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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