The Fibonacci Sequence

[Pages:13]The Fibonacci Sequence

Demonstrating the Magic of Math By Kimberly Rivera

The Fibonacci Sequence

The Fibonacci Sequence begins with a 1, followed by another 1. Later terms are found by adding together the two previous terms.

an=an-1+an-2 for a1=1 and a2=1 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, ...

The Fibonacci numbers are a complete sequence. This means that any positive integer can be expressed as the sum of various Fibonacci numbers, without

The Golden Ratio

Any term in the Fibonacci sequence divided by the previous has a quotient of approximately 1.618034.... That is, an/an-11.618034. For the first few terms, this is a very loose approximation, but as the term number (n) increases, the quotient coincides more exactly with this irrational value. The ratio between 1 and 1.618034 is known as the Golden Ratio (abbreviated ), and a rectangle with a width to height ratio of 1:1.618034 is known as the Golden Rectangle.

Rectangles and Spirals

Each term of the Fibonacci sequence can be represented with a square whose sides have a length equal to the value of the corresponding term. If one takes all of the squares from the beginning of the sequence to any point along it, the squares can be arranged into a rectangle. As more squares are added to the rectangle, the ratio of its width to its height approaches the Golden Ratio, and the rectangle approaches the dimensions of the Golden Rectangle. Furthermore, the squares within the rectangle can be arranged to form a spiral pattern if one traces from the largest square to the smallest square.

Divisibility Patterns

When examining the Fibonacci sequence, it is interesting to note: Every third term is even. Every fourth term is a multiple of 3. Every fifth therm is a multiple of 5. (Violet represents multiples of both 2 and 3. Cyan represents multiples of both 2 and 5. Orange represents multiples of both 3 and 5.)

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,

Applications to Art and Architecture

The Fibonacci spiral is can be used as a guide to placing features in art and architecture, in order to create a pleasing visual effect.

Applications to Biology

The Fibonacci numbers can be seen throughout nature.

The number of spirals on pinecones, pineapples, and certain flowers is always a Fibonacci number.

The number of branches or leaves present at certain heights on a plant is often a Fibonacci number.

The lengths of the bones in the human finger are proportionate to Fibonacci numbers.

More Applications to Biology

The sequence is also seen in the inheritance tree of the human X chromosome, the population growth of rabbits, and the lineage of a male bee.

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