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Definitions of Asymmetry

When we try to pick out anything by itself, we find it hitched to everything else in the universe.
—John Muir, My First Summer in the Sierra

 

Excerpted from Unicycle:

Asymmetry is defined by Webster’s New World College Dictionary as “lack of symmetry.” Webster’s definition of symmetry is “similarity of form or arrangement on either side of a dividing line or plane,” and “correspondence of opposite parts in size, shape, and position.”

At first sight, symmetry is easy to find wherever we look. Your window might be a rectangle divided into four panes—though each pane will frame a different part of the view out the window. So the view through the symmetrical window would be asymmetrical. Technically speaking, symmetry is often described as happening where some operation leaves the object in question unchanged in some way—even though that changing operation is connected to it. Symmetry and paradox often go hand in hand.

The window has symmetry about the vertical axis, for example, dividing it in two. This can be described as a reflection about the vertical axis or a translation where the two left-hand panes would look the same if they were slid over somehow to replace the right-hand panes. Of course we know that such a theoretical operation in practice may not work out so simply. The pane would have to be removed, and the glass of one pane might be cut to fit exactly the original square in the window and might not fit so well if reused to replace the opposite square, especially with some warping and weathering of the frame. So although “symmetry,” loosely speaking, once we start looking for it, can be sighted just about everywhere, from the flowers to the leaves to the blades of grass to the sharpest technologies, when we take a closer look—or when we stand back—things can get complicated and less than ideally symmetrical.

There are many handy categories of symmetry that we learn about in math class. For example, we will be referencing the symmetry of repetition or translation, which one finds in a ladder or a repeating series, where any rung or unit appears to be translatable to the next position without having to undergo any change.

There is point symmetry at the center of a circle, where we set the point of our compass in order to draw the circumference. The point is symmetrical as it does not seem to change in any essential way—even as the compass needle turns in the paper.

Then there is the symmetry of a flat surface that does not appear to have any variations. Wherever we are on the surface, it is the same as any other place with respect to any other place (or even with respect to some coordinates). The surface is entirely homogenous. This raises more than a few questions and paradoxes right away—like where are we? Are we anywhere at all?

There are symmetrical equations in math that are used in what has been called a “language of symmetry” or group theory. The question remains as to whether any of this “symmetry” is really sufficiently symmetrical to merit the term, even—or especially—when it is in the psyche of the mathematician in the guise of “pure math.”

In order to address these and other related questions, we will qualify symmetry with such terms as “absolute,” “pure,” or “perfect” and use them in this way interchangeably for variety. We will consider “absolute” or “perfect” symmetry as having such uniformity as to give rise to no differences, changes, or variations, whether through a given operation or not. Anything else is asymmetric.

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