Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
The computer generated ray-traced picture here is created by my brother, Brian, and here's another, based on an African violet type of plant, whereas this has lots of leaves.
The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one.
If we count in the other direction, we get a different number of turns for the same number of leaves.
The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers!
For example, in the top plant in the picture above, we have 3 clockwise rotations before we meet a leaf directly above the first, passing 5 leaves on the way. If we go anti-clockwise, we need only 2 turns. Notice that 2, 3 and 5 are consecutive Fibonacci numbers.
For the lower plant in the picture, we have 5 clockwise rotations passing 8 leaves, or just 3 rotations in the anti-clockwise direction. This time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence. We can write this as, for the top plant, 3/5 clockwise rotations per leaf ( or 2/5 for the anticlockwise direction). For the second plant it is 5/8 of a turn per leaf (or 3/8).
The above are computer-generated "plants", but you can see the same thing on real plants. One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers.
Some common trees with their Fibonacci leaf arrangement numbers are:
1/2 elm, linden, lime, grasses
1/3 beech, hazel, grasses, blackberry
2/5 oak, cherry, apple, holly, plum, common groundsel 3/8 poplar, rose, pear, willow 5/13 pussy willow, almond where n/t means there are n leaves in t turns or n/t leaves per turn.
Cactus's spines often show the same spirals as we have already seen on pine cones, petals and leaf arrangements, but they are much more clearly visible. Charles Dills has noted that the Fibonacci numbers occur in Bromeliads and his Home page has links to lots of pictures.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More.. E
_ Things to do WITH VEGETABLES AND FRUIT . _
• Take a look at a cauliflower next time you're preparing one:
1. First look at it:
■ The florets are arranged in spirals, just like the seed heads and leaves above.
■ Count the number of florets at some fixed distance from the centre. The number in one direction and in the other will be Fibonacci numbers, as we've seen here.
■ Take a closer look at a single floret. It's a mini cauliflower! Each has its own little florets all arranged in spirals. If you can, count the spirals in both directions, and they'll be Fibonacci numbers (but you expected that!).
2. Then, when cutting off the florets, try this:
■ start at the bottom and take off the largest floret, cutting it off parallel to the main "stem".
■ Find the next on up the stem. It'll be about 0-618 of a turn round (in one direction). Cut it off in the same way.
■ Now look at the stem. Where the florets are rather like a pinecone or pineapple. The florets were arranged in spirals up the stem. Counting them again shows the
Fibonacci numbers.
• Try the same thing for broccoli.
• Chinese leaves and lettuce are similar but there is no proper stem for the leaves. Instead, carefully take off the leaves, from the outermost first, noticing that they overlap and there is usually only one that is the outermost each time. You should be able to find some Fibonacci number connections.
• Look for the Fibonacci numbers in fruit.
1. What about a banana? Count how many "flat" surfaces it is made from - is it 3 or perhaps 5? When you've peeled it, cut it in half (as if breaking it in half, not lengthwise) and look again. Surprise! There's a Fibonacci number.
2. What about an apple? Instead of cutting it from the stalk to the opposite end (where the flower was), ie from "North pole" to "South pole", try cutting it along the "Equator". Surprise! there's your Fibonacci number!
3. Try a Sharon fruit (which is like an orange-coloured tomato).
4. Where else can you find the Fibonacci numbers in fruit and vegetables? Why not email me with your results and the best ones will be put on the Web here or links added to your own web pages.
13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
Look at your own hand:
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