With strings of paper dolls, you can decorate a party, make the border of a bulletin board, create a frame for your favourite picture, and of course, learn some math in the process.

Sometimes, each shape in the pattern can be a reflection of the one before it. Each second shape in the pattern can be rotated and reflected upside down. The shapes do not have to be reflected at all: each shape can simply be repeated over and over again. In fact, there are 7 distinct symmetry patterns, which can be created with paper dolls string. You can get the different symmetrical patterns by folding the paper in different ways. The design possibilities, however, within those 7 groups are endless.

Materials:
• Paper
• Scissors
• Pencil
Making Paper Dolls: (with reflective symmetry)

Procedure:
1. Fold a piece of paper in half so that the 2 shorter sides are together. See diagram below.
2. Fold the resulting piece of paper in half again but this time so that the 2 longer sides are together. (it is important to fold the paper in a certain way to get the design you want) The fold lines are shown as dashed red lines on the diagram below.
3. If your paper is long enough, fold it in half again the same way as you did in Step 2, so that the two longer sides are together.

4. With you paper still folded, cut off a square from one of the ends using scissors.

5. Take your pencil and draw a shape on your folded square piece of paper. Try an outline of a person. It is very important to make sure that your design touches both edges of the square paper where the folds are. Otherwise, when you cut out your design, the shapes will not be connected properly, and you won't end up with a string.

6. Cut out the shape, while the square is still folded. Depending on how many times you folded your paper, you should have 4 or 8 layers. Do not cut the places, where the shape meets the edges of the paper. The green lines in the diagram indicate places, where you should NOT cut the paper!! If you do, your shapes will not be connected and you won't get a string.
7. Now carefully unfold the paper, and enjoy your paper doll string.
8. You can cut out several paper dolls string from the remaining paper, and connect the string with scotch tape to make a longer string. You can also colour your paper dolls to make them more fun.

9. You have created a paper dolls string with reflective symmetry. That means that each shape is a mirror image of the shape next to it. You can draw the lines of symmetry in between the shapes to see how the shapes are reflected. The lines are shown in the diagram below. Fold the paper dolls string along these lines, and you will see that one shape will be exactly lines up with the shape next to it.

Questions:
1. How many times do you have to fold the paper to get a single string of 16 paper dolls?
2. The first paper doll is the string is the same as the third, is the same as the fifth, seventh, and so on. The same pattern can be seen with even numbered dolls. Why does happen?
Further Ideas:
1. Try folding paper in different ways to get different symmetrical patterns. For example, try folding the paper so that the shorter sides end up together every time you fold. What type of symmetry do you this time?
2. Try cutting out different designs on the paper. For example, try a butterfly, or flower, or star. Just remember that the shape has to touch both edges of the paper where the folds are

Reference: Math for Fun Projects,Andrew King. Copper Beech Books. 1999. ISBN 0-7613-0789-3