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Phloeonomus and The Fibonacci Sequence

It was a hot March afternoon when Phloeonomus awoke from his siesta. He had slept far later that usual. In fact, he had woken up quite a few times, but someone else’s counting had put him back to sleep. Now Phloeonomus listened. Yes, there was someone counting. He listened more closely.

"1, 2, 3, 5, 8, 13 ... Oh drat! What comes next? Was it 18 or 21 or 24? I’ll never get this right!"

Phloeonomus thought for a long moment. Then he thought for a bit longer. "Shouldn’t it be 1, 2, 3, four, 5?" he called out, crawling deeper into the Protea blossom.

There sat a funny white caterpillar with a fat face and a tiny head, and no legs worth speaking of. He was clearly furious and gnashed his jaws three or four times to indicate that he did not want to be disturbed.

"I said 1,2,3,5,8,13,21, or did I say 1,2,3,5,8? ... will you stop looking at me like that!"

"Who are you?", said Phloeonomus softly, "and why don’t you count properly. I think I can help. You got muddled up after leaving out four"

"I am Sphenoptera and I eat Protea seeds."

"Seeds or fruit?"

"Stop interrupting you silly little flower beetle! I eat through the fruit to get to the seed. To do that I need to know how the seeds are packed in the seedhead. And to do that I must count in a special way"

"I’m not a flower beetle. I’m a rove beetle. How do you count?"

Its really quite simple. You count one, two, like everyone does, and then you add one to two to give, ummmm, three, and then you add two and three to get, ummmmmmmm, five, and then you add three to five to get, ummmmmmmmmmmmmmm, eight, and then you add, ummm, where was I?"

"You had 1 and 2, and then went

Oh this is fun! But why do you need to count so funnily?"

"I am not counting funnily. That is the Fibonacci Sequence and it helps me to find my food. Now where was I? Five and three give me, ummmmmmmmmmmmmmm, how did you get 233 you silly little flower beetle? You trying to confuse me. Everyone knows that in a Protea head you don’t have to Fibonacci count more than 134, or was it 144? Oh dear, was it 164? Oh, yes I had eight."

"Why the Fibonacci sequence?" asked Phloeonomus, slightly confused.

"Because Fibonacci was the mathematician who first thought of adding numbers like that to count. And it so happens that if you want to pack a lot of seeds into a seedhead, then the best way to pack them in gives you spirals of Fibonacci numbers. See! And the spirals going one way are one Fibonocci number more or less than the number going the other way. See!"

"But how does this help you find your food?" asked Phloeonomus very confused.

"Well it’s like a game of battleships. The Protea does not want me to find its seeds, so it hides them in only a few of the many fruit. So I guess where they are. See! I am now going to clockwise spiral 13 and anti-clockwise spiral 21, yes 13:21! I bet I’ll find a nice juicy seed"

"But that’s silly! Plants don’t play battleships with their seeds. What if there isn’t a seed? What will you do then?"

"Oh! I’ll guess another set of Fibonacci numbers and go there. Anyway I might be lucky and find a seed on my way to 13:21. Now stop bothering me. It will take me at least two days of chewing through seeds to get to 13:21. All these empty fruit make my jaws ache!

"What a silly beetle" thought Phloeonomus, taking a sip of nectar. "But perhaps I’d also invent a game to stop me going insane if I had to chew through so many empty fruit!"

Phloeonomus talking to the grub Sphenoptera amid its tunnels through the sugarbush fruits.

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