On Division in Extreme and Mean Ratio and its Connection to a Particular Re-Expression of the Golden Quadratic Equation x 2 − x − 1 = 0

The golden quadratic x 2 − x − 1 = 0, when re-expressed as (x)(1) = 1/(x − 1), x = 1.618, can be interpreted as the algebraic expression of division in extreme and mean ratio (DEMR) of a line of length x = 1.618 into a longer section of length 1 and a smaller of length (x − 1). It can, however, also be interpreted as the formulation of the area of a golden rectangle of sides x = 1.618 and 1, and as the system of equations constituted by y = x, and y = 1/(x − 1). Based on the well-known connection existing between the first two of these interpretations, the authors address the problem of finding out the thread connecting the golden rectangle with the system of equations referred to above. The results obtained indicate first that this system, like the golden rectangle, also carries in its geometry the essential traits of DEMR; and, second, that it implicitly subsumes the simpler rectangular geometry of its alternative interpretation. The process of developing these connections brought forward a heretofore apparently unreported golden trapezoid of sides Φ, 1, ϕ and \(\sqrt{2}\).