Why Must it Be So?
The optimal method for growth, must achieve the following:
1) Permit steady, exponential growth.
2) Result in a compact, efficient layout.
3) Come from steady creation of new florets in the core (called the "meristem").
4) Be self-correcting, i.e. the pattern doesn't break down completely with the slightest error during construction.
We should expect it to follow Occam's Razor, that the simplest explanation is probably correct.
This is not a proof of why the spiral pattern is the only solution or the best. It is an explanation of how it meets all the criteria above. To create the sunflower shown above the meristem, from which the florets are formed, produces one new floret on a regular interval, the next floret being approximately 137.5° away from the last. Why 137.5°? Because:
137.5° + 222.5° = 360° and 222.5/137.5 = ϕ ≅ 1.618
So if the first floret is created at 0° at t=0, then at t=1, the second floret forms at 137.5° and floret #1 has grown. At t=2, florets 1 and 2 have grown again and the third floret is formed at 225.5°. From the diagrams (coming), you can see that as time progresses, the position of the next floret is always the place with the most space available. All others are crowded by comparison.
The number of spirals depends on the rate of growth of the florets. If the florets grow slowly, many florets will remain touching the meristem in the middle, resulting in many spirals. If the florets grow rapidly, they will be quickly pushed out by newer florets, making fewer spirals.
Why are the numbers of spirals adjacent numbers on the Fibonacci sequence? Because the rotation uses the golden ration, ϕ. For the sunflower, with 34 counter-clockwise spirals, the second floret is on the 21st spiral, the third floret is on the 6th spiral, the third floret is on the 27th spiral and so on. As the florets grow, they automatically become positioned sequentially in the spirals of opposite directions.
Because the number of spirals depends on the rate of growth and the simple rule of creating the next floret 137.5° after the last, there is no need for the meristem to "know" to grow 3, 5, 8, 13, 21 or 34 spirals. They result automatically.
