8.5 Transformational Geometry, Symmetry, and Tessellations 513 Tessellations A fascinating application of transformational geometry is the creation of tessellations . B9 D9 C9 E9 F9 G9 H9 I9 J9 K9 L9 M9 N9 O9 Q9 A9 (a) (b) F0 G0 H0 E0 I0 J0 K0 L0 M0 N0 O0 Q0 A0 B0 C0 D0 P P Figure 8.77 b) To determine whether the polygon has 180° rotational symmetry about the point P, we rotate the polygon 180° as shown in Fig. 8.77(b). Compare Fig. 8.77(b) with Fig. 8.76. Notice that other than vertex labels, the position of the polygon after the rotation in Fig. 8.77(b) is identical to the position of the polygon before the rotation (Fig. 8.76). Therefore, the polygon has 180° rotational symmetry about point P. 7 Now try Exercise 45 Exploring Reflective and Rotational Symmetries Definition: Tessellation or Tiling A tessellation , or tiling , is a pattern consisting of the repeated use of the same geometric figures to entirely cover a plane, leaving no gaps. The geometric figures used are called the tessellating shapes of the tessellation. Fig. 8.78 shows an example of a tessellation. Perhaps the most famous person to incorporate tessellations into his work is M. C. Escher (see Profiles in Mathematics ). Figure 8.78 The simplest tessellations use one single regular polygon as the tessellating shape. Recall that a regular polygon is one whose sides are all the same length and whose interior angles all have the same measure. A tessellation that uses one single regular polygon as the tessellating shape is called a regular tessellation . It can be shown that only three regular tessellations exist: those that use an equilateral triangle, a square, Elfinadesign/Shutterstock Profile in Mathematics Maurits Cornelius Escher In addition to being wonderfully engaging art, the work of M. C. Escher (1898–1972) also displays some of the more beautiful and intricate aspects of mathematics. Escher’s work involves Euclidean, non-Euclidean (to be studied shortly), and transformational geometries. Amazingly, Escher had no formal training in mathematics. In 1936, Escher became obsessed with tessellations. Escher kept a notebook in which he kept background information for his artwork. In this notebook, Escher characterized all possible combinations of shapes, colors, and symmetrical properties of polygons in the plane. By doing so, Escher had unwittingly developed areas of a branch of mathematics known as crystallography years before any mathematician had done so!
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