daniele capo

While the economy is going down I started tutoring kids (and not so kids) in math. That’s why something emerged in my mind from my past life: “square root rectangles”. Ok, the definition is ‘dynamic rectangles’, but let me call √2-rectangle a rectangle in which the width/height ratio is √2 and, more generally, √n-rectangle is the rectangle with the width/height ratio equal to √n.

You need to start the construction with a square (or, if you want a √1-rectangle w/h=1). As you know (no?) for the Pythagorean theorem the diagonal of a rectangle is √(w²+h²) which, for a square (and assuming the side as one ‘unit’), is √2, and you can now draw a rectangle with h=1 and w=√2: a √2-rectangle.

Similarly the diagonal of a rectangle with h=1 and w=√2 is √(1+2)=√3, and you have the √3-rectangle, and so on.

Given that construction you can find rectangles with side 2, 3, 4, 5, etc. (√4, √9, √16, √25 …). Isn’t that a really nice thing?

Be aware, you cannot find the infamous golden ratio that way. But you can if you follow me.

Draw a square ABCD, find its center O. Ok. Then, draw a circle centered in O, passing from A, B, C and D. The diameter of that circle is √2 and you can draw a √2-rectangle with centered on the square. The sides of the √2-rectangles falling outside the square measure (√2 - 1)/2 (right? you have half rectangle minus half square).

You can repeat the same construction, finding √3, √4, √5, etc. WAIT! √5 is what we are looking for. The sides of √5-rectangle falling outside the square (as you already know) measure (√5 - 1)/2=1/ϕ. And this is a golden rectangle! Also, notice that, for the definition of golden section if you add the square to that rectangle you have another golden one. In the figure you see the vertical golden rectangle (red), the square (orange), the big horizontal golden rectangle (red+orange) and in magenta+red you find another square.

In the next post I will try to imagine how to use it.