Quadrilateral Perspective


“Pythagoras proved that the world of sounds is ruled by exact numbers. Therefore, he oriented his research towards checking if the same occurred in the world of visual images. And there is an extraordinary accomplishment here. I look around: here am I in this colorful and wonderful Greek landscape, among natural rustic orphic grottos and the sea. Where, under this beautiful chaos a simple, numeric structure could be found?

The issue sends us back to the most primitive constants of our perceptions of the natural laws. It is evident that in order to find the answer we have to start with the universal data of experience. Our visual world is based on two experiences: gravity is vertical and the horizon is orthogonal to the first one. This association, this crossing of lines in the visual field, fixes the nature of the right angle; thus, if I rotated this sensory right angle (downward direction and both sides direction) in a fourfold pattern, I would be back to the crossing of gravity with the horizon. The right angle is defined by this operation in four stages and through it, differentiated from any other arbitrary angle”. Page 157

“Thus, in the visual world, in the vertical plane image that is presented by our eyes, a right angle is defined by its rotation in four stages over itself. The same definition applies to the horizontal plane perceived, in which, in fact, we move ourselves. Picture this world, the world of the flat earth of the maps and of the cardinal points. I am in it, looking through the straits, from Samos to Asia Minor, on the South direction. I pick up a triangular rod and point it on that direction: South (I intend to illustrate the four successive rotations of the right angle with the triangular rod). By rotating the rod one right angle it will point to the West; one more right angle rotation and it will be pointing to the North; a third one and it will point to the East and the fourth and last rotation it will point to the South, Asia Minor, our starting point.

It is not only the natural world of our perception that obeys these relations; the world constructed by us does the same. It has been like that since the times that the Babylonians constructed the Hanging Gardens and even before, when the Egyptians erected the pyramids. In a practical sense, these cultures were already aware of a square arrangement of construction, in which the numerical relations revealed and formed right angles. The Babylonians knew many of these formulas, maybe hundred of them circa 2000 B.C. The Indians and the Egyptians knew some of them. Apparently, the latter ones used the square arrangement most often as the sides of the triangle made up of three, four or five units. It was only circa 550 B.C. that Pythagoras rescued this knowledge from the world of empiric facts to the universe of that we would call today, the proof”. Page 158

“The right angle is the symmetry element that divides the plane four times. If the flat space presented another type of symmetry, the theorem would not be real; the truth would be in some other relation between the sides of other particular triangles. It should be noted that, like the matter, the space is a fundamental part of the nature, even though invisible (like the air); in this dimension lies the essence of geometry. Symmetry does not represent only a descriptive sophistication; similarly to other Pythagorean thoughts, it seeks the harmony of nature”. Page 161

From the book The Ascent of Man, by J. Bronowski
(Excerpt translated from the Portuguese version)
Livraria Martins Fontes Ltda. and Editora Universidade de Brasília
Brazilian 1st edition – 1979 – The author of this book italicized a few words


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