Fractals are geometric patterns. And no matter how small the portion of the pattern is, it will have elements similar to the overall pattern. The perimeter changes on an infinite basis as the pattern is repeated. Create a simple fractal by starting with a equilateral triangle. Divide each of the three sides into thirds. Add to this an upside down equilateral triangle so that the points pass through the center section of each side. Remove the interior lines, leaving only the overall outside perimeter line. Repeat by dividing the star point sides, around the perimeter, into thirds and adding triangles with point sizes that fit the middle third of these smaller perimeter star points. Continue this process for each succeeding smaller perimeter point sides. The resulting snowflake is known as the Koch snowflake.
Tell students that they can walk through a sheet of paper. Let them come up with suggestions as to how this can be done. Take one sheet of 8 ½-by-11-inch paper and fold the paper lengthwise in half. Along the folded edge, divide the paper using eight equally spaced lines. The length of the line is drawn to within an inch of the opposite side. Turn the paper so that the folded side is away from you. Starting in between the first two lines, draw a line centered between them. This line goes up to within an inch of the folded edge. Repeat this for the spaces between the other lines.
Turn the paper so that the folded edge is close to you again. Use scissors and cut along the full length of the lines. Cut along the fold line, but not the two end sections. The paper can now be opened and the students can walk through the one sheet of paper.
A basic feature of geometry is the straight line. Talk about how a straight line was used by the ancient Egyptians for surveying. Divide the class into groups. Give each group three tall wooden dowels. Talk about how they can find a straight line over a long distance. Have them place one dowel in the ground. A second person should walk out from this dowel and put a dowel into the ground. The third dowel should be placed in between these two. Have the students stand behind the first dowel and look ahead to the second dowel. If they see the middle dowel, tell the person with it to move it right or left. A straight line is formed when the middle dowel is moved so that it is hidden from view. Try this for multiple distances.
A 4-by-11-inch piece of paper can change simply by a twist or two. Take the strip of paper and glue the ends together. Cut the circle in half along the length of the paper and you will get two circles. Now take another 4-by-11-inch of paper and twist one end, then glue the ends together. Cut the paper in half along the length of the paper. Ask the students what they think will happen. Show how the paper is now longer. This paper with the twist is known as the Mobius strip. Repeat using more twists before gluing the ends together.