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This article is excerpted from Math Matters: Understanding the Math You Teach, Grades K–8, Second Edition by Suzanne H. Chapin and Art Johnson
One of the most famous patterns is the Fibonacci sequence, which is made up of Fibonacci numbers. Fibonacci was the nickname of Leonardo de Pisa, an Italian mathematician (1175–1245); he is best known for the sequence of numbers that bears his name. The Fibonacci sequence of numbers begins with two numbers: 1, 1. Each new number is then found by adding the two preceding numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
Mathematicians have identified many interesting relationships among the Fibonacci numbers. For example, the sum of the first three Fibonacci numbers (1 1 2 4) is one less than the fifth number (5). The sum of the first four Fibonacci numbers (1 1 2 3 7) is one less than the sixth number (8). Find the sum of the first five Fibonacci numbers. Did the relationship hold? Yes, the sum of the first five Fibonacci numbers is 12, which is one less than the seventh Fibonacci number.
The Fibonacci numbers describe a variety of phenomena in art, music, and nature. The numbers of spirals on most pinecones and pineapples are Fibonacci numbers. The arrangement of leaves or branches on the stems of many plants are Fibonacci numbers. On a piano, the number of white (8) keys and black (5) keys in each octave (13) are all Fibonacci numbers. The center of a sunflower has clockwise and counterclockwise spirals, and these spirals tend to be consecutive Fibonacci numbers. The lengths and widths of many rectangular objects such as index cards, windows, playing cards, and light-switch plates are consecutive Fibonacci numbers.
Fibonacci ratios are comparisons of two Fibonacci numbers, usually adjacent numbers in the sequence. These ratios are often expressed as decimals by dividing one number in the ratio by the other. As the numbers in the Fibonacci sequence increase, the ratios tend to approximate 1.618 (8/5 1.6, 55/34 1.6176, 233/144 1.61805). This special ratio (1.618034 . . . ) is an irrational number that occurs in many other shapes and objects. It is known as (phi) by mathematicians, and was labeled the golden ratio by the ancient Greeks. (People also call it the golden proportion.) It has been known and used for thousands of years—it is believed that it was a factor in the construction of some of the pyramids in Egypt. Rectangles whose length-to-width ratios approximate the golden ratio are called golden rectangles. Psychologists have found that people prefer golden rectangles to other rectangles; thus common objects such as cereal boxes and picture frames tend to have dimensions with a ratio of around 1.6. Which of the rectangles below do you find to be most aesthetically pleasing? Two are golden rectangles; you can check by measuring the lengths and widths and calculating the ratios.
Try this activity in your classroom!
Fibonacci Numbers and You
Objective: investigate the occurrence of the golden ratio in the human body.
The human body is characterized by golden proportions, and these ratios have
been used to draw figures accurately for centuries. Make the following measurements
(use either inches or centimeters) and calculate the designated ratios.
▲ Your height compared with the distance from the floor to your navel.
▲ The distance from the floor to your navel compared with the distance from
the floor to your kneecap.
▲ The length of your arm from the shoulder compared with the distance from
your fingertips to your elbow.
▲ The distance from your chin to the center of your eyes compared with the distance
from your chin to the tip of your nose.
▲ The length of your index finger compared with the distance from your index
fingertip to the big knuckle.
Things to Think About
One way to represent these ratios is as decimals. If your height is 68 inches and
the distance from the floor to your navel is 42 inches, the ratio is 68 to 42 or
1.619 to 1 (68 42). This is very close to the golden ratio! You may instead have
calculated a ratio of 1.7 or 1.5—not all individual proportions are exactly golden
ratios. Some of us have long legs or arms compared with our overall heights.
However, on average, the ratios will be close. Did you find other occurrences of the golden ratio? Share your thoughts with us in the comments!
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