The picture on the right shows an ordinary pine cone, as seen from its base where its stalk joined the tree.


There are two sets of spirals visible on the pine cone. One set of 13 spirals runs anti-clockwise from the centre of the cone, while the other set of 8 spirals runs clockwise, as shown in the pictures below:


13 12 2 11 10 5 9 8 7 Figure 3 Spirals clearly seen on a pine cone


You will see that the numbers 8 and 13 are consecutive numbers in the Fibonacci sequence. Arrangements that involve consecutive Fibonacci numbers appear in a wide variety of plants, including sunflowers and pineapples. In some sunflowers, for example, the spirals are arranged in groups of 55 and 34 spirals respectively. In others, the spirals are arranged in groups of 89 and 55 spirals respectively. Do you see that 34, 55 and 89 are consecutive Fibonacci numbers?


Extending patterns


The terms in a pattern are related by a rule. Once we know the rule for a pattern, we can extend the pattern. We extend patterns by adding, subtracting, multiplying and dividing a common number. Simple patterns can be extended with a constant number. However, complex patterns, such as recursive patterns, are extended with a number that changes.


120 Chapter 4: Patterns, functions and relationships Did you know?


A recursive pattern is one in which any term in the pattern (usually from the second or third terms onwards) can be described in terms of the previous term (or terms). The Fibonacci sequence is a famous example of a recursive sequence. In the Fibonacci sequence, the first two terms (0 and 1) are given. The rest of the sequence grows from these two terms.


6 5 4 3 4 6 3 1 7 2


Figure 2 Geometrics on a pine cone 8 1


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