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Advanced search. Skip to main content. All Rights Reserved. OSO version 0. University Press Scholarship Online. Sign in. Not registered? Sign up. Publications Pages Publications Pages. Search my Subject Specializations: Select Users without a subscription are not able to see the full content. Mathematics as a Science of Patterns Michael D. What happens with different sized coins? Concentric circles, those endless series of ripples and rings are fascinating.
Investigating the pattern created when one or more stones are dropped into a pool allows children to dip into concepts of gravity, force, resistance, motion and surface tension. No nature fieldtrip is complete without investigating tree rings. How do local species compare with a Giant Sequoia from Nevada which, through its rings, was found to be years old?
What important history unfolded during those lifetimes? Why are some rings thicker than others, how can we know about the climate during the years the tree was alive? Children are never too young to learn the value of tools in mathematics, especially for geometric constructions. Little hands may require some help and guidance with safety compasses, but the fascinating patterns they can create are worth the effort. Having children experiment to find out how, with a little help from their friends, they can turn a piece of string and pencil into a compass may produce a low-tech instrument but it is an exercise in lateral thinking.
Where else do we see concentric circles? On old records, compact disc surfaces? Well, yes to the first but no to the second. A strong magnifying lens will help the careful observer see that in fact the CD is made of millions of tiny dots arranged in tight spirals. Spirals start at a central point and coil around. They are easily seen on nautilus shells and ammonite fossils, in springs and the threads of screws and the tight coils of tendrils on climbing plants.
Is there a purpose to this natural and common manmade shape? If we look at and think about spiral staircase we begin to get a clue. They take up very little space and in some structures are very strong - springs are tight and tough! Spirographs, whether the wheels of varying sizes or the newer battery operated pen style, allow youngsters to create neat and regular spiral shapes. Again, string, pencil and a friend to gently pull and shorten the string length, act together to make an admirable tool.
If we pull out those magnifying lens again we will discover that spiders' webs also spiral out from the centre. Not what we expect! What we probably expect to find in a web is a radial pattern, that is to say, one in which straight lines radiate from a central point. If not in a spider's web where do we see radial patters? Where roads meet at a roundabout, on a dart board which is a combination of concentric circles and radial sections, in cactus spines where they meet the barrel and in flowers like the waterlily and the gerber daisy.
In these flowers the purpose of such an arrangement is to attract insects to their centres. The Sea Anemone is a radial animal, and so is a Starfish, its five arms allow it to move in any direction. When children make a paper flower or snowflake by folding a circle into eighths or sixths from the centre point and then cutting patterns into the folded sides and edge circumference , they are part of a long tradition.
Mathematics as a Science of Patterns
Almost years ago, Kepler wrote a book called the 6 Cornered Snowflake. He concluded upon examining the structure and pattern of snowflakes that matter is composed of identical units or atoms. Logo programs can be written to simulate Von Koch's snowflake. The edges are equilateral triangles. The pattern is one of ever decreasing size as each new generation is added. The straight lines take on an illusion of curves, in the same way as curved stitching does. Information and pictures are easily attainable on the Internet.
Princeton's mathematicians explore the science of patterns
Very young learners can use gummed paper triangles of different sizes to build a snowflake. By examining the pattern of growth they are able to estimate the number of triangles each generation needs. Similar Logo programs can be written to show other branching patterns wherein sections get progressively smaller. The same type of branching that is seen in deer antlers, fern leaves, blood vessels and TV aerials as well as trees. The results of these programmes amaze young learners and give them early exposure to the concept of fractals!
All of these patterns and we have hardly mentioned the simplest of all pattern, numeric. Even the natural world is loaded with numeric pattern: from the regular 28 days lunar cycle, the annual cycle of and a quarter days to the number of legs on animals. Yes, humans have 2, cows have 4, bees have 6, spiders have 8. Why even flowers' petals are not exempt from the power of pattern.