INVESTIGATING ESCHER'S TESSELLATIONS
Once students have experience with creating tessellating art, they are ready to learn other ways to create these symmetrical patterns. To begin, I review the regular tessellations and introduce other tessellating polygons like rectangles, parallelograms, and quadrilateral kites.
Next, I present several of Escher's tessellations, each generated by its own unique modifying rule or rules. Each student is provided with a set of corresponding worksheets, and instructed to add the parent polygon as demonstrated with the Pegasus tessellation. [Remember: Join points where more than two tessellating shapes meet.] The students study each tessellating shape in its parent polygon, looking for corresponding "bumps" and "holes", and deduce the transformations used to modify the polygon.
In the lizard of Escher's Tessellation 104, a modification to either the top or bottom side of the parent square is rotated 90 degrees to an adjacent side. Each rotation is about a vertex of the square between the related sides, inevitably alternate vertices of the square.
	In Escher's Tessellation 99, a modification to one half-side of the parent equilateral triangle is rotated 180 degrees about the midpoint of that side to the adjacent half-side. Then a modification to one of the other sides is rotated 60 degrees to the third side about the vertex between them.
The parent parallelogram of Escher's Tessellation 75 is modified by translation between one pair of parallel sides and by rotation about the midpoint of each of the other sides. In the latter, a modification to one half-side of either side is rotated 180 degrees about the midpoint of that side to the adjacent half-side.
	The parent kite-shaped quadrilateral of Escher's Tessellation 66 is modified by glide reflection between two sets of equal and adjacent sides. Each modification is flipped (L/R) and then translated (vertically) to the equal and adjacent side.
The parent parallelogram of Escher's Tessellation 97 is modified by translation between between one pair of parallel sides and by glide reflection between the other pair. In the latter, a modification to one side is flipped (L/R) and then translated (vertically) to the equal and opposite side.
As each Escher tessellation is analyzed, one or more examples of similar student artwork is presented. This Tessellating Rabbit is similar in construction to Escher' lizard of Tessellation 104. The corresponding tessellation appears in the background as a watermark.
Several examples of student tessellating art will be found in my books Investigating Patterns: Symmetry and Tessellations, Teaching Tessellating Art, and Introduction to Tessellations (co-authored with Dale Seymour). "The" source for color reproductions of all of Escher's tessellation is Doris Schattsneider's definitive book Visions of Symmetry. My Escher Gallery includes several high resolution reproductions of Escher's tessellations.