For there to be a true tessellation, three things are necessary: 1) there must be no gaps between the shapes, 2) there are no overlays, and 3) there is a pattern of shapes that repeats itself and can be extended.
A checkerboard or a honeycomb fit this criteria but Escher takes this idea one step further and makes his patterns interweave with other concepts. In Day and Night, the black and white birds not only tessellate but they also turn into the fields below. In Circle Limit III, he has two types of fish that link together through tessellation but introduces the concept of perspective as well, with the tessellation growing smaller until it fades around the edge. The boys wanted to get magnifying glasses to see if they could determine where the end was.
Build tessellations, bit by bit,
Repeating patterns, perfect fit,
Like checkerboards or bathroom tiles.
Make patterns stretch for miles and miles,
Remember: There can be no gap;
All shapes must fit, not overlap. -GEMS Build It!
To give them some hands-on experience with tessellations before attempting to make your own Escher-style art work, you can have your students fill in tessellation masters from the GEM's Build It! guide.
Your students may have a harder time with this project than most art projects, but hopefully it will then give them a greater appreciation for Escher's mastery.
Start with a square of cardstock. Cut a piece from one side of the square and taped it to the opposite side. Do the same procedure with the top and bottom. You now had a shape that you can tessellate as each bump will fit perfectly into each recess. Now trace the shape over and over onto white paper, making sure the edges touch. Color the shapes into patterns. Mount them on colored construction paper backgrounds. If you wish, you could also make a picture out of your shape, such as a bird, fish or frog, however, my students found it enough of a challenge just to have abstract tessellations.