Query: __Is There a Relationship Between Architecture
and Higher Mathematics?__ |
**ORIGINAL QUERY:** Date:
Sat, 14 Jan 2006 16:27:07 +0100
From: Dirk Huylebrouck
<huylebrouck@gmail.com>
I have a question for the Nexus
readers: "Are there any relationships between architecture
and *higher* mathematics?" By "higher" I
mean, mathematics at the level of 1st master and up. The topology
paper in the *NNJ *vol. 7 no. 2 by Jean-Michel
Kantor goes in that direction, but the reason why I propose
it is different: I recently wrote a paper (in Dutch) on "Africa
and higher mathematics". "True, die-hard" mathematicians
sometimes take architectural math as "baby math", and
of course, related to Africa, there are some down-to-earth social
(extremist) influences involved. Nevertheless, in the architecture
case, the question could be seen as a modification of Mario
Salvadori's query at the first Nexus conference for architecture
and mathematics in 1996: "Are there any relationships between
architecture and mathematics?"
*NNJ* READERS'
RESPONSES: Date: Sat, 14 Jan 2006 16:27:07
+0100
From: Kim Williams
<kwilliams@kimwilliamsbooks.com>
This is a question that I have given quite a lot of thought
to, so I want to put some of those thoughts in black and white,
and then encourage NNJ readers to respond. The kind of "baby
math" that Dirk refers to is probably that which is often
discussed in our pages: the proportional relationships involved
in the architectural orders, for instance, based on simple ratios
of whole numbers; or the ratios of architectural dimensions derived
from elementary geometrical figures such as the square or golden
rectangle. In this case, I agree: there is not much "higher
mathematics" here. But when we look at the world of ideas,
there is more going on than might meet the eye to begin with.
The relationship between architecture and ideas is the inverse
of the relationships between mathematics and ideas: where we
often find that very simple, easy-to-understand ideas underlie
some very complex mathematical structures (such as how the idea
of sensitivity to initial conditions underlies the science of
chaos), instead some very complex mathematical ideas underlying
their more elementary architectural expression. For example,
architects a few years ago were very excited about breakthroughs
in chaos, and now research in topology is providing fertile ground
for architects. However, architecture imposes two great limits
on its practitioners. One is that we human beings have some very
simple needs that must be met: we have to have horizontal surfaces
on which to walk; we feel disoriented without vertical walls;
we can feel downright frightened by non-vertical supporting elements;
architecture must be constructed. These requirements limit architectural
experimentation: we might like to hypothesize about the Möbius
strip in architecture (see the paper by Vesna
Petresin and Laurent-Paul Robert) but we probably couldn't
live in a Möbius house. But there is no limit to the architectural
imagination, and virtual architecture provides very important
tools for visualizing the application of higher mathematics to
architectural forms. Such visualization can then be modified
to meet human requirements.
Another source of higher mathematics in architecture is to
be found in mechanics. Building collapses provide particularly
good, if tragic, opportunities to study structural behaviour,
into which I suspect higher mathematics often enters. Ah, but
you say, that is engineering and not architecture. Well, I suppose
that depends on how you view architecture: in my opinion, any
aspect of the built environment is included under the umbrella
of architecture, and our aim at interdisciplinarity reflects
that fact.
Dirk's question is related to the query proposed by James
McQuillan:
Do any contemporary architects understand enough about mathematics
(beyond additive planning and accidental occurrences) to apply
it in their work? And even if they did, why should they do so,
given the collapse of ancient mimesis and related understanding
until the 18th century, which was the foundation of such applications
in the past? (Click here
to access this page in the NNJ.)
In spite of many architecture students' objections to learning
mathematics, more than just a fundamental understanding of mathematical
concepts and even a way of thinking is essential to the architectural
education. This is why we publish papers about Didactics in the
NNJ.
I hope that readers will think about and respond to these
queries: this is an important part of keeping dialogue open within
the NNJ community.
------------------------------------------------- Date:
Thu, 19 Jan 2006 12:17:41 -0500
From: Chandler Davis
<davis@math.toronto.edu>
Modern study of collapse (meaning, since 1970) often uses
modern (since 1960) deep mathematics. It is true that we don't
get many architecture undergraduates interested in our mathematics
courses, and even the mechanical engineering students may be
sort of resistant; nevertheless there is this current of study
of failure of beams, etc., and the people who do it (whether
or not they are called engineers) are doing mathematics. This
is important, but it is not what Dirk is mostly fishing for.
He wants to know, does an advanced mathematical idea go into
the architect's conception of a structure. Please let us understand
the question as referring primarily to mathematical ideas correctly
understood and relevant. Only in passing are we concerned with
misunderstood ideas, or superficial reference to ideas, or use
of exciting geometry as ornament distinct from structure(like
the Moebius band sculpture on the facade of TsEMI in Moscow).
------------------------------------------------- From:
Doug Boldt <Dboldt@lcmarchitects.com>
Date: Thu, 19 Jan 2006 11:41:41 -0600
I dispute the paragraph from your email above! We don't need
90 degree angles to be happy and feel safe. We have been conditioned
and fooled into thinking we are comfortable in our boxy prisons.
