Looking at Learning ... Again, Part 2 Web Buddies Patterns [ home]

WEB ACTIVITIES:

## What Is a Pattern?

### Part 3 — Patterns in Nature

FIBONACCI NUMBERS

Mathematicians have been studying patterns in nature for centuries. In 1202 an Italian named Fibonacci investigated the breeding potential of rabbits. If the gestation period for the rabbits is one month, every rabbit survives, and one female always has a pair of bunnies (one female and one male), the resulting total numbers of rabbits for each month are as follows:

 Generation Pairs Original 1 1st month 1 2nd month 2 3rd month 3 4th month 5 5th month 8, etc.

Further, he found that flower petals and shell spirals, and, spirals on fir cones, grew in a pattern described by the series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on. These are known as the Fibonacci numbers. The Web site below has some fascinating examples of Fibonacci numbers in nature:

http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#petals

Fibonacci numbers abound in limericks as well. For examples see http://pass.maths.org/issue10/features/syncopate/

SELF-SIMILAR PATTERNS

Have you noticed that each leaflet of a fern looks strikingly similar to the whole frond, or that broccoli and cauliflower are made of up little florets that by themselves look like miniature vegetables. Or you might have noticed how a leafless branch of a tree looks like a small version of a whole tree.

You can also draw a tree with ever-smaller repetitions of the same shape, in a binary splitting algorithim, as shown on Plus Magazine's site.

Mathematicians have recognized that small parts of many natural objects look very much like the whole object. They call this phenomenon self-similarity. Mathematicians are able to recreate many of the self-similar patterns of nature by developing a rule that can be repeated over and over (i.e. an example of a mathematical rule being repeated "go distance turn 90, cut in half, turn 90, cut in half, turn 90, cut in half, turn 90" on and on and on....) Self-similar tilings are classics.

FRACTALS

Fractals are self-similar patterns. You may have seen images like the one below — the MANDELBROT SET— which has been generated by applying a mathematical rule over and over.

• How does this view of mathematics as a way of understanding the patterns of nature fit with the way you think about mathematics?
• Make a list natural objects that you think may show self-similarity. Why do you think so?

• Share your ideas on the discussion board.

>> Go on to Part 4 — What Does It Matter if Mathematics Can Describe the Patterns of Nature?

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