Matila Ghyka The Geometry Of Art And Life Pdf
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- The Golden Number : Pythagorean Rites and Rhythms in the Development of Western Civilization
- This is an effort from the researcher for anyone interest by the Golden Ratio20190425 28065 1ywoid5
Can one find a natural aesthetic that corresponds to a universal order? If so, what importance can it have for the scientist, artist, or layman? What is the "true" significance of the triangle, rectangle, spiral, and other geometric shapes?
Published by Dover Publications in New York. Written in English. The Geometry Of Art And Life by Matila Ghyka is an interesting book that pertains primarily to ancient information regarding the fiine arts. Geometry unites all aspects of this book. In the popular Dover reprint that I read, almost half of the page book is dedicated to illustrative : Dover Publications.
The Golden Number : Pythagorean Rites and Rhythms in the Development of Western Civilization
Many have been drawn especially for this book. Introduction And it was then that all these kinds of things thus established received their shapes from the Ordering One, through the action of Ideas and Numbers. In the same way that Plato conceived the "Great Ordering One" or "the One ordering with Art," o as arranging the Cosmos harmoniously according to the preexisting, eternal, paradigma, archetypes or ideas, so the Platonic-or rather, neo-Platonic-view of Art con- ceived the Artist as planning his work of Art according to a pre- existing system of proportions, as a "symphonic" composition, ruled by a "dynamic symmetry" corresponding in space to musical eurhythmy in time.
This technique of correlated proportions was in fact transposed from the Pythagorean conception of musical harmony: the intervals between notes being measured by the lengths of the strings of the lyra, not by the frequencies of the tones but the result is the same, as length and numbers of vibra In the same way Plato's Aesthetics, his conception of Beauty, evolved out of Harmony and Rhythm, the role of Numbers therein, and the final correlation between Beauty and Love, were also bodily taken from the Pythagorean doctrine, and then developed by Plato and his School.
Let us point out at once that "symmetry" as defined by Greek and Roman architects as well as the Gothic Master Build- ers, and by the architects and painters of the Renaissance, from Leonardo to Palladio, is quite different from our modern term symmetry identical disposition on either side of an axis or plane "of symmetry".
We cannot do better than to give the definition of Vitruvius: "Symmetry resides in the correlation by measure- ment between the various elements of the plan, and between each of these elements and the whole. As in the human body. This symmetry is regulated by the modulus, the standard of common measure belonging to the work considered , which the Greeks called the number When every important part of the building is thus conveniently set in proportion by the right correlation between height and width, between width and depth, and when all these parts have also their place in the total symmetry of the building, we obtain eurhythmy.
It is also quite recently that, in the field of biology too, it was found that certain morphological intuitions of the Pythagorean and Platonist schools, and their interpretation by the Neo-Platonist thinkers and artists of the Renaissance, are confirmed by modern research. Nihil" but also in the realm of Nature. The use of Geometry in the study and classification of crystals is obvious, but it is only lately that its role in the study of Life and Living Growth has begun to be recognized.
The present work tries to present in a condensed form what we may call a "Geometry of Art and Life. The simplest asymme'trical division of a measurable whole into two parts, and Ockham's principle of economy. Generalisation of the concept of proportion. Arith- metical, geometrical and harmonic proportion. The ten types of proportion. Proportion, symmetry, eurhythmy. Rhythm in space and proportion in time. The Fibonacci Series and the Golden Section.
The c Rectangle. Phyllotaxis and "Ideal Angle" in botany. The Golden Section and pentagonal symmetry. Regular polygons and star-polygons. Rectangles: the c Rectangle and the y c Rectangle. Pentagon, pentagram, decagon, and Golden Section. Hexagon and octagon. The thirteen semi- regular Archimedian bodies. Regular prisms and anti-prisms. The two continuous star-dodecahedra of Kepler. The dodeca- hedron, the icosahedron and the Golden Section. Other remark- able volumes.
The "Chamber of the King" in the Great Pyramid. The Great Pyramid, star-dodecahedron, and the human body. Regular hypersolids in the fourth dimension. Equipartitions and partitions of space. Crystal lattices. Hexagonal and cubic symmetries. The cuboctahedron and the close-packing of spheres. The principle of least action, most general law for inorganic systems.
Pentagonal sym- metry in living organisms. Flowers and shells. Pythagorean number-mystic. The Pythagorean tradition, the pentagram, decad and tetraktys. Neo-Pythagorism and Kabbala. From the antique builders' guilds to the masons' guilds of the Middle Ages. Masons' marks and fundamental design. Masonic traditions and symbols. Proportion and dynamic symmetry. The dynamic rectangles of Hambidge and the directing circles of Moessel. Greek vases, Greek temples, and the human body.
Gothic master plans. Seurat's division- ism. Revival of Pythagorean doctrine in science and art. Mod- ern applications of dynamic symmetry in architecture, painting, and decorative art. Symphonic composition. PAGE I. The Triangle of the Pentagon 25 II. Variations on the Pentagon 33 VII. Cuboctahedron and Byzantine Cupolas 55 XV. Regular Equipartitions of the Plane 76 XX.
Mason's Marks XLV. The y 5 Rectangle, Harmonic Decompositions L. The ell Rectangle, Harmonic Decompositions Ll. Greek Bronze Mirror, Harmonic Analysis RATIO The mental operation 1 producing "ratio" is the quantitative comparison between two things or aggregates belonging to the same kind or species.
Judg- ment consists of: I perceiving a functional relation or a hierarchy of values between two or several objects of knowledge; and 2 discerning the relation, making a comparison of values, qualitative or quantitative.
