In the Classroom
Hands-On Discovery of Mirror Planes Deborah A. Moore Department of Mathematics, University of Puerto Rico, Mayagüez, PR 00681-9019 José E. Cortés-Figueroa* Department of Chemistry, University of Puerto Rico, Mayagüez, PR 00681-9019;
[email protected] In the study of chemical applications of group theory, some students find it difficult to identify the symmetry elements in a simple geometrical figure or a molecular model. Pattern blocks and mirrors can be used to identify mirror planes of symmetry. These blocks and mirrors can be purchased at low cost from catalogs that sell products for elementary mathematics education. Each student should have a set of blocks and a mirror (see Figs. 1 and 2). The objective is for each student to identify all the mirror planes in the object in Figure 1. The pattern blocks shown in Figure 2 were derived from the hexagonal prism by cutting along mirror planes of symmetry: either through two opposite hexagon vertices or through the midpoints of opposite edges. These sections can be combined with the mirror to determine where there are mirror planes
in the block in Figure 1. Figure 3 uses a horizontal mirror to create a composite virtual/real image with the full hexagonal geometric symmetry of Figure 1, but distinguishable because of a difference in color. If the section of the original figure and its mirror image combine to produce a new figure that is indistinguishable from the original figure, then the plane of the mirror is identified as a plane of reflection. If the new combined figure can be distinguished from the original, then the plane of the mirror is not a symmetry element. Figure 4 uses a vertical mirror to locate one of the three planes of reflection of Figure 1 that pass through opposite vertices. This type of exercise offers the opportunity to construct and develop the concepts of symmetry elements and operations without resorting to rote manipulations.
Figure 1. Right hexagonal prism, white above, gray below.
Figure 3. This horizontal plane of reflection creates a new figure with the same shape as the original, but it is distinguishable because of its coloring.
Figure 2. Pattern block pieces that can be used to form all needed combinations.
Figure 4. The mirror shows a figure indistinguishable from Figure 1.
JChemEd.chem.wisc.edu • Vol. 78 No. 1 January 2001 • Journal of Chemical Education
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