Abstract. Answers to a reader's query about "Why", rather than "How", mathematics is used in architecture, in the Nexus Network Journal.

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Query: Why is mathematics used in architecture?

Date: Monday, 13 January 2003 11:27:42 +0100
From: Rumiko Handa <rhanda1@unlnotes.unl.edu>

I have noticed, possibly largely from those more mathematically inclined, the tendency to limit one's inquiry to the question of "how". What Professor Peter Schneider of University of Colorado meant by distinguishing "reasoning" from "problem-solving" during the Nexus 2002 round-table, the point Dr. Alberto Perez-Gomez has made in years in his work in theory, and what Dr. Robert Tavernor pointed out by "quality" as distinguished from "quantity" all point to yet another kind of inquiry, that is, the question of "why". "Why is mathematics used in architecture?" or "Why is this particular mathematics appear in this piece of architecture?", as opposed to "How mathematics is used in architecture?", provides another important aspect of the subject. Expanding on the questions regarding of "why" will, I think, allow us to go beyond the surface of form and structure making, and toward the understanding of the ideas and the ideals that have supported architecture.

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From: Dag Nilsen <dag.nilsen@ark.ntnu.no>

I see from the answers that some immediately confused numbers with measures, which are completely different things (although you usually use numbers to convey results of measurements). The knowledge meaning ascribed to numbers must depend on a study of literary sources, and whatever we have left of traditions - maybe even "archetypes" in the Jungian sense.

About the current query - "why" - I tend to apply a rather functional view, of mathematics as a tool, but a tool that can be played with in a kind of game.

As an example I'm working on: A group of 12th C churches in the S.E. part of Norway with quite uniform plan disposition are clearly modelled on the plans of the now ruined Oslo and Hamar Cathedrals.

We may suppose that a master builder, by training and experience, would have acquired command of a repertoire of models for planning churches of various sizes and complexity. As a help to setting out the plan on the site, these may have been based on geometrical diagrams and/or simple numerical ratios. The choice of model depended on the client’s needs, while the chosen model could be scaled to accord with his budget. In theory, a suitable measure unit could then be chosen more or less arbitrarily for the necessary calculations, dependent on the wanted size of the building.

It seems to me that, if the builders used a preconceived diagram to determine the dimensions of a church, it was more in the manner of a guiding framework that could be handled with flexibility, than a fixed ideal they felt forced to stick to. There is little or no evidence in the contemporary sources to Mediaeval church building to support speculations that geometrical diagrams or numerical relationships in the design of churches were required for expressing symbolic or religious meanings. As there in Christianity is no physical dimension directly corresponding to the spiritual, no specific architectural norm can ”truly” represent the theological contents. If particular numbers, or combinations of numbers were infused with some symbolic value, or wanted as an emulation of a significant sanctuary taken as a model, the clients seem to have been quite content with approximations.

Anyway, mathematically based methods of design, to the extent they are likely to have been used, seem to have been part of the builders’ planning tools, and not something the clients were bothered with. A church can resemble another in so many aspects that they by a superficial glance seem identical, but there will always be more or less subtle differences in proportions. When analyzed geometrically, similar diagrams and patterns can be found, but not fitting in exactly the same way. If the client wanted a “copy” of an existing church, it is as if the master builder deliberately made it different from the model by varying his procedure by the ”rules of play” in what may be called a ”mathematical game”.

From: Donald Watts <wattsd@ksu.edu>

I think that mathematics is an inescapable language in the production of architecture. What "kind" of mathematics being employed is reflective of the intentions and capabilities at hand. In its highest mode, architecture may attempt to employ qualities of mathematics developed in our era and in so doing, seek a form of expression depicting current mathematical understandings of the universe. In this sense, the work of architecture seeks to depict a body of knowledge.

From: Graham Pont <pont@tpg.com.au>

My own view is that the use of mathematics is the differentia that distinguishes architecture proper from mere building. Building (an art widely practised by many species) is the larger, natural genus which includes the important subset of mathematical building (also practised by some non-humans). Perhaps the simplest form of mathematical building is that which involves the use of prcise canons or measures - hence achitecture might also be defined as 'rational building' - especially when the use of a regular canon or measure enables the introduction of symmetry (or common measures) and hence, of modules.

