Michael LeytonDepartment of Psychology Rutgers University New Brunswick NJ 08904 USA This
is the first part of a two-part series. Click on the link to
go toArchitecture and Symmetry 2: Why Symmetry/Asymmetry?
The purpose of this article is to introduce the reader to the idea that symmetries in architecture are nested. Let us consider the following example, a colonnade in a cathedral (Figure 1). What we will do is build up this structure as a nested hierarchy
of symmetries. We will proceed as follows: We start at the lowest
level of the organization, which is actually a The circle represents the cross-section of the cylinder. Now
take the generated circle, and apply to it translations in the
vertical direction, which we will denote by Translations along a line in the
horizontal plane as shown on the right in the above figure. This
generates a row of columns, as shown in that figure. Finally,
we take the row of columns and apply to it _{H}Reflections
about a mirror-plane which is parallel to the column-row, and
we obtain the reflectional pair of column-rows shown in the top
figure on this page, i.e., the entire colonnade.The sequence of operations that were used can be represented as follows: Point.Rotations.Translations_{V}.Translations_{H}.Reflectionswhere, reading from left to right, we started with a point, then applied rotations to get a column cross-section, then applied vertical translations to get a column, then applied horizontal translations to get a row of columns, and finally applied reflections to get the reflectional pair of column-rows. The important thing to notice is that these operations were nested. By this I mean the following: Each set of operations generates a level in the architecture. The levels are: - Level 1: A point
- Level 2: A circular cross-section
- Level 3: A column
- Level 4: A row of columns.
- Level 5: A pair of column-rows.
Furthermore, each level of transformations acted on the previous
level as a whole. This is easy to see as follows: The point was
acted on as a whole by Translations
to produce a row of columns; and finally the row of columns was
acted on as a whole by _{H}Reflections to produce the reflectional
pair of column-rows.Each level of transformations defines a symmetry in the architectural structure; i.e., point symmetry, rotational symmetry, translational symmetry, etc. Each level is, in fact, what is called a symmetry group in mathematics. In my research papers, I call this type of structure 'a hierarchy of nested control'. What I have shown is that the human perceptual system is organized as a hierarchy of nested control. In fact, the first research article I ever published was called "Perceptual organization as nested control." The perceptual system takes its nested structure and imposes it on the environment. What I argue is that architects exploit this psychological fact in the structure of their buildings. But the same is true of painters, and of composers. Now you might object by saying that the architectural example given above (the colonnade) is a highly regular structure, and therefore amenable to the type of analysis I have given. In contrast, you might ask, how can one describe the new types of architecture that are currently emerging, which involve irregular-shaped blocks (e.g., I.M. Pei's extension to the National Museum in Washington), and also free-form shapes (e.g., Frank Gehry's Guggenheim Museum at Bilbao)? In fact, it was exactly to analyze irregularity and free-form structures that I developed the concept of nested control. What I have shown is that, given an asymmetric design, the human perceptual system embeds this in a higher dimensional space in which it is described as a nested hierarchy of symmetries. The following is an illustration. In a sequence of psychological experiments, I conducted in the 1980's, I showed that, if people are presented with a rotated parallelogram (far left in Figure 3), they then reference it to a non-rotated one, which they then reference to a rectangle, which they then reference to a square. Thus: This means that they are actually describing a rotated parallelogram as generated in the following way. One starts with a square (far right). One applies to it a stretch to get a rectangle; then one applies to it a shear to get a parallelogram; and finally one applies to it a rotation to get a rotated parallelogram. This sequence is given thus: Stretches.Shears.RotationsEach level is, once again, an example of what mathematicians call a symmetry group. Each is in fact a symmetry of some higher-order space of shapes. Now we have said that this sequence of operations is applied
to a square. However, the square itself is built up as a nested
hierarchy of operations. We start with an individual Point.Translations.90°RotationsNow, we said that the rotated parallelogram is obtained from
the square by then applying Point.Translations.90oRotations.Stretches.Shears.RotationsThe first three operations produce the square successively from a point, and then the next three operations produces the rotated parallelogram successively from a square. It turns out that this 6-level structure is a hierarchy of nested symmetries. Each level is a symmetry of some space, and the spaces are nested in each other. This 6-level nested hierarchy of control is a very powerful structure in the human perceptual system. I have shown, for example, that it structures not only geometrical figures, such as those given above, but also motion phenomena. Now, we can go on adding higher levels of operations which will make the shape more and more asymmetric. As an example, I invented and published a free-form grammar which alters the curvature of the shape, so that it takes on more and more of an organic growth appearance. The grammar exactly analyzes, for example, Gehry's Guggenheim museum. Extensive research on this topic has been published in In contrast, all the technical mathematics will appear in
VisMath,
vol. 1 no. 3 (1999) and is reproduced by permission, courtesy
of Slavik Jablan.
Symmetry, Causality, Mind.
Cambridge MA: MIT Press. To
order this book from Amazon.com, click
here.Leyton, Michael. 2001.
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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