Golden Ratio Series
The Golden Ratio Series collection shows arrangements of geometric figures that correspond to the golden ratio. The golden section cuts a straight line into a large and a small section, so that the length ratio of the straight line to the large section is the same as the ratio of the large to the small section. The resulting ratio is denoted by the small Greek letter φ (“phi”). φ is an irrational number (such as π) with the value 1,618…
φ has an amazing relationship with the Fibonacci numbers: with the n-th power of φ, i.e. with φn, the n-th Fibonacci number can be calculated based on Jacques Binet’s formula. And: the ratio of two consecutive Fibonacci numbers tends to get closer and closer to φ the larger the Fibonacci numbers are.
Sales of Phi
The two mathematicians Alfred S. Posamentier and Ingmar Lehmann, in their book The Glorious Golden Ratio, have developed a geometric method to construct increasing powers of φ. The powers of the golden section have a property similar to the Fibonacci numbers, namely the recursion:
φn = φn-1 + φn-2
The geometric method uses a square to construct an initial golden rectangle of side lengths 1 and φ. After that, the length of the next golden rectangle will be constructed by drawing a quarter circle. This process can be continued indefinitely.
The picture Sails of Phi shows the quarter circles and the golden rectangles of the first eleven stages of construction in different tones of the complementary colors yellow and blue. The resulting yellowish sails remind me of the Sydney Opera House by the Danish architect Jørn Utzon.