Given an Archimedean Spiral, what's the maximal number of squares of maximal size you can pack onto it for the first $n$ turns?
Squares are packed in a way such that each square can touch the spiral with only 2 corner points of the same side. All squares are connected in a chain in those corner points.
Maximal size means that a set of "a bit bigger" squares won't fit to be packed in this way.
Some images I sketched:
The right one is colored each turn in other color ($n=4$). The sides of the squares are around the same size as the distance between the turnings: approximation as I'm not sure how to find the exact size of the side of the maximal squares.
If the colorful sketch is correct: then for first four turns ($n$s), we have: $2, 12, 28, 50$ squares.
Can this be mathematically calculated for some $n$? I have no idea where to start but to explore these approximation sketches I'm drawing. I haven't yet been introduced to spirals or polar coordinate systems so I'm in the dark here. This question just popped in my head when I saw stone tiles arranged as sketch on the left.