Aesthetics
History of aesthetics (pre-20th-century)
De Divina Proportione, a three-volume work by Luca Pacioli,
was published in 1509. Pacioli, a Franciscan
friar,
was known mostly as a mathematician, but he was also trained and keenly
interested in art. De Divina Proportione explored the mathematics of the
golden ratio. Though it is often said that Pacioli advocated the golden ratio's
application to yield pleasing, harmonious proportions, Livio points out that
the interpretation has been traced to an error in 1799, and that Pacioli
actually advocated the Vitruvian system of rational proportions.Pacioli also saw Catholic religious significance in the ratio, which led to his
work's title. De Divina Proportione contains illustrations of regular
solids by Leonardo da Vinci, Pacioli's longtime friend and collaborator.
Architecture
The Parthenon's façade as well as
elements of its façade and elsewhere are said by some to be circumscribed by
golden rectangles.Other scholars deny that the Greeks had any aesthetic association with golden
ratio. For example, Midhat J. Gazalé says, "It was not until Euclid,
however, that the golden ratio's mathematical properties were studied. In the Elements
(308 BC) the Greek mathematician merely regarded that number as an interesting
irrational number, in connection with the middle and extreme ratios. Its
occurrence in regular pentagons and decagons was duly observed, as well as in
the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed
exemplary that the great Euclid, contrary to generations of mystics who
followed, would soberly treat that number for what it is, without attaching to
it other than its factual properties."[23]
And Keith Devlin says, "Certainly, the oft repeated assertion that the
Parthenon in Athens is based on the golden ratio is not supported by actual
measurements. In fact, the entire story about the Greeks and golden ratio seems
to be without foundation. The one thing we know for sure is that Euclid, in his
famous textbook Elements, written around 300 BC, showed how to calculate
its value."[24]
Near-contemporary sources like Vitruvius
exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate
as opposed to irrational proportions.
A 2004 geometrical analysis of
earlier research into the Great Mosque of Kairouan reveals a consistent application of the golden ratio
throughout the design, according to Boussora and Mazouz.They found ratios close to the golden ratio in the overall proportion of the
plan and in the dimensioning of the prayer space, the court, and the minaret.
The authors note, however, that the areas where ratios close to the golden
ratio were found are not part of the original construction, and theorize that
these elements were added in a reconstruction.
The Swiss architect
Le Corbusier,
famous for his contributions to the modern
international style,
centered his design philosophy on systems of harmony and proportion. Le
Corbusier's faith in the mathematical order of the universe was closely bound
to the golden ratio and the Fibonacci series, which he described as
"rhythms apparent to the eye and clear in their relations with one
another. And these rhythms are at the very root of human activities. They
resound in man by an organic inevitability, the same fine inevitability which
causes the tracing out of the Golden Section by children, old men, savages and
the learned."
Le Corbusier explicitly used the
golden ratio in his Modulor system for the scale
of architectural
proportion. He saw this system as a
continuation of the long tradition of Vitruvius,
Leonardo da Vinci's "Vitruvian Man",
the work of Leon Battista Alberti, and others who used the proportions of the human body to
improve the appearance and function of architecture.
In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden
ratio in human proportions to an extreme: he sectioned his model human body's
height at the navel with the two sections in golden ratio, then subdivided
those sections in golden ratio at the knees and throat; he used these golden
ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches
exemplified the Modulor system's application. The villa's rectangular ground
plan, elevation, and inner structure closely approximate golden rectangles.
Another Swiss architect, Mario Botta,
bases many of his designs on geometric figures. Several private houses he
designed in Switzerland are composed of squares and circles, cubes and
cylinders. In a house he designed in Origlio,
the golden ratio is the proportion between the central section and the side
sections of the house.
In a recent book, author Jason
Elliot speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.
Painting
The drawing of a man's body in a
pentagram suggests relationships to the golden ratio.
The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram
inside a circle, implying a relationship to the golden ratio.
Leonardo da Vinci's illustrations of polyhedra
in De
divina proportione (On
the Divine Proportion) and his views that some bodily proportions exhibit
the golden ratio have led some scholars to speculate that he incorporated the
golden ratio in his paintings.
