Refer to the following information in doing Exercises 19–22. A particularly simple kind of polygon, called a polyomino, is made of squares joined edge-to-edge. The name is a generalization of “domino”; indeed, there is only one kind of domino (two squares joined at an edge to form a rectangle). There are just two trominos (short for “triominos”), the straight tromino and the L-tromino. The straight tromino has the shape of a rectangle, so it can tile the plane by translations; the L-tromino has the shape of a hexagon.
Question 20.52
22. Explain why alternative 1 of the Translation Criterion cannot be applied to the L-tromino. (Hint: Label each of the eight corners of the component squares of the tromino with the letters . Let these be our candidates for the points , , , and of the criterion.) Each of the sides of the tromino that is 2 units long has nowhere to go under a translation. Any application of the criterion must divide each side into two pieces, so their midpoints must be two of the points , , , and . Make a similar argument about two corners of the tromino. Thus, we have four points, which can be labeled consecutively , , , and , starting at any one of them. Show that none of the four possibilities “works.” (This argument can be generalized to show that trying , , , and at points other than the corners of the squares won’t work either.)