Double Square and the Golden Rectangle

above: Dan Palmateer wrote of this, “it just hit me that the conjunction of the circle to the golden rectangleA rectangle whose sides are in the ratio of the Golden Mean (1.618034) or a Fibonacci approximation to the Golden Mean. existed.”

Here we will continue in the mode of a lesson in Geometry where what is grasped intuitively has to have reason for it to be true. It occurred to me that the square in the top hemisphere is the twin of a square in the lower hemisphere, hence this has a relationship to the double squareA unit rectangle of 1 by 2, with important use for alignment (Carnac), cosmology (Egypt) and tuning theory (Honnecourt Man). rectangle. So one can (1) Make a Double Square and then (2) Find the center and (3) a radius can then draw the out-circle of a double square (see diagram below).

The diagonal from the centre would be the square root of 5 if the top square is seen as two double squares of unit size, that is (4) Identify the units as nested double squares. One can then see (5) a cross within the circle holding 12 squares, but when (6) the root 5 comes down to the right horizontal then the familiar formula (root(5) – 1)/2 = 0.618 so there are many transcendent (not Fibonacci) versions of the Golden meanThe Golden Mean is that unique ratio {1.618034}, relative to ONE {1}, in which its square and reciprocal share the same fractional part {.618034}. It is associated with the synodic period of the planet Venus, which is 8/5 {1.6} of the practical year {365 days}, by approximation. It is a key proportion found in Greco-Roman and later "classical" architecture, and commonly encountered in the forms living bodies take. within in the diagram as shown below.

The in-circle of the cross, radius 2, shows how one can divide that circle into twelve equal portions as with the ZodiacThe 12 constellations through which the sun passes in the solar year of 365.2422 days, matching the twelve squares. The out-circle shows Dan’s insight as eight golden rectangles which, overlap over the four “missing” squares of the 16 square grid, which is a simpler framework for generating this geometry as a Whole.