Fractal Geometry is the key to the future of architecture. We
are on the brink of a new Century and a new wonderful new architectural
age has begun. It is right before our eyes and no one can see
it -- just as in 1906 how many people could see and understand
that they were living in the age of Modernism. Look in any arch
magizine in 2006 and the pages are full of fractal architectural
examples.
The biggest hurdle to overcome is our understanding of fractal
geometry. I can't solve a logarithmic equation but I suspect
I understand fractal geometry or at least its application better
than most. If Fractals are the geometry of nature as Mendelbrot
proposed, then fractals rarely if ever reiterate at smaller and
smaller scales to infinity. Those pretty fractal computer generated
pictures we are so used to seeing are artifical. In natural fractals,
shaped over time by the forces of wind, water, fire, gravity,
etc., reiterate 1 time or 2 or 3 times, then the molucular composition
causes a new fractal to appear. The same is principle can be
(and is being) applied to Architecture. Unlike the rigid and
dogmatic Modern Movement, the Fractal Movement is inspired by
the limitless patterns of nature in all all its aspects, infinite
and ever changing.
------------------------------------------------- Date:
Fri, 20 Jan 2006 03:17:12 -0800
From: Lionel March <lmarch@ucla.edu>
I read your note from Dirk and thought you might like to know
that George Stiny's *SHAPE: Talking about Seeing and Doing*
is about to be published by MIT. George's work is based on extensions
to Boolean algebra known as Stone
algebras, Euclidean group transformations, lattice theory --
all of which might come under the heading of 'higher' mathematics.
As you probably know an area like fractals is subsumed by George's
shape formalism as a special case. Incidentally, Mario Salvadori
was a strong supporter of George's work and wrote a forward to
Algorithmic Aesthetics.
I recently wrote an endorsement for MIT at their request:
"Stiny is to 'shape' as Chomsky was to 'word' or Wolfram
to 'number.' In my view, though, Stiny may well prove to be the
most radical of the three. How different a place his pictorial
world is from standard textual or digital worlds: with shape
there is no vocabulary, no syntax, no bits, no atoms. As Stiny
draws, he talks. Shapes and shape rules bear the force of argument.
These drawings are to be looked at keenly, even traced and redrawn
by the reader. The supportive text illustrates what can be seen
and done, providing both a personal and intellectual history.
Through its drawings and maxims, Shape challenges much conventional
wisdom in philosophy and education, in computer science and artificial
intelligence, and in design and the visual arts."
------------------------------------------------- Date:
Sun, 22 Jan 2006 13:37:08 -0400
From: Emanuel Jannasch
<ejannasch@hfx.eastlink.ca>
As I see it, builders can “do” math in two distinct
modes. In the first, we calculate a structural form, set out
a geometric pattern, or devise some discrete structure, and try
to build accordingly. Our intentions in this mode may be aesthetic
(creating pleasant visual proportions) they may be purely instrumental
(sizing pipes), or they may be hybrid (as in shell roofs) Sometimes
deep aesthetic qualities emerge from the mathematical solution
of a technical/commercial problem, as in Maillart’s bridges.
But whatever we achieve with builder’s math, in a world
where an “applied” subject is considered inferior to
its dreamier cousins, it will always be seen as lowly or in the
phrase of your hard-core types, “baby”-ish.
In the second mode, mathematics is directly embodied in our
work, and only later does an abstract thinker devise a symbolic
description of our accomplishment. Many catenaries and funiculars
were built before mathematicians had a vantage point “high”
enough from which to describe them. Tesselations were treated
exhaustively in practice centuries before the theory was in
a podsition to concur. For millenia, boatbuilders have been bending
battens and planks to arrive at complex surfaces of least local
change in curvature which accord very neatly with their hydrodynamic
objectives. In this respect, the bent spline on the naval architect’s
drawing board is a calculator, and the bent battens enveloping
a hull form under construction are computer controlled machines,
and I mean this in a real, fuctional sense, not as a metaphor.
In both cases, the workings of the calculator were not given
abstract form until Schoenberg’s 1946 theory of mathematical
splines. In this second mode, the real world is always richer
and more complex than mere description, and the mathematicians’
approximation is destined to play catch-up.
The opposition between these two perspectives is ancient.
Medieval philosophers kept up a heated debate between the (Platonic)
“realism” of the former orientation and the “nominalism”
of the latter. I suspect replies to your query will fall into
one camp or the other, with a preponderance coming from the realist
side, as this is the inherent tendency – I would submit
– of the Nexus readership. I am reading into your query
a nominalist bias more like my own and wish you all the best
with your investigation. I look forward to your discoveries!
-------------------------------------------------
*top of
page*
Copyright ©2006 Kim Williams
Books |
*NNJ *Homepage
**Order
Nexus books!**
**Research
Articles**
**The
Geometer's Angle**
**Didactics**
**Book
Reviews**
**Conference and Exhibit Reports**
**Readers'
Queries**
**The Virtual Library**
**Submission Guidelines**
**Editorial
Board**
**Top
of Page** |