by anyarchitect on 1220343022|%e %B %Y

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The story goes like this: Pythagoras had gone to meet a friend; only to find that his friend kept him waiting in the living room for sometime. Having nothing else to do, Pythogoras did what all of us often end up doing: He kept staring at the regular tiling pattern of the floor.

Now Pythagoras (like many other Greeks in those days) believed in the purity of integers, and he may have tried expressing what he saw in the form of a rational equation. The pattern was based on putting a square at an angle within another square, which resulted in four right angled triangles at the four corners around the inner square. Pythagoras must have spent some time looking at the hypotenuses of those triangles. He probably hated them because most hypotenuses are irrational (a word which reflected the Grecian belief: they didnt find it logical that such numbers could exist^{*}) and so he may have desperately sought to fit the hypotenuses he saw onto a more sensible geometry; say onto the side of a square. And that was readily apparent in the flooring pattern he saw.

This must have led him to discover the theorem which we now know by his name. Most likely he rediscovered it, because it was discovered by several others. We, Indians, were arguably one of the oldest who had discovered the same theorem, much earlier to Pythagoras.

Look at this website which has an applet that demonstrates the kind of flooring pattern Pythagoras must have seen. Once the basic pattern is displayed in the applet, move the red dot around in that applet, and you'll get other flooring patterns — all conforming to the Pythagorean theorem.

Such flooring patterns are still used today, and I wonder how many of our young architects realize that they often trample all over Pythagoras.

Many architects tend to treat mathematics as a subject that does not directly concern them. The reasons could be many; e.g. that non-issue of a debate: is architecture art or is it science?. Anyway, let me not digress into those reasons. The Pythagorean theorem is very much around describing the relationship of the legs of the tripod of the easel of an eccentric artist, with the floor. If the artist did know the theorem, it may help him setup his easel faster.

Mathematics therefore has to do with efficiency. It is a language to express relationships succinctly. Mathematics is a tautology (a self evident truth) and not a science. Science can consist of conjectures, hypothesis and tests of hypothesis. What was once held as truth in science can become false as per new evidence. On the other hand, much of mathematics uses circular reasoning and axioms and does not depend on anything else. Mathematics is a tool to model scientific problems.

*Note that the actual definition of Mathematics may be an esoteric point. There are various other ways of defining mathematics too. Like in most other things that catches human interest there is a bit of mystery and awe in the definition of mathematics. I chose one which to me reflects its independence and universal applicability. Most agree that it is not something that changes often and is relatively independent from the sciences. Here is a website that has a nice summary.*

Architecture is a complex subject and often that complexity pulls the architects into areas where fundamental subjects such as mathematics is considered disposable (maybe delegatable is the right word) as she is busy with issues worthy of her brain. This need not be always wrong, for I am no judge on what an architect should work on. However, this could cause information regarding mathematics to percolate to the architect much later than those involved with mathematics.

The chronological timeline of the development of mathematics has important inconsistencies. i.e. It could be claimed that mathematics is not a subject where the ancient mathematicians developed some basic theories and the mathematicians who came in later build up more advanced theories on the shoulders of the ancients. There are innumerable instances of complex theories that were discovered beforehand (E.g. Fermat's last theorem) but could be proved only recently. More importantly, there has been fundamental discoveries which has happened very recently.

This has a huge implication for architects caught up in their busy lives. For example; If we did not grasp the concepts behind the Kurt Gödel's *incompleteness theorem* which was discovered only in 1931, we may be as guilty as those who thought the earth was flat. Another example that comes to the mind is complexity theory with its mathematical tool of fractals.

The whole subject of complexity was explored mainly since the middle of the last century only. It is strange when you look at it this way: Nature could have always been modeled using fractals — in fact fractals are the only way one could sensibly explain many aspects of nature. But the mathematics of fractals is all so very recent.

I would attempt to highlight a few of these fundamental concepts in this book. Hopefully, that should spark enough interest for young architects to look at the floor more thoroughly.

**References:**

- The word
*irrational*actually originated from the phrase "not a ratio". This is because of the Greeks obsession with integers. They liked a number that was either an integer or was a ratio between two integers. (Well any integer is also a ratio; the denominator being 1). Unfortunately, they encountered some numbers that could not be expressed as such a ratio. Hence they were deemed "irrational". Society took the same reasoning and the word "irrational" is now used to categorize things that do not fit in; not necessarily numbers that are not ratios of integers.