Why Symmetry? Why Asymmetry? by Michael Leyton, Page 5.

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An Illustration of Shape Evolution

The next figure shows an illustration of shape evolution. Each successive stage is given by an operation of the grammar. The sequence of operations is CM- followed by BM+ followed by Cm+ :

Let us describe what his happening in the above figure:

(1) The entire history is dependent on a single crucial process. It is the internal resistance represented by the bold upward arrow in the first shape (above).

(2) This process continues upward until it bursts out and creates the protrusion shown in the second shape (above).

(3) This process then bifurcates, creating the upper lobe shown in the third shape above. Involved in this stage is the introduction of a squashing process given by the bold downward arrow at the top of the third shape.

(4) Finally, the downward squashing process continues till it causes the new indentation in the top of the fourth shape above.

Back to Symmetry and Asymmetry

This concludes my exposition of the rules for obtaining history from curvature extrema. There are a total of eight rules.

EXTREMA-BASED RULES
1. Symmetry-Curvature Duality Theorem
2. Interaction Principle
3-8. The six rules of the Free-Form Grammar.

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, 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

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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 Symmetry, Causality, Mind. In fact, this system is given right at the beginning of the book, and after that there are another 600 pages of rule systems. Each of these systems is applied to a different type of structural feature in an organization to extract causal history from that feature.
Since I published them, my rule-systems have been applied by scientists in many disciplines to extract causal history, e.g., meteorology, radiology, chemical engineering, linguistics, etc.

Back to Architecture

Let us now return to the topic of symmetry and asymmetry in architecture. We can see, from the above discussion, that the Asymmetry Principle and Symmetry Principle lead to the following conclusions:

In addition to these general principles, we have the particular rules of the Free-Form Grammar. This will allow us, in later tutorials, 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.

To see more information about Michael Leyton's previous book, click:

Symmetry, Causality, Mind. MIT Press, paperback

Michael Leyton is president of the following two societies :

International Society for Mathematical Aesthetics

International Society for Group Theory in Cognitive Science

Home Page and email address: MLeyton@msn.com