Associated posts

The portion of the spiral in each square looks like a quarter of a circle. How well would circular arcs approximate the spiral?

The previous post included this image of a logarithm spiral passing through the corners of squares in a series of golden rectangular shapes.

For aesthetic applications, circular arcs are excellent enough and most likely simpler to work with. For a sort of three-dimensional analog of this approximation, see this post on the geometry of the Sydney Opera House.

Extremely well. Heres a plot.

The circular arc inside the blue square is outlined in blue, the arc inside the green box in green, and so on. The logarithmic spiral is plotted on leading with a dashed black line. You need to focus closely to see any distinction between the logarithmic spiral and its circular approximations.

The approximation of the logarithmic spiral by a sequence of quarter circles has a dives in curvature, which is roughly a 2nd derivative. Here its leaping from one favorable value (the mutual of the circle radius) to another favorable worth (the mutual of a smaller radius).