There are, of course, constructions already established for many shapes using the golden ratio (or the number 1.618) but it’s important, I think, for designers to draw these for themselves. I’ve actually been able to design my own constructions of various shapes and definitely understand a lot more about proportion in doing so.
This construction is of proportionally equal circles (reducing them in size by 1.618). The challenge is to do it all without measuring tools or calculators. Only a ruler’s edge and compass should be used; this way, you focus on the shapes and lines themselves rather than numbers.
I actually thought I‘d found a successful method in the drawing below but when I tried it digitally it was actually incorrect. BUT, in taking screenshots just now, I’ve figured it out!
The problem with drawing things out by hand is that your scale will be too small to check if a tiny miss-match of points is due to your compass or if it’s actually an error. Only on a digital version will you be able to be pin-point accurate and if your construction works it would be pin-point accurate on every point.
CONSTRUCTION: I started with a circle and protruding square. Then I extended lines from the centre of the circle to the corner and first quarter of the square. I then thought the two red dots (where the lines hit the circumference of the circle) would be the radius of a smaller circle (proportional to the larger). I was wrong though and, as the fourth image shows, the green circle (which would have been the correct outcome) is slightly smaller. The difference is miniscule but this is only because it’s a small scale. So, larger scale=big difference. As I said though, I’ve just figured out the correct radius to use, as shown in the last image. So one point is on the circumference and quarter into the square, and the other is defined solely by the square.
Often these constructions have relatively simple outcomes but only in drawing your way through them can you really understand it all.