How is the golden rectangle formed? - Answers

With an infinite amount of time:start with a squarejoin a square to one sideJoin a square along the edge where the two squares join (it has a side length twice the original squares side length)join another square to the side where this last added square joins the original shape.repeat step 4 (forever)The final (limiting) rectangle will be a golden rectangle with sides in the ratio of 1 : φ.The side lengths of the rectangles as created will be in the ratio:1 : 1 (the initial square)1 : 22 : 3 → 1 : 1.53 : 5 → 1 : 1.666...5 : 8 → 1 : 1.68 : 13 → 1 : 1.62513 : 21 → 1 : 61538....21 : 34 → 1 : 1.619047....34 : 55 → 1 : 1.617647....55 : 89 → 1 : 1.6181818....89 : 144 → 1.61797752.....144 : 233 → 1.6180555....233 : 377 → 1.6180257....377 : 610 → 1.6180371....610 : 987 → 1.6180327....987 : 1597 → 1 : 61803444...1597 : 2584 → 1 : 1.6180338...2584 : 4181 → 1 : 1.6180340....4181 : 6765 → 1 : 1.6180339....6765 : 10946 → 1 : 1.6180339.....The golden ratio φ = 1.6180339.....So if you start with a 1mm square, when you get to a 55mm by 144mm rectangle it is approximately a golden rectangle. By 144 mm by 233 mm it is closer still (and probably as accurate as you can measure).You may recognise the first column as the Fibonacci sequence - the ratio of each term to the next term gets closer to the Golden Ratio as the sequence proceeds.