Golden rectangle

His interest in the mathematics of art and nature led him to show how the different parts of the human body are related by the golden rectangle. Leonardo believed that artists should know the laws of nature as well as the rules of perspective. 

The cover drawing shows the set of calculated general covalence curves, in dimensionless units, with an empirical reconstruction, as circular segments, within a golden rectangle. The absolute limit to covalent interaction is... 

Because of their elements of five-fold symmetry the two large polyhedra have a close relationship with the golden ratio. For instance, three mutually perpendicular and interpenetrating golden rectangles define the vertices of an icosahedron. 

In the case of a cube it is noted that two perpendicular golden rectangles can be inscribed in a square such that each vertex divides a side of the square in the ratio I t. The edge of an inscribed tetrahedron AA) is divided by the rectangles in the ratios AC/AD = BD/AD = t. 

Bode -Titius law. The orbital radii of planets and moons in the solar system are characterized correctly by divergence angles of n7r/5 tit) on the spiral that hts into a golden rectangle, as shown diagrammatically in Figure 7.3 for n= 2. 

To good approximation, the side lengths of successive gnomons decrease in line with the Fibonacci numbers, such that Fibonacci squares cover the composite golden rectangle, e.g. 

Fig. 3 A golden spind inscribed in a golden rectangle defined by a Fibonacci sequence. The increasing size of successive squares is defined by the Fibonacci labels ( ), and the ratio of their side lengths approaches r as —> oo. The inscribed spiral approximates an equiangular logeirithmic spiral [4]...

Fig. 6 A sequence of Fibonacci squares on a scale of 1 2 serves to generate the 21 cm X 13 cm golden rectangle with its inscribed spiral. Direcily measurable radii of n a at convergence angles of 4jr/(2n — 1) terminate at the labeled points...

Fig. 1 Points generated in a golden rectangle by a Fibonacci spiral with a variable convergence angle of 4jt/ 2n — 1). Numbers indicate the distance to the spiral center in units of oq...

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