When this com- parison produces a definite measuring 1 a quantitative "weighing," the result is a ratio. To quote Euclid: "Proportion is the equality of two ratios. It is the geometrical proportion, discontinuous or continuous, which is generally used or mentioned in Aesthetics, specially in architecture. The simplest asymmetrical section and the corresponding con- tinuous proportion: The Golden Section. The Golden Section. Plato, Tirnaeus: "But it is impossible to combine satisfactorily two things without a third one: we must have between them a correlating link.
Such is the nature of proportion There are in all ten terms of proportions, established by the neo-Pythagorean School. Example c-b c 1,2, We will meet it again in the next chapter. We have seen in the Introduction that the technique by which, in a complex plan or design, the proportions were linked so as to get the right correlation or "commodulation" between the whole and its parts was called by the Greek architects and Vitruvius "Symmetry"; and the result obtained where this tech- nique was correctly applied was the "eurhythmy" of the design and of the building.
We generally associate the terms of "rhythm" and "eurhythmy" with the Arts working in the time dimension Poetry and Music and the notion of Proportion with the "Arts of Space" Architecture, Painting, Decorative Art. The Greeks did not care for these distinctions; for them, for Plato in particular, Rhythm was a most general concept domi- nating not only Aesthetics but also Psychology and Metaphysics.
And Rhythm and Number were one. For them, indeed, Architecture was not only "Frozen Music" 1 "Everything is arranged according to Number" was the condensation of the Pythagorean doctrine. And Plato, who developed Pythagoras' Aes- thetics of Number into the Aesthetics of Proportion, wrote in his Epinomis: "Numbers are the highest degree of knowledge" and: "Number is knowledge itself.
The notions of periodicity and proportion, and their interplay, can be used for succession in time as well as for spatial associations. If periodicity static like a regular beat, or dynamic is the characteristic of rhythm in time, and proportion the characteristic of what we may call rhythm or eurhythmy in space, it is obvious that in space, combinations of proportions can bring periodical reappearances of proportions and shapes, just as in a musical chord or in the successive notes or chords of a melody we may really perceive an interplay of proportions.
Francis Warrain 2 Rhythm is perceived periodicity. It acts to the extent to which such a periodicity alters in us the habitual flow of time. Pius Servien 3 Rhythm is this property of a succession of events which produces on the mind of the observer the impression of a proportion between the dura- tion of the different events or groups of events of which the succession is composed. Professor Sonnenschein Although the authors of these three distinct but excellent definitions are here thinking of Rhythm in time, we see how even in that temporal frame, proportion can play its part.
To sum up: there are proportions in time, and rhythm in space, and one could say, to cover both fields, that "Rhythm is produced by the dynamic action of Proportion on a uniform static beat or recurrence. In a sort of "Gallup Poll' asking a great number of participants to choose the most aesthetically pleasant rectangle,. Even without actually drawing the square, this opera- tion and the continuous proportions characteristic of the series of correlated segments and surfaces are subconsciously suggested to the eye; the same kind of suggestion operates in the simple case of a straight line divided into two segments according to the golden section, or when three horizontal lines are separated by intervals obeying this proportion Figure 9; for example, the horizon between the upper and lower bar of the frame in a painted seascape.
Professor Timerding sums up this subcon- scious operation and the resulting aesthetic satisfaction in the short sentence: " Hambidge, whose theory of "dynamic rectan- gles" is explained in Chapter VIII, calls the similar smaller rectangle, thus produced in the original one, his "reciprocal rectangle. To this particularity which combines the properties of additive and multiplicative, geometrical, series corresponds the geometrical illustration of the progression; that is: a series of straight seg- ments with lengths proportional to the terms of this series can be constructed by additions or subtractions of segments, by simple moves of the compass.
To quote Timerding again: "The golden section therefore imposes itself whenever we want by a new subdivision to make two equal consecutive parts or segments fit into a geometric pro- gression, combining thus the threefold effect of equipartition, succession, continuous proportion; the use of the golden section being only a particular case of a more general rule, the recurrence of the same proportions in the elements of a whole.
It is this property of producing, by simple additions, a succession of numbers in geo- metrical progression, or of similar shapes what Sir D' Arcy Thompson called "gnomonic growth" which explains the impor- tant role played by the Golden Section and the series in the morphology of life and growth, especially in the human body and in botany. We can therefore say that this "two-beat" additive series 1, 1, 2, 3, 5, 8, 13, 21,
This is an effort from the researcher for anyone interest by the Golden Ratio20190425 28065 1ywoid5
After studying the unpublished sketches of the Spanish artist Pablo Palazuelo, this essay proposes the existence of geometric patterns that order his designs. To illustrate the proposal, this paper selects three representative projects by Palazuelo that attempt to demonstrate the presence of a working process related to a mathematical substrate. Specifically, the use of an irregular tiling is proposed for the creation of the following architectural elements: the scenery for a composition by Kandinsky — , and the plan for a hotel on Princesa Street, Madrid, This study also examines the use of regular tessellations in the design of the ceilings of the Huarte Residence in Madrid, The treatises by Bourgoin and Ghyka have been selected from the library of Palazuelo as the primary theoretical bases of this study. This methodology constitutes a line of further research to analyze different architectural projects. This is a preview of subscription content, access via your institution.
The Geometry of Art and Life. Is everything chaos and chance, or is there order, harmony, and proportion in human life, nature, and the finest art? Can one find a natural aesthetic that corresponds to a universal order? If so, what importance can it have for the scientist, artist, or layman?
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