymmetries and symmetries. You then use the first rule on the asymmetries and the second rule on the symmetries. That is, the first rule says that the asymmetries go to symmetries, backward in time; and the second rule says that the symmetries are preserved, backward in time. Let us now illustrate this: In a converging series of psychological
experiments, I showed that if subjects are presented with the
first stimulus shown in One can interpret this data by saying that, given the initial object, subjects are inferring the process-history that produced it. That is, the presented object was produced by starting with a square, stretching it, then shearing it, and then rotating it. We shall now see that what the subjects are doing is using the Asymmetry Principle and Symmetry Principle. To see this, we must, as I said, first partition the presented shape—the rotated parallelogram—into its asymmetries and its symmetries. Consider first the asymmetries. There are in fact three of them: (1) the distinguishability between the orientation of the shape and the orientation of the environment; (2) the distinguishability between adjacent angles; (3) the distinguishability between adjacent sides. As we can see from Figure 1, what subjects are doing is removing these three distinguishabilities, backwards in time as prescribed by the Asymmetry Principle. That is, successively, the orientation of the shape becomes the same as that of the environment, the sizes of the adjacent angles becomes the same, and the sizes of the adjacent sides become the same. To repeat: Asymmetries become symmetries backward in time—as predicted by the Asymmetry Principle. Now let us use the Symmetry Principle. It says that the symmetries must be preserved, backward in time. Well, the rotated parallelogram has two symmetries: (1) opposite angles are indistinguishable in size; and (2) opposite sides are indistinguishable in length. Observe that both of these symmetries are preserved backward in time—thus corroborating the Symmetry Principle. Now, those of you who have seen my book might say to me, "There seem to be 100s of rules in your book. How can you say that there are actually only two rules?" Well, the reason is that, as I said earlier, the term symmetry means indistinguishability under transformations: reflectional symmetry is indistinguishability under reflectional transformations; rotational symmetry is indistinguishability under rotational transformations, and so on. Thus you obtain the different kinds of symmetry by instantiating the different kinds of transformations in the definition of symmetry. The different rules of the book are obtained by instantiating different transformations within the Asymmetry Principle and Symmetry Principle. Notice that it is by doing this instantiation process that you obtain the different sources of memory that can exist in an organization. In the paper so far, I have given you only an intuitive sense of the instantiation process. What I want to do now is show you how it works, in depth. We are going to examine the extraction of memory from a particular asymmetry called curvature extrema. We will see later that curvature extrema are violations of rotational symmetry in the outline of a shape. So lets look at curvature extrema. What is a curvature extremum? Well, first we note that curvature, for curves in the 2D plane, is simply the amount of bend. The straight line has no bend, and therefore has no curvature. As you successively increase bend, you are increasing curvature. Finally, observe that on a shape such as a finger, there is a point that has more bend than the other points on the line (the finger tip). This is a curvature extremum. We will start be elaborating two successive rules by which the curvature extrema can be used to infer processes that have acted upon a shape. The input to the rules will be smooth outlines of shapes such as embryos, tumors, clouds, etc. So the rules will infer the history of such objects — that is, convert them into memory. The inference, from curvature extrema to historical processes will be seen as requiring two stages: (1) Curvature extrema Symmetry axes, and (2) Symmetry axes Processes. We first consider stage 1.
As
we have seen, an essential aspect of the inference of history
is symmetry. For a simple shape, a symmetry axis is usually defined
to be a straight line along which a mirror will reflect one half
of the figure onto the other. However, observe that, in complex
natural objects, such as the branch of a tree, a straight axis
might not exist. Nevertheless, one might still wish to regard
the object, or part of it, as symmetrical about some curved
axis. For example, a branch of a tree tends not to have a
straight reflectional axis. Nevertheless, one understands the
branch to have an axial core that runs along its center.How can such a Then move the circle continuously along the two curves, c
To illustrate: Consider the shape shown in
The principle was advanced and extensively corroborated in Leyton [1984, 1985, 1986a, 1986b, 1986c, 1987a, 1987b, 1987c], in several areas of perception including motion perception as well as shape perception. The argument used in Leyton [1984, 1986b] to justify the principle involves the following two steps: (1) A process that acts along a symmetry axis tends to preserve the symmetry; i.e. to be structure-preserving; (2) Structure-preserving processes are perceived as the most likely processes to occur or to have occurred.
