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Golden Proportion
The problem that follows gives an overview of the mathematics involved in demonstrating the beauty of the "Golden Proportion" and indicates how the application of that principle makes sense in determining subject placement in any rectangular work of art. Those not interested in the mathematical interpretation should skip the solution and simply refer to the conclusion.
PROBLEM: Consider a line segment with a length of one. Divide this segment into two segments, a shorter segment, x, and a longer segment, 1-x. The "Golden Proportion" is the one where the ratio of the shorter segment to the longer segment (x : 1-x) is equal to the ratio of the longer segment to the whole segment (1-x : 1).
The mathematical equation that needs to be solved for x is:
x/(1-x)=(1-x)/1
Without going into a full blown solution to the equation, and since my keyboard does not have a square root key, let me simplify things by indicating that the only place to divide the segment of length one is such that the shorter segment is of length .382 and the longer segment is of length .618. It is easy to note the truth of the matter by substituting those numbers for x and 1-x.
(.382/.618)=.618/1
Sure enough, this is true. Try adjusting these values up and down a bit and notice that the two ratios are never equal. The only ratios that are equal are the ones described above. A proportion, defined as two ratios being equal to one another, only exists for these figures. Since this is a universal truth the Greeks referred to this proportion as being "divine" or "golden".
The "Golden Proportion" formed the basis for many of man's creations. The rectangle is a basic form used in art, and the Greeks believed that the most "beautiful" rectangle must therefore utilize the ratio of length to width found in that proportion. Hence, the "Golden Rectangle" with length one, and width .618 exists.
Draw a diagonal within the "Golden Rectangle." Note that the interior is divided into 2 congruent triangles, each with congruent corresponding angles. The ratio of the short side to the long side of each triangle is .618 and the ratio of the long side to the short side is 1.618. These numbers, pretty familiar by now, indicate that these triangles are beautiful by Greek standards.
Now construct a perpendicular from the far vertex to the diagonal. The interior is now divided into several triangles of different size, but note that each is similar to the other, indicating that their corresponding sides are proportional and their corresponding angles are congruent. In fact all of the triangles are "Golden Triangles."
Draw another perpendicular and more of these beautiful triangles are produced within the interior of the "Golden Rectangle."
Note the points of intersection within the rectangle's interior. Each point, called a "saddle point", indicates a position within the interior that is placed in a spot pleasing to the eye. It pleases us as viewers in an abstract sense because it forces us to recognize the "Golden Proportion" within the framework of the rectangle.
In fact there are 4 such "saddle points" available within the rectangle. Either one of these may be utilized as the location of the primary subject. Any secondary elements of the image should be placed at another saddle point or on a diagonal line that exists between saddle points.
Return to "COMPOSITION"
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