Strolling along Baggot Street in Dublin recently, I noticed a plaque at the entrance to the Ibec head office. It was a logo with a circular pattern of dots. According to Ibec’s website, the logo brings “dynamism” and hints at Ibec’s “member-centric ethos”. In fact, it is more interesting than this.
The logo is based on the spiral patterns found in many flowers and plants. Examining it we find 34 clockwise and 21 counter-clockwise spirals. These numbers are sequential entries in a famous number sequence called the Fibonacci sequence. This is no coincidence.
Iterative process
In 1202, mathematician Leonardo of Pisa, usually known as Fibonacci, published a book,
Liber Abaci
, in which he described the sequence now known as the Fibonacci numbers. They are easily defined by an iterative process: starting with 0 and 1, each entry is the sum of the previous two. Thus, the sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and 144.
As the values increase, the ratios of successive Fibonacci numbers – like 8/5, 13/8 and 21/13 – tend to a definite limit, about 1.618, called the golden number.
It was known to the Greeks through their study of proportions and the geometry of the pentagon. If we divide a circle into two arcs whose ratio is the golden number, then the shorter arc subtends an angle of about 137.5 degrees. This is called the golden angle.
Distinctive spiral patterns are found on many plants. For example, the hexagons on pineapples fit together in interlocking families of helical spirals. The numbers of spirals are successive Fibonacci numbers. Sunflowers, which belong to the daisy family, usually have 55, 89 or 144 petals, and spiral patterns are evident in their seeds.
Biologists described long ago how phylla (leaves, petals, seeds, etc) are arranged on plants: this branch of botany is called phyllotaxis. But explaining why these patterns form is much more difficult than describing them, and it is only recently that real progress has been made.
The seeds of a sunflower are arranged in a manner that makes efficient use of the available space, giving maximum room for each seed to flourish and minimising wastage of space. As a new seed sprouts forth at the growth tip of a plant, it naturally tends to grow where there is most open space. Each seed is displaced from the previous one by the golden angle.
Golden angle
But why the golden angle? Recent research shows that the angle emerges naturally as a feature of the dynamics of plant growth. Some years ago, Dublin-born Alan Newell, at the University of Arizona, applied elasticity theory to continuum models of growing cacti shoots. But mechanics could not answer all the questions.
Recently, Newell has shown that biochemistry, mechanics and geometry all play a role in generating the observed patterns. The growing seeds exert forces on each other, triggering the production of auxin, a growth-enhancing plant hormone. The solutions generated by the model of auxin concentration are found to be very similar to the patterns found in real flowers. This illustrates in a striking manner how nature is capable of producing optimal packing strategies.
Peter Lynch is professor of meteorology at University College Dublin. He blogs at thatsmaths.com