To order from Amazon.com, click
here.Reviewed by
The hallmarks of Leyton's generative theory are transfer and
recoverability. The ability to " If all of this sounds very mathematical, it most certainly is! Before we go into too many mathematical details, however, let's consider how Leyton's theory relates to art and architecture. This allows us to cite and discuss some additional tantalizing quotes -- Leyton's captivating text is full of them. "[T]he design process is one of successive symmetry-breaking" (p. 44); actually, we later learn that the more informative way to view this is as asymmetry-building.[2] Think of an architect designing a floor plan. He or she might start with a rectangle, which itself is a square having undergone a breaking of the 90-degree rotational symmetry. Each additional design element will break additional symmetries, and of course, the end result may not seem symmetric at all. However, Leyton's constructive, generative method for recording the design process will capture each symmetry-breaking (or better yet, asymmetry-building), so that the whole design process can be recovered. " Leyton explains the aesthetic structure of science by his
basic statement that "there is an Chapter 15 is entitled "A Mathematical Theory of Architecture" and begins with the following text (p. 366):
While Leyton's theory may not inform the design process for most architects, it does have some important ramifications, not only for designers who appreciate being in tune with the mathematical underpinnings of their profession, but also for architectural theorists and specialists in virtual architecture.[4] The ability to encapsulate a design in the programmable format made possible by Leyton's theory could open up new approaches and possibilities. While defining a single group to specify a whole architectural work may seem so complete as to be unwieldy, software could be developed to facilitate both the creation and the subsequent utilization of the encoded design process. Leyton's theory is quite compelling as it applies to the mathematical specification of the primitives. Chapter 10 focuses on the surface primitives, e.g., plane, cube, sphere, cylinder, and torus. The representation of each, as specified by Leyton's theory, is intuitively clear, appropriately clean, and completely convincing. In Chapter 16 Leyton defines the solid primitives, i.e., the same sorts of objects as the surface primitives but viewed not just as the outer shells. The specifications of the solid primitives are more complicated than those of their surface counterparts, requiring some additional machinery that Leyton defines as part of his theory. He points out, in fact, that the complexity of the solid symmetry groups "give crucial information not just for the physical sciences" but also other applications such as mechanical CAD and assembly planning (p. 418). Let us consider the mathematics behind Leyton's theory. It
is based on a fairly specialized operation from group theory,
namely the h) where
each g_{i} is from G and h
is from H, and h has a certain action on all of
the g_{i}.[6] The details of this action are suppressed
for simplicity, but in some sense the group H controls
the n copies of the group G and transfers information
between those copies; recall that transfer of information is
a key aspect of Leyton's generative theory of shape.Realizing that mathematics, like any language, is designed to convey information, one might imagine that the wreath product allows for considerably more information storage (and therefore recoverability of information, the other key aspect of Leyton's theory) than the direct product, simply due to the considerably larger size of the wreath product. More information storage in turn allows for the generative specification of more complex shapes. Of course, this also means that Leyton's theory takes considerable mathematical savvy to understand.[7] An example is in order. One primitive that Leyton uses instructively
at various places in the book is the cylinder. He explains (p.
13) that the standard formulation of a cylinder is the direct
product of a circle and the real number line, the circle being
"swept" under the action of the line to create a cylinder.[8] In Leyton's
theory the direct product is replaced by the wreath product.
The result is that at each point on the sweeping line, the corresponding
circle can be rotated independently from all of the other circles
making up the cylinder; that is, by design, the group representation
dictated by the theory has As for physics, Leyton informs us (p. 467) that he plans on a second volume that will focus on the conservation laws of physics. Chapter 20 is an intriguing preview, and it certainly whets the reader's appetite. The standard approach towards developing conservation laws in quantum mechanics is to determine whether appropriate differential operators commute, and Leyton casts this into his theory by providing a correspondence between commutators and wreath products. It should be noted that this is one aspect of a very ambitious goal: "One of the purposes of our generative theory of shape is to replace the different sets of laws in the different sciences by a single set of scientific laws that are universal across the sciences" (p. 52). Volume II should go a long way towards determining whether or not Leyton will ultimately be successful. As one can imagine, while creating a theory capable of being
used to define complex shapes is difficult enough, setting out
such a theory in an understandable form is quite a task in and
of itself. Leyton has done a fine job in this regard, incorporating
many examples and figures throughout the text, dissecting large
theoretical constructs into smaller, more understandable pieces,
and restating aspects of the theory in various informative ways.
Still, the book is not easy reading, and to be fair, it is certainly
beneficial to have some exposure to group theory, which is usually
not seen until advanced undergraduate courses for students majoring
in mathematics. Moreover, examples from the other fields cited
above may also be quite challenging for the non-specialist. Leyton
is clearly a polymath, and the more the reader knows about these
fairly diversified fields, the easier it will be to understand
return to text[2]
From p. 242: "In asymmetry-building, one does not loose
[sic] the initial symmetry ground-state, i.e., there is no actual
symmetry-breaking." [3]
This principle is restated on p. 75 with the addition of " [4]
One example of virtual architecture is virtual reality, where
users immerse themselves in simulated environments by wearing
specialized sensory equipment that interfaces with computer systems.
Even if this is not yet -- or never becomes -- mainstream, however,
it should be realized that more and more people are spending
more and more time "living" in virtual architecture
simply via the popularity of first-person video games and the
like. (I would like to credit Jean Brangé, whom I met
during preparations for the Nexus
2002 Round Table Discussion, for conveying these ideas to
me.) [5]
For those interested in a succinct general discussion of the
wreath product, try the definition given under PlanetMath
at http://planetmath.org/encyclopedia/WreathProduct2.html. [6]
To be precise, the number [7]
Leyton provides an appendix reviewing normal subgroups and semidirect
products, which can be used as a litmus test for further reading,
depending on whether or not the reader has the mathematical background
to understand the appendix! [8] Actually, the direct product would be taken of the
[9]
At least for the foreseeable future, in fact, it seems that the
theory is most applicable to areas such as CAD, robotics, mechanical
design and manufacture, and human perception. [10]
Michael Leyton
is a very friendly and approachable individual who would welcome
contacts from serious researchers. The Mathematical Intelligencer and the Nexus
Network Journal as well as a book (with Kim Williams) entitled
On Ratio
and Proportion (a translation and commentary of Silvio
Belli, Della proportione et proportionalità), and
taught a course at SBC entitled Architecture and Mathematics.
Steve's overall aim is to explore and extol the mathematics of
beauty and the beauty of mathematics.
top of
pageCopyright ©2004 Kim Williams Books |
NNJ HomepageSpring
2004 indexAbout the
ReviewerOrder
Nexus books!Research
ArticlesThe
Geometer's AngleDidacticsBook
ReviewsConference and Exhibit ReportsReaders'
QueriesThe Virtual LibrarySubmission GuidelinesTop
of Page |