Pineapples and Fibonacci Numbers P B Onderdonk The Fibonacci Quarterly vol 8

# Pineapples and fibonacci numbers p b onderdonk the

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Pineapples and Fibonacci Numbers P B Onderdonk The Fibonacci Quarterly vol 8 (1970), pages 507, 508. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. Leaf arrangements (13 of 20) [12/06/2001 17:12:15]
The Fibonacci Numbers and Golden section in Nature - 1 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. Leaves per turn 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). Leaf arrangements of some common plants (14 of 20) [12/06/2001 17:12:16]
The Fibonacci Numbers and Golden section in Nature - 1 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.

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