René Descartes La Geometrie (french / english ) is an historically important work since it signified one of the early attempts at uniting geometry and algebra but also because it’s an instructive read on how you can have a different perspective on the nature of numbers.
In the actual text he starts out by characterizing addition, multiplication, and taking square roots as geometric operations and lines and it is from this last operation that I will derive this post. To find (construct) a square root (radical) of a numbers represented as a length proportional to another unit length yo can follows a rather simple procedure. I have sketched it here
It’s not an original construction of course. It can be found in Euclid (Book VI Prop 13) but it’s a nice concrete representation to have if you somehow doubt that you can truly take square roots of any (positive) real number and not just those which occur in certain right triangles. Still what I was really looking for when I was working on this was a way of providing a visual construction which in a transparent way relates proportions of lengths and proportions of area. Since I can take the square root in this construction as the side of a square with corresponding are I get the following illustrations
From this we might observe a sight curiousity in how the square being for 1 and 2 not entirely contained in the half-circle but which for some value in between 2 and 3 ‘slips into the circle’. Now for what number does this happen. I.e for what proportions do we get the following case?
If is the side of the square then from the construction we know that and dammit it’s the equation for the golden ratio . Explicitely
Might as well highlight the golden rectangle.
Of course I should have seen this coming since this is indeed one of the known ways you can construct golden rectangles and ratios but still, it was kind of funny.