Mathematics & Art - Rahul Rajendran - XB -
The
Golden Rectangle is considered to be one of the most pleasing and beautiful
shapes to look at, which is why many artists have used it in their work.
The
two artists, who are perhaps the most famous for their use of the golden ratio,
are Leonardo Da Vinci and Piet Mondrian.
It
can be found in art and architecture of ancient Greece and Rome, in works of
the Renaissance period, through to modern art of the 20th Century. The Golden
Rectangles present in the following Figures, are quite obvious. However,
various features of the Mona Lisa have Golden proportions, too.
The
Parthenon was perhaps the best example of a mathematical approach to art.
Further
classic subdivisions of the rectangle align perfectly with major architectural
features of the structure.
Art
Master Piece: Mona Lisa
Mona Lisa, is one of the most famous paintings
in the world, and is a very good example of Da Vinci's use of the golden ratio
in art.
If
you draw a rectangle around Mona Lisa's face, that rectangle will turn out to
be golden. The dimensions of the painting itself also form a golden rectangle.
As well, the proportions of Mona Lisa's body exhibit several golden ratios. For
example, a golden rectangle can be drawn from her neck to just above the hands.
According
to one art expert, Seurat "attacked every canvas by the golden
section". His Bathers has obvious golden subdivisions.
FIG
4 BATHERS
Piet
Mondarian
Piet
Mondrian is a modern Dutch artist, who lived in 1872 - 1944. Although at
the beginning of his career, Mondrian painted many landscapes, he later on
moved to an abstract style in his work. Mondrian is famous for using horizontal
and vertical black lines as the basis for a lot of his paintings. Like Da
Vinci, Mondrian believed that mathematics and art were closely connected. He
used the simplest geometrical shapes and primary colours (blue, red, yellow) to
express reality, nature and logic from a different point of view. (Mondrian's
point of view lies in the fact that any shape is possible to create with basic
geometric shapes as well as any colour can be created with different
combinations of red, blue, and yellow) The Golden Rectangle is one of the basic
shapes, which keeps appearing in Mondrian's art:
Mondrian painted the following compositions in
Red, Yellow and blue in 1942 and in 1926. There are many golden rectangles in
this work.
Fig.5
Mondarian (1942)
Fig.6 Mondarian (1926)
The
more recent search for a grammar of art inevitably led to the use of the golden
section in abstract art. La Parade, painted in the characteristic multi-dotted
style of the French neo-impressionist Seurat (1859-1891), contains numerous
examples of golden proportions.
Fig.7 Georges Seurat, La Parade
Leonardo da Vinci (1451-1519). Leonardo
had for a long time displayed an ardent interest in the mathematics of art and
nature. He had earlier, like Pythagoras, made a close study of the human figure
and had shown how all its different parts were related by the golden section.
Fig.8
Study of Human Proportions Fig.9
Study of Facial
According to Vitruvious
One of the strongest advocates for the application
of the Golden Ratio to art and architecture was the famous Swiss-French
architect and Painter Le Corbusier (Charles-Edouard Jeanneret, 1887- 1965).
Originally,
Le Corbusier expressed rather skeptical, and even negative, views of the
application of the Golden ratio to art, warning against the “replacement of the
mysticism of the sensibility by the Golden Section.”In fact, a thorough
analysis of Le Corbusier’s architectural designs and “Purists” paintings by
Roger Herz-Fischler shows that prior to 1927, Le Corbusier never used the
Golden ratio. This situation changed dramatically following the publication of
Matila Ghyka’s influential book Aesthetics of Proportions in Nature and in the Arts, and his Golden Number,
Pythagorean Rites and Rhythms (1931) only enhanced the aspects of Φ
even further.
Le
Corbusier’s fascination with Aesthetics and with the Golden Ratio had two
origins. On one hand, it was a consequence of his interest in basic forms and
structures underlying natural phenomenon. On the other, coming from a family
that encouraged musical education, Le Corbusier could appreciate that
Pythagorean craving for a harmony achieved by number ratios. He wrote: “More
than these thirty years past, the sap of mathematics has flown through the veins
of my work, both as an architect and painter; for music is always present
within me.” Le Corbusier’s search for a standardized proportion culminated in
the introduction of a new proportional system called the “Modulor.”
The
Modulor was supposed to provide “a harmonic measure to the human scale,
universally applicable to architecture and mechanics.” In the spirit of
Vitruvian man and the general philosophical commitment to discover a proportion
system equivalent to that of natural creation, the Modulor was based on human
proportions.
A
six-foot (about 183-centimeter) man, somewhat resembling the familiar logo of
the “Michelin man,” with his arm upraised (to a height of 226 cm; 7’5”), was
inserted into a square. The ratio of the height of the man (183 cm; 6’) to the
height of his navel (at the mid-point of 113 cm; 3’8.5”) was taken precisely in
a Golden Ratio.
