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Patterns

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Patterns in mathematics may be either numerical or visual. Three common numerical patterns (also referred to as sequences) are arithmetic, geometric, and exponential.

Common Numerical Sequences

In an arithmetic sequence, a common difference exists between a term and its previous term. For example, the sequence {1, 2, 3, 4, 5} has a difference of 1 between every term, making it arithmetic.

Geometric sequences have a common ratio, that is, a multiplying number, between every term. For example, the sequence {3, −6, 12, −24} is geometric because it has a common ratio of −2.

Finally, an exponential sequence has a base, or a number, that is raised by an increasing power. For instance, the sequence {1, 2, 4, 8, 16} = {2⁰, 2¹, 2², 2³, 2⁴} is exponential with 2 as the base.

Visual Sequences

Visual sequences may consist of geometric objects, a form often used in standardized tests. Usually, the sequences can be equated to a numeric pattern. For example, the number of sides in each figure can numerically state the visual sequence shown below. The arithmetic sequence {3, 4, 5} explains this visual sequence.

Another example of patterns from numbers is from the Fibonacci sequence {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …}, in which each term is calculated by adding the previous two terms. This mathematical pattern also occurs, for example, in pinecones in which the number of petals in the right diagonal rows is eight and in the left diagonal rows thirteen.

Predicting a Sequence

The answers to the questions outlined below help to determine the pattern and to predict the next (or any) term of a sequence.

1. Is the sequence numeric or visual? What is the common difference, ratio, or base?
2. Is there more than one pattern? Is one term calculated by using the previous terms?
3. What is the next term in the pattern?

For example, the sequence {1, −1, 1, −1} can be identified as geometric because there is a common ratio of −1; hence, the next term is 1.

Another example, the visual sequence shown below, illustrates two patterns occurring at one time.

The long orientation of the diamond alternates between horizontal and vertical positions, whereas the triangle inside is rotating 90 degrees clockwise. Therefore, the sequence is predictable, and the next term should be as shown below.

In contrast, the sequence {1, 2, 5, 6, 11, 13} is neither arithmetic nor geometric because there is no common difference or ratio, and it is not exponential because no base is being raised by an increasing power. In addition, any combination of previous terms does not equal a following term. Therefore, the sequence has no pattern and is not predictable.

Patterns do not necessarily occur in a single sequence. One of the most important applications of patterns is in chaos theory. Natural systems, like weather, which are chaotic and display little pattern, can be modeled with computers by generating millions of different possibilities. When all the different scenarios are analyzed together, a pattern may occur that exhibits the most likely possibilities. This process of analysis and pattern recognition allows meteorologists to understand the chaotic behavior of weather and to make more accurate weather predictions.

SEE ALSO CHAOS; NATURE; SEQUENCES AND SERIES.

Michael Ota

Bibliography

Field, Michael. Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature. Oxford, U.K.: Oxford University Press, 1995.