Fractal Folds

Purpose To create paper sculptures of fractal shapes by cutting and folding paper. The emphasis will be on making Cantor sets.

Materials 12 by 18 inch colored construction paper, ruler, pencil, and scissors.

Background

Procedure

Conclusion Only our imagination limits the variety of sculptures that can be made in this fashion. Some represent fractals, others do not. Mathematical questions can be asked about some; all can be enjoyed visually.

Here are some questions that can be asked for all the constructions presented here, and for their variations. We provide solutions for the Sierpinski gasket example.

1. (a) Compute the lengths of the cuts, assuming the process is continued for infinitely many steps, producing the limiting shape. Here is the answer.

(b) Compute the length of the edges of the limiting shape. (Each cut creates three edges, each fold creates one.) Here is the answer.

2. Folding creates some open sides. Suppose these are covered with paper squares. Find the area of the limiting shape. Here is the answer.

3. Find the volume of the limiting shape. Here is the answer.

4. Is there a relation between edge length, area, and volume of the limiting shape?

Here are three pictures from our summer, 2002, workshop. Click each picture for a larger version in a new window.

Here is a gallery of fractal paper sculptures made by Stephen Smith's high school class in December, 2002.

Here is a gallery of fractal paper sculptures made by Tova Feldmanstern and Jennifer Michelstein as part of their spring, 03, course project.