The role of mathematics in architecture wold make an excellent conference topic (I mean the general theory or philosophy of mathematics and architecture, as opposed to particular examples of their interrelations).

From: Matt Insall <montez@fidnet.com>

I am not an architect.
I am not a philosopher.
I am not an artist.
I am not a geometer.

What I have to say is not full of references,
because I am not in a position today to look them all up.
Please accept it for what it is:
I have my views that come from years of reading and listening
to accounts of how and why things were done.

The Hindus,
Egyptians and Arabs
used geometry to design their buildings because
they thought geometry was the absolute law for the right shape
of things in the universe.
They thought God geometrized.
More importantly,
they thought God geometrized "Euclideanally".
It was not until Bolyai,
Gauss and Lobachevsky found
models of noneuclidean geometry that
the erudite began to stop believing that Euclidean
geometry was physical law.

Euclidean or noneuclidean,
it does not matter.
The subject was mathematics.
It produced beauty and strength of design.
It produced sucessful architectures.
That is one of the "why's".

when certain types of purely
utilitarian segments of our world society
took over,
mathematics became associated with a spiritually voided
and artless system of thought.
the use of mathematics in the other disciplines may enjoy a
I only hope it is not too late for us
to teach those outside mathematics and
artistic brothers the beauty that can be realized
when one sees it as the wonderful world that it is.

Until then,
the main reason for using
mathematics in architecture is,
I believe,
related to the engineering applications
that are based almost entirely on the
least aesthetic aspects of our discipline.
These involve making sure that a pipe is long enough
to reach from one location to another,
or being able to convince the potential
client that the planned project will not exceed
available funding.

The value of the beauty of the mathematics behind
and in the design can then come forth,
and take its rightful place in center stage again.

There are some mathematicians involving themselves in this,
I know.

I applaud them,
and wish them well.

From: Malynda L. <wellnesscoach1@yahoo.com>

It makes sense because the people who are more mathematically inclined,therefore left brained, are less likely to think in the abstract. As a result they do not stray from the norm or what is generally accepted. Most scientist believe that everything is a result of science and nature. Most don't specifically believe that there is a God, or conscious mind behind creation. "Why" is the question of those who believe that a conscious mind is behind creation in all of it's forms, whether in architecture or nature. "How" is the question of those who know that everything follows a pattern and want to understand it better.

From: Jørgen Holten Jensenius <jorgen@jensenius.no>

I tried to answer the questions "why" and "how" ratios and geometrical figures were used in medieval wooden churches of Norway in my PhD thesis from 2001 called (in Norwegian): "Wooden Churches before the Stave Churches. An investigation into the planning and design of Churches prior to c. 1100 CE". My conclusions on "why" are 1) To make a design of the building on forehand without drawings, 2) To transfer knowledge of form from prototypes, churches abroad for example, 3) To follow the ratios of for example the Temple of Ezekiel , 4) To satisfy the symbolic wishes of the theology-inclined patron. In my post-doc I continue the same path of study, with the wooden churches in Norway built between 1130-1350, a number of 28.

From: Jonathon Giebeler <giebeler@infinito.it>

Perhaps it is best to think about what we are trying to do with architecture. What should it be ideally? Math is of course needed as a practical tool for engineering and measurement, but when we talk of aesthetics we are of course talking about a different use of math.

There are actually many other uses for math and the readers will probably respond according to their goals. Those obsessed with the "sacred," and the romance of the past will perhaps answer that math is needed and used to achieve sacred principles and relationships: the importance of 3, the fact that 8 stands for the resurrection, 12 as a divine number, etc, golden mean, etc, etc.

Those who approach things from a more intellectual or mathematical point of view may focus on the order of composition and rationality that mathematics brings. These minds probably favor the rational Renaissance as the highest point possible.

