madarchodo behan ke lodo


bakchod



betichod



kuch nahi hai yaha

Although many readers of this archive might find an article on mathematics and architecture a little surprising, in fact architecture was in ancient times considered a mathematical topic and the disciplines have, up to the present time, retained close connections. Perhaps once one realises that mathematics is essentially the study of patterns, the connection with architecture becomes clearer. Salingaros writes in [27]:-
Historically, architecture was part of mathematics, and in many periods of the past, the two disciplines were indistinguishable. In the ancient world, mathematicians were architects, whose constructions - the pyramids, ziggurats, temples, stadia, and irrigation projects - we marvel at today. In Classical Greece and ancient Rome, architects were required to also be mathematicians. When the Byzantine emperor Justinian wanted an architect to build the Hagia Sophia as a building that surpassed everything ever built before, he turned to two professors of mathematics (geometers), Isidoros and Anthemius, to do the job. This tradition continued into the Islamic civilization. Islamic architects created a wealth of two-dimensional tiling patterns centuries before western mathematicians gave a complete classification.
The first mentioned type of architecture Salingaros mentions in this quote is the pyramid and here we have marked disagreement between experts on the how much geometry and number theory the architects used. For example the Great Pyramid was built at Giza in Egypt around 2575 BC for King Khufu. Much has been written on the measurements of this pyramid and many coincidences have been found with , the golden number and its square root. There are at least nine theories which claim to explain the shape of the Pyramid and at least half of these theories agree with the observed measurements to one decimal place. This is a difficult area, for there is no doubt about certain astronomical alignments in the construction of the pyramid. Also regular geometric shapes were sacred to the Egyptians and they reserved their use in architecture for ritual and official buildings. That they had a goddess of surveying, called Seschat, shows the religious importance placed on building. However, no proof exists that sophisticated geometry lies behind the construction of the pyramids.
One has to make decisions as to whether the numerical coincidences are really coincidences, or whether the builders of the pyramids designed them with certain numerical ratios in mind. Let us look at just one such coincidence involving the golden number. The golden number is (1 + √5)/2 = 1.618033989 and an angle based on this will have size arcsec(1.618033989) = 51° 50'. Now the sides of the Great Pyramid rise at an angle of 51° 52'. Is this a coincidence? F Röber, in 1855, was the first to argue that the golden number had been used in the construction of the pyramids. Many authors have followed Röber, or produced even more elaborate versions of how and the golden number have been utilised by the Egyptians. The authors of [23], however, suggest reasons for the occurrence of many of the nice numbers, in particular numbers close to powers of the golden number, as arising from the building techniques used rather than being deliberate decisions of the architects. Arguments of this type have appeared more frequently in recent years.

Even if deep mathematical ideas went into the construction of the pyramids, I think that Ifrah makes a useful contribution to this debate in [4] when he writes:-
I once knew a professor of mathematics who ... tried to persuade his students that abstract geometry was historically prior to its practical applications, and that the pyramids and buildings of ancient Egypt "proved" that their architects were highly sophisticated mathematicians. But the first gardener in history to lay out a perfect ellipse with three stakes and a length of string certainly held no degree in the theory of cones! Nor did Egyptian architects have anything more than simple devices -- "tricks", "knacks" and methods of an entirely empirical kind, no doubt discovered by trial and error -- for laying out their ground plans.
The first definite mathematical influence on architecture we mention is that of Pythagoras. Now for Pythagoras and the Pythagoreans, number took on a religious significance. The Pythagorean belief that "all things are numbers" clearly had great significance for architecture so let us consider for a moment what this means. Taken at face value it might seem quite a silly idea but in fact it was based on some fundamental truths. Pythagoras saw the connection between music and numbers and clearly understood how the note produced by a string related to its length. He established the ratios of the sequence of notes in a scale still used in Western music. By conducting experiments with a stretched string he discovered the significance of dividing it into ratios determined by small integers. The discovery that beautiful harmonious sounds depended on ratios of small integers led to architects designing buildings using ratios of small integers. This led to the use of a module, a basic unit of length for the building, where the dimensions were now small integer multiples of the basic length.
Numbers for Pythagoras also had geometrical properties. The Pythagoreans spoke of square numbers, oblong numbers, triangular numbers etc. Geometry was the study of shapes and shapes were determined by numbers. But more than this, the Pythagoreans developed a notion of aesthetics based on proportion. In addition geometrical regularity expressed beauty and harmony and this was applied to architecture with the use of symmetry. Now symmetry to a mathematician today suggests an underlying action of a group on a basic configuration, but it is important to realise that the word comes from the ancient Greek architectural term "symmetria" which indicated the repetition of shapes and ratios from the smallest parts of a building to the whole structure. It should now be clear what the belief that "all things are numbers" meant to the Pythagoreans and how this was to influence ancient Greek architecture.

Let us look briefly at the dimensions of the Parthenon to see how the lengths conform to the mathematical principles of proportion of the Pythagoreans. In 480 BC the Acropolis in Athens was totally destroyed by the Persians in the Second Persian War. To understand the timescale, let us note that this was about the time of the death of Pythagoras. After the Greek victory over the Persian at Salamis and Plataea the Greeks did not begin the reconstruction of the city of Athens for several years. Only after the Greek states ended their fighting in the Five Years' Truce of 451 BC did the conditions exist to encourage reconstruction. Pericles, the Head of State in Athens, set about rebuilding the temples of the Parthenon in 447 BC. The architects Ictinus and Callicrates were employed, as was the sculptor Phidias.