Up and down like a daisy

By Burkard Polster and Marty Ross

The Age, 28 April 2008

Get fit, take the stairs! And, to keep it interesting, walk up the stairs differently each day, by mixing 1-step and 2-step jumps. How many different ways can you do this?

If you are walking up a flight of five steps, then there are 8 different ways: 11111, 1112, 1121, 1211, 122, 2111, 212, and 221. Similarly, if you have flights of 1, 2, ... up to 9 steps, then you can count that there are 1, 2, 3, 5, 8, 13, 21, 34, and 55 methods, respectively. And, the pattern is clear: every number is the sum of the previous two.

It is easy to see why this pattern appears. Suppose you have 10 steps to climb. If your first move is a 2-step, then you have 8 steps to go, and you’ve already counted that there are 34 methods of completing those steps. On the other hand, if your first move is a 1-step, then you have 55 ways to complete the remaining 9 steps. So, there are a total of 34+55=89 ways to climb the 10 steps.

This simple sequence of numbers is called the Fibonacci sequence, after the famous 12th Century Italian mathematician. Fibonacci stumbled across these in a completely different context, whilst contemplating how rabbits might breed.

The Fibonacci sequence turns out to be the mathemagical key to scores of natural phenomena, including the spiral patterns in plants. Next time you look at a daisy, a sunflower or any other flower with distinct spiral patterns, count the number of spirals winding left and the number of spirals winding right. Almost always the two numbers you come up with will be consecutive Fibonacci numbers.

Along the outside of the daisy pictured, there are 55 left-winding spirals and 34 right-winding spirals. As well as these two readily apparent spirals, you can often identify other groups of spirals on the inside. The number of spirals in the next most visible group will be the next smaller Fibonacci number, in this case 21, and so on.

The explanation of why the Fibonacci numbers appear in plants is complex, and a great example of deep mathematics lurking in the real world. The individual seeds appear one at a time at the centre of the flowerhead, and are pushed to the outside by seeds that appear later. This leads to a very tight packing of the seeds, and it can be shown that this tight packing is achieved courtesy of the golden ratio, φ = (1+√5)/2. This “most irrational of all numbers” is intimately related to the Fibonacci sequence.

Here is the relationship in action. How many different ways can you walk up the 1254 steps to the observation deck in Melbourne’s Rialto Towers? Instead of churning out 1254 terms of the Fibonacci sequence, we find that the number of ways is also equal to φ¹²⁵⁵/√5, rounded to the nearest whole number. Not that the rounding really matters, since this number is approximately 9 x 10²⁶¹.