Probably each viewpoint or approach has its own truth. For me I ask myself the question again, what should architecture be? And, I answer this with a single word: beautiful. Then I ask myself, what is beautiful - what is beauty? And, beauty is hard to define; so, I search for examples...I think of music and the way that it reaches to my core; I think of sunsets; I think of walking through a forest and the peace I sense; I think of flowers and the way that each pedal fits together; I think of vibrant paintings; I think of all these things and many other things, and then I ask myself, what makes them so beautiful?

All of these things have in common elements of composition, and these elements combine to create the sensation of beauty. There is unity, proportion, rhythm, harmony, nuances, etc. It is not so easy to break apart the composition, because when something is truly beautiful each part is less itself and more of the whole. But, if you did, you would find that one part is proportion, and proportion is of course related to math.

The notes of the musical scale, the leaves of a tree, the division of a face and body, the field of colors in a painting, all of these things have a structure and proportion - they have much more also, but proportion is one necessity.

So why is math important for architecture? Or better yet, why is it important for beautiful architecture? Because, beauty is composed, a part of composition is proportion, and proportion relates to mathematics. It doesn't have to be about symbolism, theory, or philosophy. It can simply be about creating beauty in and of itself. Mathematics and proportion do not create beauty by themselves, but they are a necessary part of the compositional whole.

From: Han Vandevyvere <Han.Vandevyvere@asro.kuleuven.ac.be>

In response to the question raised by prof. Rumiko Handa I would like to give a personal appreciation of some of the reasons for which mathematics is used in architecture. This appreciation is based on what I have seen, read and analysed until now.

Probably a major motive for incorporating mathematical or geometrical schemes in architecture since ancient times has been the discovery of numerical principles guiding the 'manifest world'. If we look at the amazing appearance of numbers such as phi, pi or the Fibonacci series in nature, or if we look at some astonishing phenomenae in mathematics and geometry themselves, then we often remain in surprise. It is very exciting to discover these principles, and it gives us a feeling as if we were looking into the secret building code of creation. It raises automatically questions about the 'why', beside those dealing with the 'how'. In this respect we may say that mathematics or physics provide an answer to the 'how', but that the 'why' is in the field of meta-physics.

I think that historically speaking there has always been a relationship between physics - in a very broad sense - and metaphysics, and that architecture with its implicit geometrical character has been an ideal contender of metaphysic representation. Therefore what we find as a body of knowledge in a society, will often be represented symbolically in its built artefacts. Architecture is so interesting to represent this knowledge body because you can show or say certain things much more efficiently with a geometrical scheme than e.g. with a written text.

A second element which I think should be taken into account is the sacred nature of this knowledge and its representation. In our desacralized world it is not easy to empathize with this attitude, but I think a lot of architecture carries a message that is to be read by the "instructed reader". Just as not anyone would be allowed to go to the centre of a temple, unless being e.g. a priest, not anyone was supposed to access metaphysical knowledge unless having gathered the credits for it. I think this is why we find few written or other direct sources on e.g. the geometrical principles underlying Egyptian, Greek-Roman or gothic architecture.

A mathematical or geometrical message is moreover very much self-protecting: you already need a rather complex key to read it properly. Another important advantage of geometry is that it tranmits its message in a non-verbal way, so that the builder is assured that it will be readable for eternity - or at least as long as the building stands.

Of course there are other motives for which one can have embedded mathematics in a design. I intend to point to one aspect because I think it is an important one to consider.

From: dotun oke <dotwuzhia@yahoo.com>

what will architecture be without mathematics?
1.it is needed for the aspect of structural design.
2.cost analysis, etc.
the list could go on and on

From: Alexey P. Stakhov <anna@nest.vinnica.ua>

Answering the questions "Why is mathematics used in architecture?" and "How mathematics is used in architecture?" we should answer to the questions: "What is Mathematics?" and "What is modern situation in Mathematics?". According to definition of the famous Russian mathematician academician Kolmogorov, "Mathematics is a science about quantitative relations and space forms of real world".

