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Golden Section

  Golden Section (or Golden Mean) was the 19th-century name given to a system of mathematical proportion, which had been discussed since ancient Greek times and had been in use for even longer (examples have been found in the planning of Stone Age temples in Britain and in Egyptian pyramids of the 3rd millenium  BCE). The Golden Section is a way of dividing a line so that the ratio of the smaller section is to the larger as that of the larger is to the whole; or a way of dividing a rectangle into two areas in such a way that the ratio of the smaller area is to the larger as that of the larger is to the whole. It can be expressed as: 5:8::8:13 or as a:b::b:c (a + b).

The Golden Section is the only ratio that is also a proportion. This uniqueness led early Greek numerologists (who thought that the entire universe was founded on number relationships) to assign to it a kind of philosophical perfection. The Pythagoreans, for example, were fascinated by the isosceles triangle (in which the base forms a Golden Section with the longer side) and the pentagram, which is formed from three triangles, contains many Golden Sections, and was assumed by them (and by numerologists and black magicians ever since) to have mystical properties. Later writers, including Plato, Omar Khayyám (in his role as astronomer and mathematician) and the 16th-century Italian mathematician Luca Paccioli, devoted much attention to the Golden Section. Their interest was in finding some kind of relationship between its mathematical nature and its assumed philosophical or mystical ‘perfection’: Plato, indeed, assigned to it a kind of moral force, claiming that works of art and buildings using it exerted a calming influence on the beholder.

This may not be as far-fetched as it seems. Sculptors and architects, not only in the West, where it has been taught since Greek times, but throughout the world, have used the Golden Section in works that are regarded as particularly harmonious and pleasing to the eye. In painting, again worldwide but consciously in the West since Renaissance times, it is used to form relationships between different areas of the painting surface. In Japanese landscape painting, for example, the division is often diagonal across the picture area, the ‘upper’ triangular area forming a Golden Section with the ‘lower’. In Western landscape painting the division can be horizontal or vertical. The break between land and sky, for example, or the disposition of a stand of trees, is often so organized. In all such work—and this, perhaps, bears out Plato\'s contention—the effect of the Golden Section is greatest when it is subliminal, when we seem to discover harmonious proportion in the painting for ourselves without analysing it to see how it is achieved.

In a similar manner, musical form was sometimes organized in the Baroque period in Europe to make what were claimed to be the sound equivalents of the Golden Sections (for example, Purcell\'s ‘Golden’ Sonata). Such pieces have a perceptible and carefully organized proportion between the various blocks of sound which comprise each movement\'s form, but since the Golden Section is a visual and not an aural phenomenon, the use of the name in music is purely fanciful. MG PD KMcL

Further reading H.E. Huntley, The Divine Proportion: a Study in Mathematical Beauty.



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