POLYGONS AND TESSELLATIONS
A simple connect-the-dots exercise is used to introduce the concept of a polygon. The 50-sided polygon in the solution resembles a reptile. The creature is an adaptation of the one that appears in Escher's lithograph Reptiles. In the graphic, several identical lizards interlock in a jigsaw puzzle configuration or tessellation. A set of 15 large soft foam lizards is available from the Imagination Project (telephone 888-477-6532 or 513-860-2711).
Reptiles - M. C. Escher
To introduce students to regular polygons, each is provided with a pair of plastic mirrors hinged together with cloth tape. [Suitable 2 ¼" x 3 ½" mirrors are available from ETA/Cuisenaire.] When the assembly is opened like a book, the mirrors stand alone. If the students place their hinged mirrors on the broken lines in the figure below, they will view a reflection-based equilateral triangle.
If the students move their hinged mirrors towards one another, a square will eventually be visible. If they continue to move the mirrors towards one another, they will see, in turn, a regular pentagon, a regular hexagon, a regular heptagon, and so forth. After investigating the measure of each polygon's interior angle, the students discover that only three of the regular polygons tessellate the plane. (The measure of their interior angle divides 360 exactly.)
If a third mirror is added to the hinged mirror assembly, the resulting equilateral triangle prism can be used to generate tessellating art with reflectional symmetry. (The configuration of mirrors is maintained with an elastic band.) A generating triangle for Escher's Lizard/Fish/Bat tessellation (Tessellation 85) appears below.
	Tessellation 85 - M. C. Escher (click for enlargement)
Students can create a generating triangle for a tessellation with reflectional symmetry by drawing identical or distinct curves from the center of an equilateral  triangle to its vertices. Two examples follow.