Letter from the Guest Editor |
The golden section, commonly denoted by *phi*, or f, is almost certainly the most controversial
subject in the interdisciplinary field served by the *Nexus
Network Journal*. Its use as a design tool is not controversial.
Indeed, it is hard to argue with the considerable potential for
useful proportional relationships available to the designer,
based on the mathematical properties of the golden section, whereby
additive and multiplicative properties are so beautifully married
because of the fact that f^{2}
= f + 1 (the square of the Golden
Section is equal to the Golden Section plus Unity).
No, the golden section is controversial because of the large
number of questionable claims about applications of the golden
section to a wide range of objects, both natural and human-made.
In the last issue of the NNJ, we presented a controversy over
the conjectured use of the golden section in Palladio's Villa
Emo (see Palladio's Villa Emo: The Golden
Proportion Hypothesis Rebutted by **Lionel March**, and
Palladio's Villa Emo: The Golden Proportion
Hypothesis Defended by **Rachel Fletcher**). The current
issue is dedicated completely to the golden section, and we hope
to foster the kind of meaningful debate exemplified by these
two articles from the last issue.
Any scholarly discussion involving the golden section in architecture
should be grounded in an understanding of the evidence available
in architectural literature. To this end, **Marcus Frings**
offers The Golden Section in Architectural
Theory. Starting with the *Ten Books* of Vitruvius and
ending with Le Corbusier's *Modulor*, this thoroughly-researched
essay reviews the appearance of the golden section - and lack
thereof - in architectural writing. It provides a great service
to those yearning for an accurate historical summary, and the
reference list comprises a comprehensive compendium for further
study.
The substantial role of the golden section in the climactic
thirteenth book of Euclid's *Elements* is indisputable.
Indeed, the "division in extreme and mean ratio" (the
ancient Greek name for the golden section) is inherent in various
proportional relationships involving the five Platonic solids,
as explicitly shown by Euclid. **Christopher Glass** draws
from this Euclidean tradition for his design of The
Pythagopod. This project for a residence is reminiscent of
Buckminster Fuller's geodesic dome, but the clever interlocking
of the five Platonic solids necessitates proportions based on
irrational quantities important to classical mathematicians,
namely Ö2 and f.
I would like to acknowledge the generous assistance of Dr.-Ing.
Klaus Kremer with issues relating to the O2C-Player
that makes the interactive animated image possible on Christopher
Glass's page.
**John Sharp** provides the mathematics behind the facts
and fiction concerning Spirals
and the Golden Section. This accessible and well-illustrated
account describes the mathematics of Archimedean and logarithmic
spirals and reviews various approximate spiral constructions
based on geometric objects involving the golden section, including
the widely celebrated golden rectangle spiral, based on squares
adjoined to increasingly larger golden rectangles. The relationships
between the approximate spirals and their true logarithmic spiral
counterparts are detailed, and the commonly cited (but incorrect)
association of the golden rectangle spiral to the Nautilus shell
is thoroughly debunked.
The "metallic means" are the (positive) solutions
to the family of quadratic equations,
*x*^{2} - *nx* -1 = 0,
one mean for each positive integer *n*. The first metallic
mean ( *n* =1) is the golden mean (golden section), and
the next two are often called the silver mean and bronze mean,
respectively. Because of the simplicity of this family of quadratic
equations, the metallic means find various applications in science.
More True Applications of the Golden Number,
by **Dirk Huylebrouck** and **Patrick Labarque**, provides
some new applications of the golden section and its relatives,
in the diverse contexts of colour theory, bicycle gears, and
optimality problems involving the scaling of rectangles.
For the Geometer's Angle, **Marcus the Marinite** has drawn
from his considerable portfolio of constructions pertaining to
the golden section in order to offer R-Tiles,
a group of shapes, tiles, patterns, tessellations, and relationships
that make up a unique geometric system based on Öf.
Following the precedent he set in the last issue, Marcus is joined
by a collaborator who offers a complementary essay for the Geometer's
Angle. In The Indefinite Dyad and
the Golden Section: Uncovering Plato's Second Principle,
**Scott A. Olsen** hypothesizes and outlines an interpretation
explaining Plato's principle of the "Greater and Lesser"
(Indefinite Dyad), together with the "Divided Line"
construction, by means of a novel association to the golden section.
In the book review section, **John Sharp** provides a personal
reading of a perceived dichotomy between mathematics and geometry
in Richard Padovan's new philosophical analysis of proportion,
Proportion: Science, Philosophy,
Architecture. Padovan's important new work raises questions
about the nature of proportion in architecture and design; while
not dedicated solely to the golden section, his book certainly
includes a great deal of material on it. The second review marks
the first of a journal article in the *NNJ*, namely Nigel Reading's Dynamical Symmetries:
Mathematical Synthesis between Chaos Theory (Complexity), Fractal
Geometry, and the Golden Mean, which appeared in the popular
"glossy" magazine Architectural Design and has become
regularly cited by young architects and theorists. **Michael
J. Ostwald** and **Stephen R. Wassell** argue that Reading's
article is mathematically vague and often incorrect, more pseudo-mathematical
and metaphysical than scientific, and, most importantly, essentially
devoid of substance regarding architectural design methodology.
This is the *Nexus Network Journal*'s first special issue
dedicated to a particular subject, and it is my pleasure to have
accepted Kim Williams's invitation to take the lead on the editing.
I would be delighted to receive any feedback about this issue,
so please email me with
any comments, positive or negative.
Steve Wassell, Mathematics Editor
**The correct citation for
this article is:**
Stephen
R. Wassell, "Letter from the Editor", *Nexus Network
Journal, *vol. 4, no. 1 (Winter 2002), http://www.nexusjournal.com/letter_editor_v4n1.html |
Copyright ©2002 Kim Williams
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