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 repeating any of the Fibonacci numbers. The Golden Ratio
Any term in the Fibonacci sequence divided by the previous has a quotient of approximately
1.618034…. That is, an/an-1≈1.618034. 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 훗), 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, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 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 The number of branches or The lengths of the bones in the pinecones, pineapples, and certain leaves present at certain human finger are proportionate flowers is always a Fibonacci heights on a plant is often a to Fibonacci numbers. number. Fibonacci number. 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. Applications to Business
Following a significant change in the value of the stock market, it is expected to retrace a certain portion of the increase (advance) or decrease (decline) before stabilizing or reversing trend. This is called retracement. Quite often, the amount of pullback is 23.6%, 38.2%, or 61.8% of the original advance or decline. These values are ratios generated from the Fibonacci sequence, in which an/an+3≈.236, an/an+2≈.382, and an/an+1≈.618. For this reason, a retracement of 23.6%, 38.2%, or 61.8% of the original change is called a Fibonacci retracement, and a retracement of 61.8% is knows as the golden retracement. This knowledge allows us to anticipate a trend reversal in the stock market when the retracement has reached one of these levels. Applications to Computer Algorithms
Pseudorandom Number Generator The Fibonacci sequence is used in the following computer science-related algorithms and processes:
● Euclid’s algorithm, which determines the greatest common divisor of two integers
● The pseudorandom number generator, which creates a set of numbers with similar properties to those of a random set of numbers
● Planning poker, a process used in developing computer software that uses Scrum methodology Fibonacci Heap Data Structure More Applications to Computer Algorithms
● The polyphase merge sort algorithm, which divides a set of terms into two lists, whose numbers of terms are two consecutive Fibonacci numbers
● The Fibonacci heap data structure
● The Fibonacci search technique, which operates more quickly than the binary search technique, by finding possible positions of the desired item within a sorted array
● The Fibonacci cube, a graph used in parallel computing Works Cited
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