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Using only a straightedge and compass to create tessellations follows in the tradition of Islamic designs.

Geometric Concepts in Islamic Art

by Issam El-Said and Ayse Parman
Dale Seymour Publications
P.O. Box 10888, Palo Alto, CA 94303
ISBN 0-86651-421-X

is a wonderful source of information. On page 3 they write,

"In the construction of geometric patterns in some traditional Islamic crafts (e.g., wood inlaying, metalwork, ceramic designs) the compasses and ruler are the only two major instruments used. In principle, this method is reminiscent of the rope stretching techniques of surveying, using peg and rope for a pair of compasses, in the planning of buildings in Ancient Egypt. Man has found through the utilization of geometry (meaning literally land measure), based on the circle, a perfect method to shape areas without resorting to complicated mathematical calculations such that, after the development of mathematics (the decimal system), this method, complete in itself, remained unaltered."

Objectives: [NCTM Standards Grades 6-8: Geometry, Measurement and Connections]

1. To construct a triangle using a straightedge and a compass.

2. To construct a tessellation of triangles using only a straightedge and a compass.

3. To use a two-dimensional tessellation to create a three-dimensional figure.

4. To view the relationships between

tessellation of triangles
tetrahedron
octahedron.

Materials

1. paper
2. compass
3. straightedge
4. cardboard to protect the working surface
5. scissors
6. tape
7. colored pencils (optional)

Procedure:

1. Instruct the students to place the cardboard on their desks under the paper in order to protect the surfaces of the desks.

2. Mark a starting point.

Use the straightedge to draw a horizontal line extending from the starting point.
Adjust the compass to the desired length of the sides of the triangle.

(suggestion: 1 to 2 inches)

Place the stylus of the compass on the starting point.
Swing and mark on the horizontal line.
Pick up the stylus, place on the new mark and repeat the process.

3. Now that there are 4 marks ticked off on the horizontal line, pick up the compass and put the stylus on the starting point, swing up and draw an arc.

Pick up the compass and place the stylus on the first tick mark on the horizontal line, swing back (to the left) and draw an arc. [The two arcs should intersect.]
Without picking up the compass swing to the right and draw another arc.
Continue this process until there are three sets of arcs.

4. The pattern of where the triangles have been constructed should be apparent at this point.

Using the straightedge construct a line segment from the "starting point" to the first set of arcs.

5. Continue constructing line segments.

6. If the task is to create a tessellation, then this process could be continued over the plane.

If the task is to create a tessellation that will result in a net that can be cut out of paper and folded to form a tetrahedron or an octahedron, then please go to the next page.