But the suggestion that his Mona Lisa,
for example, employs golden ratio proportions, is not supported by anything in
Leonardo's own writings.
Similarly, although the Vitruvian Man is often shown in connection with the golden
ratio, the proportions of the figure do not actually match it, and the text
only mentions whole number ratios.
Salvador Dalí, influenced by the works of Matila Ghyka,[35] explicitly used the golden ratio in
his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A
huge dodecahedron, in perspective so that edges appear in golden ratio to one
another, is suspended above and behind Jesus and dominates the composition.
Mondrian has been said to have used the golden section extensively
in his geometrical paintings, though other experts (including critic
Yve-Alain Bois) have disputed this claim.
A statistical study on 565 works of
art of different great painters, performed in 1999, found that these artists
had not used the golden ratio in the size of their canvases. The study concluded
that the average ratio of the two sides of the paintings studied is 1.34, with
averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).On the other hand, Pablo Tosto listed over 350 works by well-known artists,
including more than 100 which have canvasses with golden rectangle and root-5
proportions, and others with proportions like root-2, 3, 4, and 6.
Book design
Depiction of the proportions in a
medieval manuscript. According to Jan Tschichold:
"Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned
in the Golden Section."
There was a time when deviations
from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section
were rare. Many books produced between 1550 and 1770 show these proportions
exactly, to within half a millimeter.
Industrial design
Some sources claim that the golden
ratio is commonly used in everyday design, for example in the shapes of
postcards, playing cards, posters, wide-screen televisions, photographs, and
light switch plates.
Music
Ernő Lendvaï
analyzes Béla Bartók's works as being based on two opposing systems, that of the
golden ratio and the acoustic scale
though other music scholars reject that analysis.
In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals
1:2:3:5:8:5:3:2:1.French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries
de la Rose+Croix. The golden ratio is also apparent in the organization of
the sections in the music of Debussy's
Reflets dans l'eau (Reflections in Water),
from Images (1st series, 1905), in which "the sequence of keys is
marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the
phi position."
The musicologist Roy Howat
has observed that the formal boundaries of La Mer
correspond exactly to the golden section.Trezise finds the intrinsic evidence "remarkable," but cautions that
no written or reported evidence suggests that Debussy consciously sought such
proportions.
Pearl Drums positions the air vents on its Masters Premium models based
on the golden ratio. The company claims that this arrangement improves bass
response and has applied for a patent on this innovation.
Though Heinz Bohlen
proposed the non-octave-repeating 833 cents scale
based on combination tones, the tuning features relations based on the golden ratio.
As a musical interval the ratio 1.618... is 833.090... cents (
Play (help·info)).
Nature
Adolf Zeising, whose main interests were mathematics and philosophy,
found the golden ratio expressed in the arrangement of branches along the stems
of plants and of veins in leaves. He extended his research to the skeletons of
animals and the branchings of their veins and nerves, to the proportions of
chemical compounds and the geometry of crystals, even to the use of proportion
in artistic endeavors. In these phenomena he saw the golden ratio operating as
a universal law.
In connection with his scheme for golden-ratio-based human body proportions,
Zeising wrote in 1854 of a universal law "in which is contained the
ground-principle of all formative striving for beauty and completeness in the
realms of both nature and art, and which permeates, as a paramount spiritual
ideal, all structures, forms and proportions, whether cosmic or individual,
organic or inorganic, acoustic or optical; which finds its fullest realization,
however, in the human form."
In 2010, the journal Science
reported that the golden ratio is present at the atomic scale in the magnetic
resonance of spins in cobalt niobate crystals.
However, some have argued that many
of the apparent manifestations of the golden mean in nature, especially in
regard to animal dimensions, are in fact fictitious.
Optimization
Perceptual studies
Studies by psychologists, starting
with Fechner,
have been devised to test the idea that the golden ratio plays a role in human
perception of beauty. While Fechner found a preference for rectangle ratios
centered on the golden ratio, later attempts to carefully test such a
hypothesis have been, at best, inconclusive.
.jpg){kind=link}
No comments:
Post a Comment