The first rule is the Symmetry-Curvature Duality Theorem,
which states that to each curvature extremum, there corresponds
a unique symmetry axis terminating at that extremum. The second
rule is the Interaction Principle, which states that each of
the axes is a direction along which a process has acted. The
implication is that the boundary was deformed along the axes;
i.e., each protrusion was the result of Under this analysis, processes are understood as creating the curvature extrema; i.e., the processes introduce protrusions, indentations, etc. into the shape boundary. This means that, if one were to go backwards in time, undoing all the inferred processes, one would eventually remove all the extrema. Observe that there is only one closed curve without extrema: the circle. Thus the implication is that the ultimate starting shape must have been a circle, and this was deformed under various processes each of which produced an extremum.
Let me make the following comments: (1) The reader will notice that, on each shape, each extremum is marked by one of four symbols: M+, M-, m+, m-. This is because there are mathematically four kinds of curvature extrema: Positive Maxima (M+); Negative Maxima (M-), Positive minima (m+); and Negative minima (m-). (2) When one surveys the shapes, one finds that there is the following simple rule that relates the type of extremum to an English word for a process:
Remember as you look at the figures that the process arrows are inferred by the two simple rules we gave above: The Symmetry-Curuvature Duality Theorem, and the Interaction Principle. The 21 shapes are shown in the following figures: Shapes with
4 extrema: ( Let us now go more deeply into the structure of these histories.
Observe now that, since the later shape is assumed to emerge
from the earlier shape, one will wish to explain it, as much
as possible, as the outcome of what can be seen in the earlier
shape. In other words, one will wish to explain the later shape,
as much as possible, as the As a simple first cut, let us divide all extrapolations of processes into two types: (1) Continuations (2) Bifurcations (i.e., branchings). What we will do now is elaborate the only forms that these two alternatives can take. We first look at continuations and then at bifurcations.
Exactly the same argument applies to any of the Now recall that there are four types of extrema, CONTINUATION AT m+A Observe what happens to the extrema involved. Before continuation,
in the left-hand shape, the relevant extremum is Cm+ : m+ ® 0
m- 0The above string of symbols says "Continuation at Observe that, although this operation is, formally, a rewrite
rule on discrete strings of extrema, the rule actually has a
highly intuitive meaning. Using our Semantic Interpretation Rule,
we see that it means: CONTINUATION AT M-As noted earlier, we need to consider only one other type
of continuation, that at a Now, recall that our interest is to see what happens when
one continues a process at a Now let us observe what happens to the extrema involved. Before
continuation, in the left-hand shape, the relevant extremum is
Therefore the transition between the left-hand shape and the
right-hand shape can be structurally specified by simply saying
that the . Thus we have:CM- : M- ® 0
M+ 0The above string of symbols says "Continuation at Observe once again, that, although this operation is a formal
rewrite rule on discrete strings of extrema, the rule actually
has a highly intuitive meaning. Using our Semantic Interpretation
Rule, we see that it means: i
BIFURCATION AT M+Consider the Observe what happens to the extrema involved. Before splitting,
one has the BM+ : M+ ® M+
m+ M+The above string of symbols says "Bifurcation at Observe that, although this transition has just been expressed
as a formal re-write rule on discrete strings of extrema, the
transition has, in fact, the following highly intuitive meaning:
BIFURCATION AT m-Consider now the Observe, again, what happens to the extrema involved. Before
splitting, one has the single extremum Bm- : m- ®
m- M- m-The above string of symbols says "Bifurcation at Observe that, although the transition has just been expressed
as a formal re-write rule using discrete strings of extrema,
the transition has, in fact, a highly intuitive meaning: BIFURCATION AT m+ and M-Bifurcations at these two extrema turn out to be very easy to understand: They are simply the introduction of a protrusion and the introduction of an indentation, respectively. It will therefore not be necessary to diagram them.
Recall however that, although these operations are expressed as formal re-write rules on discrete strings of extrema, they describe six intuitively compelling situations, as follows:
These situations are illustrated in the previous figures.
