If the image of a shape after reflection is exactly the same as the shape itself, we say the shape has bilateralsymmetry. The mirror line
that leaves the shape unchanged is called the line of symmetry of the shape and the reflection transformation is a symmetry of the shape. Bilateral symmetry is very common in nature; for example, the human body and
many kinds of leaves have bilateral symmetry. Artists often use bilateral symmetry to give balance and harmony to their work.
16. For each of the following bilateral designs, locate the line of symmetry.
17. Which of the following designs have bilateral symmetry? For those that do, give the line of symmetry.
18. Make your own designs by folding a piece of paper in half and then cutting out a shape. When you unfold the shape, you will have a design with bilateral symmetry. Where is the line of symmetry?
19. Do the designs you created for questions 14 and 15 have bilateral symmetry? Where are the lines of symmetry?
20. Find some examples of bilateral symmetry in nature and locate the line of symmetry for each object.
21. Find some examples of bilateral symmetry around you—advertising logos, familiar objects, or artistic designs—and locate the line of symmetry for each design.
If a figure or design has more than one mirror line, we say that it has dihedral symmetry. If there are 3 mirror lines, we say the figure has 3-fold dihedral symmetry or has dihedral symmetry of order 3, and so on.
23. What do you notice about how all the mirror lines of a specific design are related to each other? Can you explain why this relationship holds in these designs? Will this relationship always hold in a dihedral design?
24. Which of the following designs have dihedral symmetry and which have only cyclic symmetry? Give the order of rotation and, if the design is dihedral, give the number and location of lines of symmetry.
25. Draw several designs with 3-fold, 4-fold, and 6-fold dihedral symmetry. Which of your designs also have cyclic symmetry?
26. Will a design with dihedral symmetry also have cyclic symmetry? How can you characterize designs that have only cyclic symmetry?
27. For a design that has dihedral symmetry and cyclic symmetry, how is the number of mirror lines related to the order of rotation of the design?
28. For a design that has dihedral symmetry and cyclic symmetry, how is the center of rotation related to the mirror lines?
29. Artists sometimes use cyclic designs and sometimes use dihedral designs.
What is the main difference in terms of visual effect between dihedral designs
and cyclic designs?
Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations.
Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.
Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
References
For All Practical Purposes: Introduction to Contemporary Mathematics,
W.H. Freeman and Company, New York, 1997
Understanding Congruence, Similarity, and Symmetry Using Transformations and
Interactive Figures: Visualizing Transformations, NCTM Principles and Standards
for School Mathematics: E-example 6.4. http://standards.nctm.org/document/eexamples/chap6/6.4/index.htm
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