"Golden" Rectangle

As was mentioned in our previous pages, the golden section is very widely used in geometry. We begin our travel on geometrical properties of the golden section from the "golden" rectangle, which has the following geometrical definition (Fig.1). The "golden" rectangle is called such rectangle, in which the ratio of the larger side to the smaller one is equal to the golden proportion, that is:

Let's consider the simplest case AB = t and BC = 1.

'Golden' rectangle
Figure 1. "Golden" rectangle.

Let's find now at the line segments AB and DC the points E and F, dividing the corresponding line segments AB and DC in the golden section. It is clear that AE = DF = 1, then we have:

Let's link now the points E and F with the line segment EF and we will call this line segment as the "golden" line. The "golden" line EF divides the "golden" rectangle ABCD into two rectangles AEFD and EBCF. As all sides of the rectangle AEFD are equal among themselves, this rectangle is a square.

Let's consider now the rectangle EBCF. As its larger side BC = 1, and the smaller one it follows from here that their ratio BC: EB = t and, therefore, the rectangle EBCF is the "golden" one! Thus the "golden" line EF divides the initial "golden" rectangle ABCD into the square AEFD and the new "golden" rectangle EBCF.

Let's draw now the diagonals DB and EC of the "golden" rectangle ABCD and EBCF. From a similarity of the triangles ABD, FEC, BCE it follows that the point G divides by the "golden section" both the diagonal DB and the "golden" line EF. Let's draw now the new "golden" line GH in the "golden" rectangle EBCF. It is clear that the "golden" line GH divides the "golden" rectangle EBCF on the square GHCF and the new "golden" rectangle EBHG. Moreover, the point I divides by the golden section the diagonal EC and the side GH. Repeating multiply this procedure we will get an infinite sequence of squares and "golden" rectangles, which converge in a limit to the point O.

Note, that such infinite repetition of the same geometrical figures, that is squares and "golden" rectangles produces no realized aesthetic feeling of harmony and beauty. It is considered, what exactly this circumstance is the cause why many subjects of the rectangular form (match box, burn-in lighter, books, suitcases) frequently have the form of the "golden" rectangle. Later we will tell about application of the "golden" rectangle in the architecture and painting.