Let us describe what his happening in
I will refer to these rules as the extrema-based rules. They pick certain features—the curvature extrema—and extract causal history from those features; that is, they construct memory from those features. What I want to do now is show you that these eight rules are an instantiation of the theory of process-inference, or memory construction, that was given in the first part of the paper. Before I do this, it is necessary to understand first that curvature variation is a form of rotational asymmetry. To understand this, imagine that you are driving a car on a racing track which is in the shape of one of the curvilinear shapes I showed you. On such a track, there is no alternative but to keep on adjusting the steering wheel as you are driving. This is because curvature is changing at all points. In contrast, if you are driving on a track that is perfectly circular, you would have to set the wheel only once, at the beginning, and never have to adjust it again. This is because the curvature is the same at all points on a circle. Another way of saying this is that a circle is rotationally symmetric— that is, in going around the circle, each section is indistinguishable from any other. Therefore, we can now see that what curvature extrema do is introduce rotational asymmetry in the shape. So much for the asymmetry in a curvilinear shape. The Asymmetry Principle will be applied to this asymmetry and remove it backward in time. What about the symmetry in such a shape. The Symmetry Principle will be applied to that and preserve it backward in time. Observe that despite all the asymmetry in such a shape, there is a form of symmetry. It is reflectional symmetry. It is exactly captured by the symmetry axes. Now, having understood this, let us now go through our eight rules, and see how they are an instantiation of our scheme for process-inference, or memory-construction. What was our scheme? It was in fact illustrated several times in the paper already: You first partition the situation into its asymmetry and symmetry components, and then you apply the Asymmetry Principle to the asymmetry component, and the Symmetry Principle to the symmetry component. Let us now do through the eight rules and show how each is designed to carry out a role in this scheme: First we have the Symmetry-Curvature Duality Theorem. This theorem corresponds each curvature extremum to a symmetry axis. We can now understand that what the theorem is now doing is, in fact, describing the exact relationship between the asymmetry component and the symmetry component. It says that, for each unit of asymmetry, that is, for each curvature extremum, you will find a unit of symmetry, a symmetry axis. In other words, the role of the theorem is to carry out the initial partitioning stage in the inference process. Now let us look at the next rule. It is the Interaction Principle, which says that processes have to have gone along the symmetry axes. This has the effect of preserving the symmetry axes over time. In fact, brief consideration reveals that this principle is merely an example of the Symmetry Principle— the injunction to preserve symmetries backwards in time. Now lets move onto the six rules of the Free-Form Grammar. What do these six rules do? The answer is that they describe the six only possible ways in which curvature variation can increase in a shape. In other words, they are the six only possible instantiations of the Asymmetry Principle when the asymmetry is curvature variation. So now we can understand exactly how this entire system of eight rules instantiates our scheme for the extraction of memory: The Symmetry-Curvature Duality Theorem specifies the partitioning of the shape into its asymmetry and symmetry components; the Interaction Principle is an instantiation of the Symmetry Principle; and the Free-Form Grammar is an instantiation of the Asymmetry Principle. What I have done in this part of the paper is shown just one
of the rule-systems that I developed in my book
Classical architecture
aimed at removing memory.Contemporary architecture
aims at creating memory.In addition to these general principles, we have the particular rules of the Free-Form Grammar. This will allow us, in future articles, to do careful analyses of Frank Gehry's Guggenheim Museum at Bilbao as well as the free-form buildings of Greg Lynn. Furthemore, our other memory rule-systems, derived from the Asymmetry Principle and Symmetry Principle, will enable us to analyze the non-free-form buildings of the Deconstructivist Architects, and Lebbeus Woods.
Leyton, M. 1986a. Principles of information
structure common to six levels of the human cognitive system.
Leyton, M. 1986b. A theory of information
structure I: General principles. Leyton, M. 1986c. A theory of information
structure II: A theory of perceptual organization. Leyton, M. 1987a. Leyton, Michael.
1999. Leyton, Michael. 2001.
VisMath,
vol. 1 no. 4 (1999) and is reproduced by permission, courtesy of Slavik
Jablan.
Group Theory—Mathematics and the Liberal Arts Group Theory Discussion Forum Introduction to Group Theory The Paintings and Sculptures of Michael Leyton Wellblack Art: Michael Leyton
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