PYTHAGORAS' MATHEMATICS IN ARCHITECTURE AND HIS INFLUENCE ON GREAT CULTURAL WORKS (original) (raw)
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2006
There is always a mystery on pre-modern architecture practice on the relation between dimensions and ratios. The reasons of using certain proportions used on the design of religious buildings/ spaces are the result of the application of numerical symbolism and Pythagorean triangle. Thus, the paper will be focused on the unity of theory in premodern architecture practice via giving some special examples of pre-modern architecture through the human history, such as Antique Egyptian and Antique Greek temples, Roman churches, Gothic cathedrals, and so on.
Geometry in Greece The Use of Geometry by Ancient Greek Architects
the development of practical and theoretical geometry by the ancient Greeks was a significant cultural accomplishment, and it proved critical for the evolution of Greek architecture. Interest in geometry in Greece may be divided into three phases. the earliest phase emphasized philosophical and religious applications, such as the speculations of two pre‐Socratic philosophers who used geometric models to articulate their views of the cosmos, thales of Miletos and anaximander of Miletos, both of them active in the first half of the sixth century bce. the second phase was centered in hellenistic alexandria, where the most important compiler, euclid, was active in the third century bce. Our existing corpus of writing about geometry has been preserved from this period. the third phase of exploration of geometry took place in the late hellenistic period, when archimedes of Syracuse and apollonius of perga were prominent among geometers; this period also was remarkable for practical developments in mechanics and engineering (heath 1921: I.345–348, II. 346–352, thomas 1941). the text of Vitruvius illustrates the wide range of interests of such authors; although he regarded himself as an architect and a designer, Vitruvius was, as we would define the role, an engineer (rowland and howe 1999: 21). he wrote his still‐ preserved treatise, De architectura, about 20 bce and dedicated it to the emperor augustus. the development of early Greek geometry took place concurrently with several significant cultural events in the seventh century bce. the reign of psammetichus I (664–610 bce) marked the beginning of increased contact between Greeks and egyptians (hahn 2001: 66–69). During the reign of amasis (570–526 bce), Cyprus fell first under egyptian rule, then that of the assyrians. the capture of the city of Sardis by persia in 547/6 bce opened a direct link between the Greek world, persia, and India (Burkert 2004: 49–55, 70–74). thales observed the solar eclipse of 585 bce, and his book on the cosmos is datable to 547 bce. although individuals such as thales and, later, pythagoras (circa 550–495 bce) could have traveled to egypt and Babylon to gain specialized knowledge, the geometry that existed in egypt and Babylon was not the philosophical geometry of the pre‐Socratics; instead, it featured pragmatic solutions for calculating areas, needed because of the ever‐changing topography due to the annual flooding of the Nile (heath 1921: 122–126). the egyptian rhind Mathematical papyrus documents the interest in area calculations. But the calculations included inaccuracies and approximations, and the level of accuracy was not what we would accept today. Yet the Greeks did not have to go far to meet egyptians or Babylonians as these people were coming to them. It is no coincidence that the early Greek geometers thales and anaximander were from Ionia and that, in the following generation, pythagoras was originally from Samos, just off the Ionian coast.
Historia Mathematica, 1989
In this article, two questions are posed: Just how reliable is the evidence concerning Pythagoras's mathematical studies, and can we reconstruct his contribution to mathematics? All known fragments of evidence by fourth-century B.C. authors on Pythagoras's mathematical investigations are examined, and it is shown that all the discoveries they mentioned belong to the sixth century B.C. The opinion that the Pythagoreans ascribed their own discoveries to Pythagoras is refuted, and it is shown that we are able to establish logically his contribution to mathematics.Der Aufsatz behandelt die Frage, ob es sichere Zeugnisse über Pythagoras' mathematische Beschäftigungen gibt und ob wir auf dieser Grundlage seinen Beitrag zur Mathematik rekonstruieren können. Im Aufsatz werden Zeugnisse der Autoren aus dem 4 Jh. v.u.Z. über Pythagoras' mathematische Forschungen gesammelt, und es wird gezeigt, daß alle seine Entdeckungen wirklich dem Ende des 6 Jh. v.u.Z. angehören. Im Aufsatz wird die ältere Meinung abgelehnt, daß die Pythagoreer ihre Entdeckungen dem Pythagoras zugeschrieben haben, und es wird gezeigt, daß wir in der Lage sind, seinen Beitrag zur Mathematik abzugrenzen.
