Do not change the opening of the compass. Repeat step 3 as many times as needed to go almost completely around the island, so that the last distance back to P 0 is less than the compass opening. An excellent homework assignment that can be given as a follow up to this activity is for the students to graph the results with compass opening on the horizontal axis and perimeter on the vertical axis. They can then write a paragraph discussing their observations. In this activity students will generate fractal stages that are initiated from a segment.
Students can use square or isometric grid paper to sketch the first two iterations on a segment.
The figure used to replace the line segment is called step 1 or the generator. To sketch step 2, the next iteration is performed by replacing each of the four segments with a reduced copy of the generator. Students should conjecture as to the number of segments that would be used to draw step 3, the third iteration. A continuation of this process leads to the Koch curve. Some students may also be interested in considering changes in length of the figure and area under the figure from one stage to the next although this will be covered in greater depth in the next lesson.