Axis of Symmetry
The axis of symmetry is a critical concept in geometry and mathematics that defines the line around which a shape or function can be reflected to produce a congruent image. This transformation maintains the size and shape of the original figure, which is why it is classified as a congruence transformation. Shapes that exhibit this property are described as having reflectional symmetry, or bilateral symmetry, where one half mirrors the other across the axis. This concept extends beyond geometric shapes to numerical functions, especially when plotted on a Cartesian coordinate plane.
Identifying an axis of symmetry involves locating the line that, when used for reflection, results in an identical representation of the original function. For instance, linear functions can have vertical or horizontal axes of symmetry, while specific non-linear functions may have a unique axis. Functions that meet certain symmetry criteria, such as being identical when evaluated at positive and negative inputs, are termed even functions, with the y-axis often serving as their axis of symmetry. The idea of symmetry in shapes and functions has practical applications in various fields, including art, design, and psychology, where it is often associated with aesthetic appeal.
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Axis of Symmetry
A reflection is one of the most basic geometric transformations. It is a congruence transformation, because the size and shape remains constant despite the transformation. A reflection is defined by an axis of symmetry around which the reflection takes place. See Figure 1.
If there exists a line of symmetry (see Figure 2) around which the final shape or image is exactly identical to the original shape or image, it is said to have reflectional symmetry, also called mirror or bilateral symmetry. Humans often use bilateral symmetry in design, as evidenced by the architecture and craftsmanship of the earliest human civilizations. Modern psychological evidence suggests that such symmetry in facial features is related to the human notion of beauty.
These relationships for shapes can also apply to relationships of numbers. Numerical functions, which can be represented visually on a Cartesian coordinate plane, can also possess bilateral symmetry. For numerical relationships that possess this form of symmetry, the line of symmetry used to define this symmetrical relationship is called an axis of symmetry.
Overview
Identifying an axis of symmetry for a numerical relationship involves identifying the line of symmetry about which a function or shape can be reflected to obtain a result that is identical to the original shape or function. One of the most basic examples for consideration would be a constant function, such as f(x) = 3 (Figure 3). For an infinite linear function, any line that is perpendicular to the linear function could serve as an axis of symmetry for the function. In the case of a constant function, a horizontal line on the Cartesian plane, any vertical line can be used as an axis of symmetry to obtain an equivalent function.
The function f(x) = x (Figure 4) is also symmetrical. The axis of symmetry in this case is not vertical, but it must be perpendicular to the line f(x), so any straight line with a slope of −1, such as g(x) = −x, will serve as an axis of symmetry.
Though the previous two examples are linear equations, the principle also applies to many non-linear equations. In the case of f(x) = x2 (Figure 5), there exists only a single line, the y-axis at x = 0, about which f(x) can be reflected to get a resulting image that is identical to the original image.
Certain symmetrical relationships within mathematical functions have particular significance. When the axis of symmetry is x = 0, the function results in f(x) = f(−x). Such functions are called even functions. Examples include cos(x) and x4.
Bibliography
McKellar, Danica. "Romantic Walks on the Beach: Congruence Transformations." Girls Get Curves. New York: Hudson Street, 2012.
Sautoy, Marcus. Symmetry: A Journey into the Patterns of Nature. New York: Harper, 2008.
Stewart, Ian. Why Beauty Is Truth: A History of Symmetry. New York, NY: Basic, 2007.
Than, Ker. "Symmetrical Bodies Are More Beautiful to Humans." National Geographic. National Geographic Society, 18 Aug. 2008. Web. 7 Jan. 2015. <http://news.nationalgeographic.com/news/2008/08/080818-body-symmetry.html>
Weyl, Hermann. Symmetry. Princeton, NJ: Princeton UP, 1980.
Wilders, Richard, and Lawrence VanOyen. "Turning Students into Symmetry Detectives." Mathematics Teaching in the Middle School. 17.2 (2011): 103-107.
Williams, Maggie. "Finding Lines of Symmetry." Illuminations: Resources for Teaching Math. National
Council of Teachers of Mathematics. Web. 2 Jan. 2015 <http://illuminations.nctm.org/Lesson.aspx?id=1800>.