**ORIGINAL QUERY:**

**Date: Monday, 13 January 2003 11:27:42 +0100**

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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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Its the way structures are

not wobbly neither bad

its geometry , shapes ,

sizes , lines , curves ,

That takes over our mind.

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 www.amazon.com/exec/obidos/ASIN/3540427171/thenexusnetworkj/), 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 (seeby 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.From the Golden Mean to ChaosThis 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.

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 www.goldenmuseum.com/ 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).

the list could go on and on

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.

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.

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