MesMath

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.