The Golden Rectangle — Cut a Square, Get Another (φ)
Cut a square off a golden rectangle and what remains is another golden rectangle. Repeat and a golden spiral winds inward — the ratio φ that nature and art adore.
Do not write an equation yet. Shake the figure first.
Remove a square, then another
The remainder shrinks but copies the same φ aspect ratio again and again.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
A golden rectangle: side ratio φ:1 (≈1.618:1).
Return to the exact question
Show the self-similarity of a golden rectangle (ratio φ:1) and explain φ = (1+√5)/2.
Why this approach works
A golden rectangle has side ratio φ ≈ 1.618. Cut off the square on the short side and the remainder is again golden — repeat and ever-smaller golden rectangles spiral to a point, the quarter-circles forming the golden spiral. Self-similarity is φ’s very definition: φ − 1 = 1/φ, i.e. φ² = φ + 1, giving φ = (1+√5)/2. The ratios of consecutive Fibonacci numbers (1,1,2,3,5,8,…) tend to φ.
Proof
Removing a square leaves a rectangle of the same ratio (self-similar). The condition φ−1=1/φ gives φ²=φ+1, so φ=(1+√5)/2; the quarter-circles form the golden spiral.
Try it yourself
Use Fibonacci sides: 144×89, then 89×55, then 55×34 … each ratio ≈ 1.618, approaching φ.
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