Math (562419), страница 5
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The square demonstrates the same rotation and reflection symmetries as the Ethiopian cross and the Cypriot mosaic design. See figure 5. A description of the symmetry motions can be simplified. For the square, instead of thinking of the one-quarter turn, one-half turn, and three-quarter turn as different motions, the one-quarter turn can be considered as the unit motion. Thus, the one-half turn is the one-quarter turn applied twice, and the three-quarter turn is the one-quarter turn applied three times. Students can verify this result by manipulating tracing paper or transparencies as previously described. In general, the smallest rotational symmetry of an object is represented by r and successive rotations by r[sup 2], r[sup 3], r[sup 4], and so on. For the square, r is the one-quarter turn and r[sup 2], r[sup 3], and r[sup 4] represent turns of two-quarters, or one-half; three-quarters; and four-quarters, respectively.
Although more than one line of reflection often exists, specifying only one line is sufficient. Reflection with respect to this line can be represented by m. The remaining reflections can then be created by combining rotation and reflection motions. In general, a sequence of r's and m's indicates that these symmetry motions are applied sequentially to an object, with the order in which they are applied being read from right to left. In this article, the symbol "diamond" is used to indicate the sequential application of motions. For the square, we can define the line of reflection to be the vertical one, as shown in figure 6. The sequence r diamond m indicates a reflection through this line followed by a rotation of one-quarter turn counterclockwise, which is equivalent to a reflection through the original diagonal AC. This sequence is shown in figure 7.
GROUP SYMMETRIES OF THE SQUARE
A symmetry group is a special case of a mathematical group, but great diversity exists among the members of any one symmetry group. In spite of the differences, the implicit mathematical characteristics that determine group membership allow even the untrained eye to recognize the unity. Figure 8 shows examples from Ethiopia and Cyprus that are members of one symmetry group; all the designs contain exactly four rotation symmetries and exactly four reflection symmetries. For item (c), consider the inner cross. For items (e), (f), and (g), consider only the outlines and not color or internal design variations. Members of a symmetry group that contains only the four rotation symmetries of the square are shown in figure 9 (p. 368). For the mosaic design, consider only the pattern outline and not color variations.
IDENTITY AND INVERSE OPERATIONS
A complete discussion of symmetry groups includes two additional operations that can be applied to a figure. The identity symmetry motion, denoted by "1," leaves the original figure unchanged. An inverse symmetry motion returns the object to the original figure. In the square, for the basic rotation unit r of one-quarter turn counterclockwise, the inverse rotation is denoted by r[sup -1] and is a three-quarter turn counterclockwise. Thus, r[sup -1] diamond r = 1; that is, applying a counterclockwise three-quarter turn after applying a counterclockwise one-quarter turn leaves the original figure unchanged.
THE SYMMETRIES OF THE EQUILATERAL TRIANGLE
The equilateral triangle contains symmetries analogous to the reflection and rotation symmetries of the square. Figure 10 shows examples of designs that contain only threefold rotation symmetry. The rotocenter is the point of intersection of the angle bisectors of the triangle; the unit of rotation, r, is a one-third turn, or 120 degrees. A rotation of two-thirds of a turn, or 240 degrees, is represented as r[sup 2]. Figure 11 is a sketch of the triangle showing these rotation symmetries. The equilateral triangle also contains three lines of reflection, as shown in figure 12. Students can convince themselves that these lines are lines of reflection by drawing the lines on an equilateral triangle, cutting out the triangle, and folding along the lines.
Many designs that contain threefold rotation symmetry also contain the reflection symmetries of the equilateral triangle. To illustrate both rotation and reflection symmetries combined, a figure that has reflection symmetry through its center is placed in each third of the triangle, as in figure 13. The mosaic design shown in figure 14 is from Kourion in Cyprus; it illustrates the rotation symmetry of the equilateral triangle. The original is not well preserved, but the intended threefold symmetry of the pattern is evident. All figures that contain both the rotation and reflection symmetries of the equilateral triangle belong to a single symmetry group; see figure 15 for examples.
As with the square, indicating one line of reflection and one rotation is sufficient for the equilateral triangle. This line of reflection, m, can be the perpendicular bisector drawn from vertex A in the originating A ABC, as shown in figure 12. The rotation unit of 120 degrees is represented by r. If the lines of reflection m, m[sub 1], and m[sub 2] remain fixed and do not change position as the triangle is rotated, reflection in line m[sub 1] can be expressed as "m r," that is, a one-third turn followed by a reflection in m. Similarly, reflection in line m[sub 2] can be expressed as "m diamond r[sup 2]," that is, two one-third turns and reflection in m. Thus, all symmetries of the equilateral triangle can be expressed as a set of six motions in terms of r and m: [1, r, r[sup 2], m, mr, mr[sup 2]}. The symbol "diamond" can be omitted when the meaning of the sequence of motions is clear. Students should convince themselves that this set of six motions expresses all symmetries contained within the equilateral triangle. Again, a model with tracing paper can make the experience more concrete.
AN EXTENSION EXPLORATION
Students in advanced classes can explore consecutive applications of the symmetry motions in more depth and in abstract form. These applications can be related to a mathematical group that is a collection of elements and an operation applied to the elements that satisfy the following characteristics: (1) the set of elements is closed with respect to the defined operation; (2) an identity element exists; (3) for each element in the set, an inverse element exists; and (4) the operation is associative. Taking as the set of elements the symmetry motions of the equilateral triangle and the operation diamond as the application of the motion read from right to left, table ! (p. 370) shows the outcomes of applying diamond to the set {1, r, r[sup 2], m, mr, mr[sup 2]} with itself. The convention for reading the order of operations is row by column.
