# Symmetry

30 31 fABulous fiBonAcci golden angles and a golden number Around the end of the 12th century a young Italian customs officer became intrigued by and gave his name to a number series that has fascinated mathematicians ever since. Nicknamed Fibonacci, Leonardo of Pisa had discovered the cumulative progression where each number is the sum of the preceding two numbers, i.e., 1, 1, 2, 3, 5, 8, 13, 21, 34 etc. He also recognised that this series has some very special mathematical properties. The Fibonacci numbers are frequently involved in plant growth patterns, notably in petal and seed arrangements. Flower petals are almost invariably fibonaccian in number fircones use series of 3 and 5 or 5 and 8 intertwined spirals pineapples have 8 rows of scales winding one way, 13 the other wayand so on. The series is also found in phyllotaxy, the configurations of leaves and branches in plants. The Fibonacci series is focussed on phithat is to say, as they get higher the ratio between successive numbers gets closer and closer to this golden number. There is a related quality too in the format of successive primordia in phyllotaxy which use the golden angle of 137.5 360 phi2. This arrangement provides the most efficient use of space in the succession of branches, leaves and flowers. Fibonacci patterns are not restricted to organic formations they have been observed in many aspects of the physical world, from nanoparticles to black holes. 1. 2. 3. 4. 5. 1. Phyllotaxis order 138 in a cactus. 2. Order 85, 8 leaves forming in 5 anticlockwise turns, with every 8th leaf above another 3. Another example of 85 phyllotaxis. 4. A rare case of Lucas phyllotaxis, order 117. 5. A sunflower head demonstrating 8955 Fibonacci phyllotaxis on a Fermat spiral. Count the spirals each way.