Modulor
man
The
total height (from the feet to the raised arm) was also divided in a Golden
ratio (into 140cm and 86 cm) at the level of the wrist of a downward-hanging
arm. The two ratios (113/70) and
(140/86) were further subdivided into smaller dimensions according to the
Fibonacci series
Le
Corbusier developed the Modulor between 1943 and 1955 in an era which was
already displaying widespread fascination with mathematics as a potential
source of universal truths. In the late 1940s Rudolf Wittkower's research into
proportional systems in Renaissance architecture began to be widely published
and reviewed. In 1951 the Milan Triennale organized the first international
meeting on Divine Proportions and appointed Le Corbusier to chair the group. On
a more prosaic level, the metric system in Europe was creating a range of
communication problems between architects, engineers and craftspeople. At the
same time, governments around the industrialized world had identified the lack
of dimensional standardization as a serious impediment to efficiency in the
building industry. In this environment, where an almost Platonic veneration of
systems of mathematical proportion combined with the practical need for systems
of coordinated dimensioning, the Modulor was born.
Golden Ratio in Music
In addition to existing in nature, art and
architecture, it has been hypothesized that great classical composers like
Mozart had an awareness of the Golden Ratio and used it to compose some of his
famous sonatas. Surprisingly, the Golden Ratio appears in a couple of different
aspects of music. It appears in particular intervals in the western diatonic
scale as well as the arrangement of a piece of music itself. In order to
clearly understand the relationship between the Golden Mean and music, it helps
to have a working knowledge of the fundamentals of western music theory
and a basis knowledge of sound.
The Golden Ratio appears in the relationship
of the intervals or distance between the notes. Each of these intervals or note
pairs creates either a tonic (consonant) sound or a dissonant sound, in which
the listener desires to hear it followed by a tonic sound to
"resolve" the tension created by its unstable quality. The
interesting part of this, according to H.E. Huntley, author of the Divine Proportion,
is that it is a relationship of the consonance and dissonance of the rhythmic
"beats" that occur in the sound waves of the resonant frequencies
between notes in the diatonic scale.
Huntley
goes on to explain that the reason that we prefer visual aspects of a Golden
Rectangle over a perfect square is measured in the amount of time it takes for
the human eye to travel within its borders. This period of time is in same
proportion (Phi) to the beats that exist in specific musical intervals. Unison
(two notes of the same frequency being played simultaneously) is said to be the
most consonant, having a rhythmic quality that is similar to the time interval
that is perceived by the eye when viewing a perfect square.
The
octave has a similar consonant quality that could be represented visually by
two squares of equal size. A correlation could be made between the consonant
properties of the interval of an octave to the first two squares in the golden
rectangle or the first two numbers in the Fibonacci sequence which are
represented as 1,1. From there the relationship reflects a ratio of 8:5 in the
interval of a Major 6th (an approximate Golden Ratio of 1.6), in the first and
sixth notes of a diatonic (Major) scale. The ratio of this interval is related
by the rhythmic beats that are created by the respective frequencies of the
sound waves and interpreted as sound in the human ear. Suffice it to say, that
the interval of a Major 6th is supposed to be the most aesthetically pleasing
since it contains the golden ratio.
Musical scales are based on Fibonacci numbers
Here are 13
notes in the span of any note through its octave.
A scale is comprised of 8 notes, of which the
5th and 3rd notes create the basic foundation
of all chords, and are based on whole tone which is
2 steps from the root tone, that is the
1st note
of the scale.Note too how the piano keyboard of C to C above of 13 keys has 8
white keys and 5 black keys, split into groups of 3 and 2.
Another
aspect of the golden ratio in music is illustrated in compositions by Mozart.
Mozart's piano sonatas have been observed to display use of the golden ratio
through the arrangement of sections of measures that make up the whole of the
piece. In Mozart's time, piano sonatas were made up of two sections, the exposition
and the recapitulation. In a one hundred measure composition it has been noted
that Mozart divided the sections between the thirty-eighth and sixty-second
measures. This is the closest approximation that can be made to the Golden
Ratio within the confines of a one-hundred measure composition. Some scholars
have debunked this theory since further analyses of such compositions have
shown that the Golden Ratio was not consistently applied within the subsections
of the same compositions. Others state that this does not prove that he did not
utilize the Golden Ratio, only that he did not apply it to all aspects of
particular compositions. Whether he applied the Golden Ratio intentionally or
used it intuitively is not known but studies seem to indicate the latter.
The
Golden Ratio has also appeared in poetry in much the same way that it appears
in music. The emphasis has been placed on time intervals. Some have even stated
that the meaning of chosen words is less important than its rhythmic quality
and the intervals between words and lines that serve to create the overall
rhythm of a poem.
Probably
the most compelling display of the Golden Ratio is in the many examples seen in
nature. The Golden Ratio and the Fibonacci sequence can be seen in objects from
the human body to the growth pattern of a chambered nautilus. Examples of the
Fibonacci sequence can be seen in the growth pattern of a tree branch or the
packing pattern of seeds on a flower. Ultimately, this aspect is what has
earned the Golden Mean its representation as the Divine Proportion
It
is the prevalence of the Golden Ratio in nature that has influenced classic art
and architecture. The great masters developed their skills by recreating things
they observed in nature. In the earliest of cases, these artists and craftsmen
probably had no knowledge of the math involved, only an acute awareness of this
pattern repeated around them. It was the mathematicians that unlocked the
Secrets
of the Golden Ratio. Their work has led to the understanding of the complex
mathematical underpinnings hidden within the Golden Mean.
Musical instruments are often based on phi
Fibonacci
and phi are used in the design of violins and even in the design of high
quality speaker wire.
