The proportion that survives a square cut
A golden rectangle is one whose long side is about 1.618 times its short side. Cut off a square with sides as long as the rectangle is wide, and the remainder is another golden rectangle—just turned around. The reason is built into the defining proportion, not a coincidence of drawing.
Why the leftover has the same shape
Call the short side 1 and the long side φ, the conventional symbol for the golden ratio. After removing a 1-by-1 square, the leftover rectangle measures 1 by φ − 1. For it to match the original proportions, its longer-to-shorter ratio must again be φ. That requires 1/(φ − 1) = φ, or φ² = φ + 1. The golden ratio satisfies this relationship, so the cut preserves the shape.
That equation also explains why the process can repeat: each remainder has the same ratio, so another square can be removed, leaving another golden rectangle. The pieces get smaller, but their proportions stay constant.
An endless pattern in a finite rectangle
On a diagram, repeated cuts produce a chain of squares whose sides shrink by the same factor at each step. Curved arcs drawn through those squares can suggest a spiral, though the arcs themselves are not the rectangle’s defining feature; the unchanging ratio is.
In practice, a drawn rectangle can only approximate the golden ratio, and measurements or cuts will be imperfect. But mathematically, the self-similarity is exact: however many times the cut is repeated, the leftover keeps the original proportions.
