The rectangle of the Whirling Squares: Hambidge called the following rectangle illustrated in Figure I15 below, the rectangle of the whirling squares because the squares within it are constructed whirling around a central axis, or pole. This rectangle is constructed, not from the diagonal of a square as you have done in the past 3 exercises in dynamic symmetry, but from the halfdiagonal of a square.
For construction, draw a square, ACDB and at the halfway point of line BD, X, draw a diagonal line, XC. With a compass point at X, and the compass marker at C, bring the curved line down to the base line of BD. Extend lines AC and BD to the edge of the curved line. Draw line FE enclosing the arc, CF. This is what Hambidge calls a whirling squares rectangle, and he also refers to it as a ''1.618 rectangle'' because that is its ratio. This is also the basis of the Greek "Golden Rectangle." Within the basic whirling square rectangle, above, whirling squares can be constructed (See Figure I16 below). You will note the diminishing squares whirling around the axis, at X. Squares number 1 through 6 are each in a series of diminishingsized squares. This rectangle is usually abbreviated as the "WS" rectangle (whirling square). Notice particularly that located on the line BE at D is the center of the whirl that the squares spiral around.
As an interesting aside, this whirling squares rectangle with its ratio of 1.618 contains all of the functions of the progression of the mathematical biological science system known as phyllotaxis. Phyllotaxis itself is based on a series of mathematical progressions known as the Fibonacci Series and also known as the Summation Series. This term is apt because the ratios in dynamic symmetry and the mathematical series are developed from the sums of their last components (See formula below). The whirling squares rectangle and the following spiral composed within it are the visual geometrical renderings of the mathematical progression series. (See Figure I17 below) Almost incredibly, these correspond to the whirl of a seashell, the heart of a sunflower and the way leaves grow on a stem, among countless similar biological manifestations.
The mathematical progressive spiral can be obtained from the whirling squares rectangle simply by drawing a quadrant arc at each of the squares that swirl around the whirling squares axis (or pole). Again, this spiral dynamic ratio corresponds to the biological ratios found in nature. This visual "root" progression scheme of 1.618, which nature uses in its construction of form in the plant world is, as stated above, known in biology as phyllotaxis; in mathematics as the Fibonacci Series and the Summation Series, and can also be constructed in the dynamic symmetry system with the use of the whirling squares rectangle. The above progression system is socalled because the succeeding terms of the system are obtained by the sum of the two preceding terms, beginning with the lowest whole number: 0+1=1; 1+1=2; 1+2=3; 2+3=5; .... Thus the summation series yields 1, 2, 3, 5, 8, 13, 21, 34, etc. Dynamic symmetry as a geometrical and mathematical progression is also constructed with these ratios, but for this method of icon analysis, we do not need to use the mathematical formulas, or formal geometric proofs. The rectangle below is formed from the rectangle of the whirling squares. To the WS rectangle is added another "reciprocal" of .618 on the opposite side of the square (or "unity") in the center. Rectangles GASH and CEFD are reciprocals of the square ACDB. The reciprocal of a rectangle is a figure similar in shape to the major rectangle but smaller in size. The reciprocal of any number is obtained by dividing that number into 1 or unity. In the case below, CEFD and GAHB are reciprocals of the square ACDB. A reciprocal is always smaller in size or number than its "parent.'"
This rectangle has the property of added design features. Its base is the diameter of a semicircle. GEFH is a "rootfive" rectangle (ratio 2.236) and the two overlapping whirling squares rectangles are GCDH and AEFB with 1.618 ratio each. The two reciprocals of .618 each plus the square of 1 add up to 2.236, the "rootfive" rectangle. These figures and their interesting ratios need not concern us with their mathematical aspects, but they do lend themselves to rigid proofs should the student care to test them. The foregoing is but a small example of the almost endless number of geometric arrangements which can be made using "root" rectangles. As will be shown in the following paragraphs, these arrangements not only surprisingly underlie many of the world's most pleasing icons, they also are found in other arts, crafts, architecture and especially, and most surprising of all, are widespread in nature itself. Now let us turn to the analysis of five Byzantine and Russian icons of the fourteenth to sixteenth centuries and also to a review of perspective as it deals with the unusual form of Byzantine iconographic "reverse" perspective which we shall further explore in part III.
