Sequence of Golden Rectangles/Equiangular Spiral

Theorem

A golden rectangle can be divided into a square and another golden rectangle.

That golden rectangle can in turn be divided into a square and another golden rectangle.

The sequence can be continued indefinitely.

The points where the vertices of successive squares of this sequence meet can be joined together by an equiangular spiral.

This equiangular spiral can be approximated by quarter circles constructed as shown.

The equiangular spiral tends towards the point of intersection of the diagonals of the golden rectangles.

Proof

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Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$