Root rectangles obtained outside a square: In the illustration below, rectangle ABEF is a "roottwo" rectangle, which we developed as Figure I5, above, the side AF divided by the side AB will equal a ratio of 1.4142 which is the square root of 2. Whatever the dimensions of the "root" rectangles their individual ratios will remain constant. Roottwo rectangle Ratio: 1.4142 Rootthree rectangle Ratio: 1.732 Rootfour rectangle Ratio: 2.000 Rootfive rectangle Ratio: 2.236 Figure I10. "Root" rectangles obtained from outside of a square.
"Root" rectangles obtained inside a square: The "root" rectangles constructed below (Figure I11) are within a square (or "unity"), by essentially the same simple geometrical method as for the outside the square. Arc AB (the dotted line) is a quadrant arc with the center D and the radius line DB. DC is a diagonal to the square ACBD, and it cuts the quadrant arc at F. A line, parallel to the top of the square, is drawn through F. This line determines an internal "roottwo" rectangle and DE forms its diagonal. Next, the diagonal to the internal "roottwo" rectangle cuts the quadrant arc at H. Then, DG becomes a "rootthree" rectangle, the diagonal of which cuts the quadrant arc at J. DI is a "rootfour' rectangle and its diagonal cuts the quadrant arc at L. DK is a "rootfive" rectangle and its diagonal cuts the quadrant arc at M. All the "root" rectangles may be thus obtained within a square.
The development of "rootfive" rectangle themes All "root" rectangles have inherent themes from which further design and compositions can be developed. The literature on dynamic symmetry contains much information concerning this, and includes the mathematical ratios that accompany them. This is interesting to know, but it is not necessary to describe these various ratios/themes in our present icon composition analysis. In our later icon analyses, use is made of the simple geometry of the rectangles and their design lines, which we will discuss in further paragraphs. Below are some examples of design themes and forms of the "rootfive" rectangle. As stated above, each "root" rectangle has these inherent themes and as the size of the rectangle changes the themes change subtly with it  that is, the same design within a "roottwo" rectangle will accommodate a different pattern than the same design theme within a "rootfive" rectangle. In Figure I12 below, it can be seen how geometric detail can be created within a "rootfive" rectangle. Notice the arrangement and detail development possible in the subordinate rectangles where every rectangle within is a "rootfive" rectangle repeatedly divided and subdivided down into smaller "rootfive" rectangles. (Our compass arcs have been omitted for clarity)
In Figure I13 below, is represented the theme of dividing into repeatedly smaller halves within a "rootfive" rectangle. These smaller and smaller segments help form different designs within the rectangle. (Again, we have omitted the compass arcs for the sake of clarity)
From some of these simple themes in the "root" rectangles you may recognize, later, the bases for the design of Byzantine and Russian icons that have survived from the distant past. In Figure I14 below, is demonstrated the "rootfive" rectangle divided into three equal sections vertically. This is known as the vertical triad position within a "root" rectangle. Notice how this arrangement results in another wellproportioned rectangle overall. The icon, The Entombment, which will be discussed in detail in this paper, is designed within a "roottwo" rectangle but in a horizontal triad position. (See Figure II3)