The outcomes in table 1 can be simplified to the symmetries shown in table 2 (p. 370). Students can verify that the equilateral triangle with the set of six symmetries and the operation of diamond satisfy the properties of a mathematical group. The outcomes from combining the six symmetries can be written in terms of the original set of symmetries. A study of the table verifies the properties of closure, identity, and inverse. Associativity can be explored by considering a number of examples, such as (m diamond r[sup 2])diamond r = m diamond (r[sup 2] diamond r) = m. Students can conclude that the property of associativity appears to hold, even though it has not been proved.
DEVELOPING INTEGRATED UNITS WITH ART, HISTORY, AND OTHER ACTIVITIES
Symmetry groups can also help students see mathematics as a human activity that overcomes the sterility that is sometimes associated with it. The mathematical developments shown in this article offer an opportunity to develop an interdisciplinary unit among the mathematics, social studies, and art teachers. For the mosaics of Cyprus, the historical link could be studying the Roman world during classical and early medieval times. A link with art could include studying how mosaic designs are created on paper and transferred to tiles or onto pavement. Students could create their own mosaic designs on graph paper and then render them onto unit squares of wood or cardboard with small colored tiles set into mastic, using grout to fill in any remaining spaces. For the crosses of Ethiopia, the historical link could be studying the adaptation of Christianity by an African culture.
Students can also create their own designs to illustrate the different types of group symmetry. By working collaboratively with a defined set of symmetries, each group of students can create a design to illustrate the given set of symmetries. Graph paper, straightedges, and compasses are all that are needed, although computer software can serve as a modern tool. The differences among the resulting designs illustrate the common underlying mathematical concepts and the potential for diversity in their interpretation.
CONCLUSION
A mathematical group is often difficult for students to understand. Symmetry groups furnish a visual image for this abstract concept and a cultural environment in which it can be embedded. The designs that are members of any one symmetry group are both the same and different. The similarities exist because of the universality of the underlying mathematical principles; the differences exist because of the differences in the cultures that produce them. The mosaic designs of Cyprus and the religious art of Ethiopia are radically different with respect to the media that were used to create them and the uses to which they were put. But the significant mathematics that is at the base of their creation is the same and should not be taken lightly. As Stevens (1996, 168) quotes Herman Weyl,
[o]ne can hardly overestimate the depth of geometric imagination and inventiveness reflected in these patterns. Their construction is far from being mathematically trivial. The art of ornament contains in implicit form the oldest piece of higher mathematics known to us.
The visual images that lead to an informal definition of the concept of a symmetry group can lay the foundation for more formal definitions and higher levels of abstraction. For all students, the examples shown can provide a concrete visual image and intuitive notion of the mathematical unity that underlies a mathematical group.
TABLE 1 Application of Consecutive Motions of Symmetries of the Equilateral Triangle
Legend for Chart:
A - diamond
B - 1
C - r
D - r[sup 2]
E - m
F - mr
G - mr[sup 2]
A B C D
E F G
1 1 r r[sup 2]
m mr mr[sup 2]
r r r[sup 2] r[sup 3]
rm rmr rmr[sup 2]
r[sup 2] r[sup 2] r[sup 3] r[sup 4]
r[sup 2]m r[sup 2]mr r[sup 2]mr[sup 2]
m m mr mr[sup 2]
mm mmr mmr[sup 2]
mr mr mr[sup 2] mr[sup 3]
mrm mrmr mrmr[sup 2]
mr[sup 2] mr[sup 2] mr[sup 3] mr[sup 4]
mr[sup 2]m mr[sup 2]mr mr[sup 2]mr[sup 2]
TABLE 2 Application of Consecutive Motions of Symmetries of the Equilateral Triangle Simplified
Legend for Chart:
A - diamond
B - 1
C - r
D - r[sup 2]
E - m
F - mr
G - mr[sup 2]
A B C D
E F G
1 1 r r[sup 2]
m mr mr[sup 2]
r r r[sup 2] 1
mr[sup 2] m mr
r[sup 2] r[sup 2] 1 r
mr mr[sup 2] m
m m mr mr[sup 2]
1 r r[sup 2]
mr mr mr[sup 2] m
r[sup 2] 1 r
mr[sup 2] mr[sup 2] m mr
r r[sup 2] 1
Fig. 1 Symmetrical designs from Cyprus and Ethiopia
ASTRONOMICAL MATH
Source: Mathematics Teacher, Dec99, Vol. 92 Issue 9, p786, 7p, 1 chart, 12 diagrams Author(s): Ryden, Robert
High school mathematics teachers are always looking for applications that are real and yet accessible to high school students. Astronomy has been little used in that respect, even though high school students can understand many of the problems of classical astronomy. Examples of such problems include the following: How did classical astronomers calculate the diameters and masses of Earth, the Moon, the Sun, and the planets? How did they calculate the distances to the Sun and Moon? How did they calculate the distances to the planets and their orbital periods? Many students are surprised to learn that most of these questions were first answered, often quite accurately, using mathematics that they can understand.
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