## Designer Fractions - Symmetry

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Symmetry is a very important concept in mathematics which you observe frequently in nature.As long as you study mathematics, you will study symmetry.

There are several types of symmetry. We are going to investigate line symmetry. Think about your own body. If you drew a line down the center of your body from the top of your head to the bottom of your feet, your body on one side of the line would be almost identical to the part on the other side of the line. If they were exactly identical, we'd call the line a line of symmetry. Would a line across your waist be a line of symmetry?

You can think of a line of symmetry as a fold line. If you fold a figure on the line of symmetry, the figure will fold right in half, with both sides right on top of each other.

Some figures have more than just one line of symmetry, and some have none. Consider the 3 colored designs below. First investigate the symmetry including the colors inside. In other words, red would have to fold onto red. How many lines of symmetry does each figure have?

Would your answers be different if you considered just the outline of the figures? Why don't you give it a try? How many lines of symmetry would each figure have if you ignored the inside and just considered the outline?

### Symmetry Can Save You Work

Remember the problem where we calculated the fraction of the figure that was red, the fraction that was blue, and yellow and green? We had to count the total number of triangles in the whole figure. Or did we? Since you have a line of symmetry, the top half is identical to the bottom, so you only have to count the top (or bottom). There are 3 yellow triangles in the top, and 12 triangles in all in the top. So the fraction of yellow is 3/12. Before we counted 6/24. Those are equal, so we can use symmetry to save us work.

In fact, since the figure has two lines of symmetry, and if you can work with fractions, you can really save yourself some work. Now you can look at just a fourth of the figure, and see that there are two blue triangles out of six triangles in all (You have to add the 1/2 yellow and 1/2 red to make six total). That is equal to the 8 out of 24 in all - 1/3.

### Look for Symmetry around You

Look around you right now. Look at your computer screen. Would it have any lines of symmetry? What about this W? What about I? What about the clock? Can you find a leaf that has line symmetry? If you start looking for lines of symmetry, you'll find them everywhere.

### Symmetry on the Web

Lots of information is on the web about symmetry. You can explore these sites to find out more.

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lanius@math.rice.edu