6th International Conference on Geometry and Graphics MONGEOMETRIJA 2018, 2018
The architecture and mathematics of ancient India based on the same principles of traditional geometry. The most important geometric constructions, which underlain the traditional Hindu architectural theory, were regular square grids, known as Vastu-Purusa Mandalas, and different Pythagorean triangles applied to proportioning of altars. In the paper it is demonstrated that these two as it may seem quite distant fields of geometry constitute mutually complementary processes. The regular square grids generate all possible Pythagorean triangles and the triangles in turn generate the more and more sophisticated square grids.
Geometry in nature and Persian architecture
Building and environment, 2005
Nature displays profound preference for certain specific ratios to design her life-forms. These are geometric relationships that are transcendent and originated from Sacred Geometry. The view that geometry had a ritual origin is a part of a wider view that civilisation itself had a ritual origin, and therefore the history of utilisation of Sacred Geometry by man goes back to many centuries ago. The Pythagorean tradition, and the Egyptian and Babylonian sciences from which it derived, and Persian mathematics, a part of which reflects a Pythagorean intellectuality, are based on the sacred conception of numbers and their symbolism. In the traditional world, geometry was inseparable from the other sciences of the Pythagorean Quadrivium, namely arithmetic (numbers), music and astronomy. Traditional geometry is related to the symbolic configurations of space. Geometric forms such as the triangle, square and various regular polygons, the spiral and the circle are seen in the traditional perspective to be, like traditional numbers, as aspects of the multiplicity of the Unity. Architecture itself has always had a sacred meaning to all traditional civilisations through millennia, by which means man has tried to provide for himself a manifestation of heavens. Persian architecture always emphasised on Beauty, and by means of Sacred Geometry Persians measured the proportions of heaven and reflected them in the dimensions of buildings on the earth. A comprehensive utilisation of proportions in Persian architecture, such as in the design of plans, elevations, geometric and architectural patterns, and mechanical and structural features, can be proved through geometrical analysis of Persian historical buildings. In this paper, the sacred conception of geometry and its symbolism in the Pythagorean tradition, and Sacred Geometry and proportions in natural life-forms will be explained. The use of the science of geometry in design of a number of Persian historical buildings will be presented. The geometric factors upon which the design of these buildings, from both architectural and structural viewpoints, is made will be discussed.
New Light on Ancient Architecture and Mathematics
Nexus Network Journal, 2021
Editor-in-Chief Kim Williams examines the way intersections of architecture and mathematics in ancient architecture are being brought to light thanks to increased information and new technologies, and introduces the articles in Nexus Network Journal vol. 23 no. 2 (2021). Keywords Nexus Network Journal • Methods of laying out • Methods of analysis Despite centuries of studies and scholars' best efforts at understanding, many mysteries still shroud the fragments of ancient buildings and cities that have come down to us. Today, new light is being shed on artifacts that have eluded earlier attempts at explanation. This is due in part to a new wealth of information placed at our fingertips thanks to advances such as drone photography and satellite observation, made available by sites such as Google Earth and GeoEye. It is also due to new technologies capable of capturing massive data, leading to new means of visualization and analysis such as point clouds. However, most advances in our understanding are due to researchers' persistence in studying, deconstructing, reconstructing and decoding, using whatever means provides them with the most information. This issue of the Nexus Network Journal presents a collection of research on built artifacts of epochs going as far back as 1500 BC, and ranging geographically from Europe to Egypt, from the Middle to the Far East. The topics addressed fall, broadly speaking, into two main groups. The first is concerned with techniques of laying out, that is, how mathematical knowledge was applied in disposing elements into a meaningful patterns of order. The second deals with methods of analysis that allow us to identify and build upon such patterns of order.