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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 Let us describe what his happening in the above figure: (1) The entire history is dependent on a single crucial process. It is the (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.
This concludes my exposition of the rules for obtaining history from curvature extrema. There are a total of eight rules. EXTREMA-BASED RULES
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.
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 .
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: 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 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 |