The pointy end of pineapple numbers

There’s nothing more Australian than to celebrate Australia Day by visiting one of Australia’s Big Things. This year, a Maths Master took his family on a pilgrimage to the Big Pineapple in Woombye, Queensland. 

It is big indeed, which pleased the kids. And, your Maths Master was also very pleased. We’ll explain why. 

The skin of the monster fruit is made of hexagons, stuck together as in a bathroom tiling and then wrapped  to form a cylinder. Connecting hexagons in line, we can picture three different types of bands winding around the pineapple, as highlighted below. It turns out that the Big Pineapple has 13 green bands, 13 blue bands and 26 red bands.

You can also see such bands in real pineapples, but the numbers are different. It is well-known that a real pineapple exhibits consecutive Fibonacci numbers, such as 5-8-13 or 8-13-21.

What is famously special about the Fibonacci numbers is that, starting with 1 and 1, each number is the sum of the two previous ones: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on. Fibonacci numbers often occur in nature, and the numbers of spirals in pineapples and sunflowers are commonly quoted examples. We wrote briefly of this in an earlier column. 

All this is really fantastic. Except, a lot of it isn’t true.

Recently, your Maths Masters have been spending a lot of time in greengrocers. We did indeed find a pineapple with Fibonacci numbers, the one with 5, 8 and 13 bands pictured above. However, we only found this pineapple after checking out 26 duds. Or, 27 if you count the Big Pineapple. 

Is there something wrong with Australian pineapples? Should the food authorities order a massive recall? Probably not.

Fibonacci numbers are mathematically beautiful, but people tend to be carried away by their enthusiasm, and wishful thinking can sweep caution aside. So, a truthful claim that “some specimens of some plant species exhibit Fibonacci numbers” can morph into the very false declaration of a universal law. (We’ve previously written of such mythmaking in regard to the Fibonacci numbers’ close cousin, the golden ratio). And, it is simply not the case that all or most pineapples exhibit Fibonacci numbers. Indeed, for many pineapples, the hexagons are so irregular as to make the counting of the bands practically impossible.

However, all is not lost. As a model for our pineapples, let’s take a closer look at how a perfect cylinder can be tiled with perfectly regular hexagons. Cutting open and unwrapping the cylinder produces a rectangle, and then it is not too difficult to count the number of bands of each type. For example, the unwrapped cylinder below has 5, 8 and 13 bands: a good model for our Fibonacci pineapple above. 

As the Big Pineapple demonstrates, the band numbers do not have to be Fibonacci numbers. Nonetheless, it is still possible to prove a very beautiful Fibonacci-like theorem:

Tile a cylinder with hexagons, and count the number of bands of each type. Then the largest number is the sum of the two smaller numbers.

For example, though Woombye’s Big Pineapple is not Fibonacci, its band numbers do satisfy the equation 13 + 13 = 26. In fact, given any three numbers with A + B = C, it is always possible to make a “pineapple” with those numbers of bands.

So, expecting all pineapples to be exactly Fibonacci is too much to ask, but nicely grown pineapples with well-formed bands do still obey a simple and beautiful rule.

We hope to write more about this addition rule in a future column. Here, we’ll finish by suggesting why the addition rule gives some hope of finding Fibonacci numbers in pineapples and other banded plants. (In fact, sunflowers and other plants tend to grow more regularly than pineapples, and they exhibit Fibonacci numbers much more predictably).

Big pineapples grow from baby pineapples, and as they grow the numbers of bands change. What tends to happen at each stage (don’t worry about the details!) is that the two largest band numbers will stay the same, and the new band number is the sum of these two.

This means that the band numbers in a baby pineapple tend to determine the future band numbers. And so, if our baby pineapple exhibits Fibonacci numbers, such as 2, 3 and 5, and if the pineapple grows nicely, it will continue to exhibit Fibonacci numbers.

But of course not all babies are so amenable. A quick calculation shows that Woombye’s Big Pineapple must have been an extremely peculiar baby.

Extra Credit: This week, your homework is to sneak into greengrocers and supermarkets, and to count pineapple band numbers. See if the addition rule actually works, and see if you can find any Fibonacci pineapples. We’d be delighted if you’d post your findings as comments below.

Cricket Maths: On tonight’s broadcast of the Allan Border Medal (8:30 on Fox and 10:30 on Channel 9), you might catch a glimpse of the Maths Masters. Check out if they succeeded in inserting prime numbers into a cricket show.

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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