The following questions review the main ideas of this chapter. Write your answers to the...

The following questions review the main ideas of this chapter. Write your answers to the questions and then refer to the pages listed by number to make certain that you have mastered these ideas.

What is a regular tiling? pg. 66 What is the difference between an edge-to-edge tiling and one that is not an edge-to-edge tiling? pg. 66 Why are there no regular tilings for n-gons if n is greater than 6? pg. 67 How many regular tilings are possible? pg. 67

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Now that we have examined polygons and their angle measures, we are ready to take a closer look at tilings. A regular tiling is a tiling composed of regular polygonal regions in which all the polygons are the same size and shape. Figure 2.10 shows three regular tilings. You may have seen tilings similar to these on floors or countertops.

Notice that the polygonal regions in the tilings in Figure 2.10 have entire sides in common. Such tilings are called edge-to-edge tilings. Tilings need not be edge to edge. One example of a tiling that is not edge to edge was shown in Figure 2.4.

By sliding alternating rows to the right or left so that the top vertex of each triangle intersects the midpoint of the base of the triangle directly above it, we form a new regular tiling as shown in Figure 2.11(b). This new tiling is not edge to edge because some adjacent triangles do not share an entire edge. We have seen regular tilings using three kinds of regular polygons: equilateral triangles, squares, and hexagons. How many other regular tilings are possible? For example, could we tile a floor with regular pentagons? With regular octagons? To answer this question, note that in every edge-to-edge tiling, the vertex angles of tiles must meet at a point (see Figure 2.10). This point is a vertex for each polygonal tile that joins other tiles at that point. An enlarged view of such a point is shown in Figure 2.12. We will consider only edge-to-edge regular tilings in the discussion that follows.

In the case of a regular (edge-to-edge, using regular polygonal regions) tiling, the vertex angles at a point must all have the same measure. For example, in the tiling with equilateral triangles in Figure 2.10(a), six 60° angles are formed at each vertex, as shown in Figure 2.13(a). In the square tiling in Figure 2.10(b), four 90° angles are formed at each vertex, as shown in Figure 2.13(b). In the hexagonal tiling in Figure 2.10(c), three 120° angles are formed at each vertex, as shown in Figure 2.13(c).