Nets  Straightedge/Compass
Constructions
Back to Paper Polyhedra Nets
Note: If the task is simply 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, you may go directly to the next page.
Using only a straightedge and compass to create tessellations follows in the tradition of Islamic designs.
Geometric Concepts in Islamic Art
by Issam ElSaid and Ayse Parman Dale Symour Publications
P.O. Box 10888, Palo Alto, CA 94303
ISBN 086651421X
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 Grade 68 Standards: Geometry, Measurement, Connections]
 To construct a triangle using a straightedge and a compass.
 To construct a tessellation of triangles using only a straightedge and a compass.
 To use a twodimensional tessellation to create a threedimensional figure.
 To view the relationships between
tessellation of triangles
tetrahedron
octahedron.
Materials
 paper
 compass
 straightedge
 cardboard to protect the working surface
 scissors
 tape
 colored pencils (optional)
Procedure:
 Instruct the students to place the cardboard on their desks under the paper in order to protect the surfaces of the desks.
 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.
 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.
 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.
 Continue constructing line segments.
 If the task is to create a tessellation, then this process could be continued over the plane.
If the task is simply 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, go on to the next page.
