514 CHAPTER 8 Geometry or a regular hexagon as the tessellating shape. Fig. 8.79 shows each of these regular tessellations. Notice that each tessellation can be obtained from a single tessellating shape through the use of reflections, translations, or rotations. Figure 8.79 Exploring Tessellations We will now learn how to create unique tessellations. We will do so by constructing a unique tessellating shape from a square. We could also construct other tessellating shapes, using an equilateral triangle or a regular hexagon. If you wish to follow along with our construction, you will need some lightweight cardboard, a ruler, cellophane tape, and a pair of scissors. We will start by measuring and cutting out a 2 in. by 2 in. square from the cardboard. We next cut the square into two parts by cutting it from top to bottom using any kind of cut. One example is shown in Fig. 8.80. We then rearrange the pieces and tape the two vertical edges together as shown in Fig. 8.81. Next we cut this new shape into two parts by cutting it from left to right using any kind of cut as shown in Fig. 8.82. We then rearrange the pieces and tape the two horizontal edges together as shown in Fig. 8.83. This completes our tessellating shape. 2" 2" Figure 8.80 Move this piece to the right side. Figure 8.81 Figure 8.82 Move this piece below. Figure 8.83 We now set the cardboard tessellating shape in the middle of a blank piece of paper (the tessellating shape can be rotated to any position as a starting point) and trace the outline of the shape onto the paper. Next move the tessellating shape so that it lines up with the figure already drawn and trace the outline again. Continue to do that until the page is completely covered. Once the page is covered with the tessellation, we can add some interesting colors or even some unique sketches to the tessellation. Figure 8.84 shows one tessellation created using the tessellation shape in Fig. 8.83. In Fig. 8.84, the tessellation shape was rotated about 45° counterclockwise. An infinite number of different tessellations can be created using the method described by altering the cuts made. We could also create different tessellations using an equilateral triangle, a regular hexagon, or other types of polygons. There are also other, more complicated ways to create the tessellating shape. The Internet has many sites devoted to the creation of tessellations by hand. Many computer programs that generate tessellations are also available. Figure 8.84 Instructor Resources for Section 8.5 in MyLab Math • Objective-Level Videos 8.5 • Animation: Exploring Reflective and Rotational Symmetries • Animation: Exploring Tesselations • PowerPoint Lecture Slides 8.5 • MyLab Exercises and Assignments 8.5
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