By the 17th century, the well-known astronomer Johann Kepler observed that the Fibonacci numbers are commonly found in plants. For example, a white lily is a one petal flower, euphorbia have 2 petals, trillium have 3, columbine have 5, bloodroot have 8, black-eyed Susan have 13 and Shasta daisies have 21. Do those numbers sound familiar? Other daisies might have 34 or 55 petals and larger sunflowers often have 89 or 144 petals. Later, in 1790, Bonnet first noted that the scales of pinecones and pineapples contain families of interlaced spirals, one winding clockwise and the other, counterclockwise. The same spirals appear in the heads of daisies and sunflowers. Depending on the species, the pairs of spirals number 21 and 34, 34 and 55, 55 and 89, or actually any two adjacent Fibonacci numbers.

This phenomenon is common throughout the plant world, but why? Understanding why was a multidisciplinary effort. Several ideas were proposed over the years, many based on the notion of natural selection. For example, the observed arrangement of spirals could provide dense packing of seeds, maximizing offspring, or perhaps the arrangement of leaves and petals allows sunlight to nourish as many as possible. The key is to determine how Fibonacci numbers promote the survival of mature plants. A daisy whose flower has 22 spirals going in one direction and 33 in the other should have about the same chance of producing viable seeds as one whose flower has 21 spirals in one direction and 34 in the other. Yet the first case is virtually unheard of while the second is quite common.

To understand we need to examine plant growth more closely. Consider the schematic diagram of the sneezewort plant . At each growth level both the number of branches and the number of leaves is a Fibonacci number.

Plants tend to grow from a single central apex. Emerging from the apex, small bumps called primordia appear. These primordia will eventually become leaves, petals, sepals or the like. They grow outward in spirals. By 1837, the pioneer crystallographer Auguste Bravais and his brother Louis observed that successive primordia make an angle of about 137.5°, as seen from the apex. This angle is supplementary to 360°/Ø (about 222.5°). Is it mere coincidence that the Golden Ratio, Ø, is involved?

In 1992, Douady and Couder from the Laboratory of Statistical Physics in Paris offered an important breakthrough. They developed a laboratory model of plant growth. According to their model, the dynamics of plant growth can account for the Fibonacci numbers. The model shows that the angle at which the primordia move away from the apex determines the spiral. If the angle is a rational fraction of 360°, then there is no spiral. Primordia would line up and, consequently, leaves and petals would eventually block the light of their predecessors. Irrational multiples of 360° create spirals that never result in overlapping leaves or petals. It makes sense that the least overlap would come from some irrational multiple of 360°, and it does! In particular, Ø is the irrational number with the simplest expression as a continued fraction.

The Douady-Couder model shows that spirals at this angle create the most efficient packing, expending the least amount of plant energy. Remember that quotients of successive Fibonacci numbers approach Ø. These quotients most closely approximate Ø. Thus, the number of spirals coming from the apex would have to be a Fibonacci number. The particular one depends only on the size of the primordia, that is, how many primordia fit around the apex before they overlap and join an existing spiral.

**Lesson Plan 5**