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Mathematical Skills Across The Curriculum
Fibonacci numbers
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The Fibonacci numbers are Nature's numbering system.
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They appear everywhere in Nature, from the leaf arrangement in
plants, to the pattern of the florets of a flower, the bracts of a
pinecone, or the scales of a pineapple.
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The Fibonacci numbers are therefore applicable to the growth of
every living thing, including a single cell, a grain of wheat, a hive of
bees, and even all of mankind.
Fibonacci Sequence
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it.
•The 2 is found by adding the two numbers before it (1+1)
•Similarly, the 3 is found by adding the two numbers before it (1+2),
•And the 5 is (2+3), and so on!
Task (2 minute) : Can you figure out the next few numbers?
Here is a longer list:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657,
46368, 75025, 121393, 196418, 317811, ...
Fibonacci Spirals: If you construct a series of squares with lengths equal to the
Fibonacci numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each square,
it forms a Fibonacci spiral.
Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
Fibonacci numbers create a
mathematical pattern found
throughout nature
Some of the examples are listed on next few
slides.
Fibonacci spiral in sea shell:
Many examples of the Fibonacci spiral can be seen in nature, including in
the chambers of a nautilus shell.
The sunflower seed pattern always shows Fibonacci numbers
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …).
Below are the three most natural ways to find spirals in this pattern.
Note that the black pattern is identical in all the images on this page.
Only the coloured lines indicating the selected spirals are different.
Here is a passion flower (passiflora incarnata) from the back and front:
Back view:
The 3 sepals that protected the bud are
outermost, then 5 outer green petals
followed by an inner layer of 5 more paler
green petals
Front view:
The two sets of 5 green petals are outermost,
with an array of purple-and-white stamens.
In the centre are 5 greenish stamens (Tshaped) and uppermost in the centre are 3
deep brown carpels and style branches)
Leaf arrangements
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.
Starting at the leaf marked "X",
we find the next lower leaf turning clockwise.
Numbering the leaves produces the patterns
shown here on the right.
Romanesque Broccoli/Cauliflower (or Romanesco):
Each floret is peaked and is an identical but smaller version
of the whole thing and this makes the spirals easy to see.
How many spirals are there in each direction?
Here is a picture of an ordinary cauliflower. Note how it is almost a pentagon in
outline. Looking carefully, you can see a centre point, where the florets are
smallest. Look again, and you will see the florets are organised in spirals around
this centre in both directions.
How many spirals are there in each direction?
Pine cones show the Fibonacci Spirals clearly. Here
is a picture of an ordinary pine cone seen from its
base where the stalk connects it to the tree.
Can you see the two sets of spirals?
How many are there in each set?
Here is another pine cone. It is not only smaller, but has a different spiral
arrangement.