History of mathematics shows that mathematics developed under influence on practical needs of natural sciences. However, according to opinion of many famous mathematicians (in particular, John von Neumann) the tendency to be separated from vital problems of natural sciences is particular feature of modern mathematics and there exists a real danger of transformation of modern mathematics to "Art for Art". That is why the representatives of many natural sciences and arts began to search own ways in development of mathematics and the question "Why is mathematics used in architecture?" is not accidental.

Searching answer to the question "Why is mathematics used in architecture?" we should determine a concept of "Architecture". We can use the following definition of this concept given in the Great Soviet Encyclopedia: "Architecture .. is a system of buildings and structures forming a space medium for life and activity of people, and also art to create these buildings and structures in correspondence with the laws of beauty". It follows from this definition that there exist two aspects of Architecture notion. On the one hand, Architecture is a particular kind of Technology intended for "forming a space medium for life and activity of people". And creating his buildings and structures architect should know "Laws of Mechanics" ensuring mechanical strength and stability of his buildings and structures.

But Architecture is a kind of Fine Art and architect should create his "buildings and structures in correspondence with the laws of beauty". It means that architect should be guided by Laws and Principles of Harmony and Beauty in his creativity. Architecture as a kind of Fine Art is connected closely to other kinds of Fine Art, in particular, to Music, Sculpture and Painting. Sometimes Architecture is called "Frozen Music".

In this connection there arises an idea to create a new mathematics, the Mathematics of Harmony, adapted very well to studying physical phenomenon and based on the Golden Section. It is impossible to state all scientific achievements of Harmony Mathematics in this brief essay. If you wish to study Mathematics of Harmony more in detail I would like to invite you to visit my Museum of Harmony and Golden Section and to read my essay "Museum of Harmony and the Golden Section: Mathematical Connections in Nature, Science and Art" submitted to the International Essay Contest (University of Toronto, Canada).

From: Nat Friedman <artmath@albany.edu>

Geometry is the language of architecture. It may be classical Greek or Frank Gehry. A building is three-dimensional geometry. You can't describe a building without geometry. When I first started to look at architecture in downtown Chicago, I saw that it was mainly about rectangles-windows , doors, building outlines, etc-a synphony of rectangles-just rectangles. Seemed somewhat simplistic. Especially the steel and glass apartment buildings-very minimal geometry. I liked the minimality. So as I understand the question-it doesn't make sense to ask it. It is really about the how. How do you use mathematics? The "new" geometry is more curvey-not rectangular.

This reminds me of a question that came up at one of my conferences here. I was talking about form and space and someone asked me what is space? I said space is space. Then I said that if you asked me "why is space?", that is a good question. For me, sculpture is form, space, and light. Space is where the light goes! That was my answer to this why question.

From: Vera W. de Spinadel <vspinade@fibertel.com.ar>

A great part of this question can be easily answered! In fact, if you ask why is geometry used in architecture, it is quite obvious that since geometry is the instrument to design, and to design is to build an organized spacial structure, the knowledge of spacial forms is a necessary condition to deliver an architectonic message of the best quality. Moreover, with the introduction of computer design softwares, the possibilities of combinations of geometrical systems are enormously increased. And geometry is for the architect a disciplinal means, an essential implement in the "consideration" of the forms that intervene in the "composition" of space.

But if you consider the rest of mathematics, why is it used in architecture? Let us try to give two examples of possible answers: 1) because there is a "generative theory of shape" (see the book with that title by Michael Leyton), developed using the most abstract concept of symmetry groups, which is used as an intelligent means of describing the entire complex structure of a building; 2) because if you consider the fascination that numbers have in architectonic design, we may conclude that we have an innate proportion science that obliges us to measure dimensions comparing one with another.The fascination starts with the integer numbers as indicated by the egiptian "sacred triangle", a right-angled triangle with the two cathetus and the hypotenuse in the proportion 3, 4 and 5. But quickly goes from magic triangles and perfect squares to the square root of 2 rectangle and to the golden rectangle, where the numbers considered are irrationals and demand rational approximations to be applied in the real construction (see From the Golden Mean to Chaos by Vera W. de Spinadel). Among these two examples there is a myriad of applications of mathematical subjects, going from number theory to fractals and frontiers of chaos, to architecture.


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