Before we proceed we need to say what
symmetry is. Symmetry seems to be one of those things that you
can recognize but not define exactly. Consider the following
examples of symmetry.
- The letters A, C, S and X
- A red table with a blue lamp next to a blue table with a red lamp
- The equation x2 + y2 = r2
How can there be a single definition to cover all these cases?
Definition (informal) of Symmetry:
Given a set of objects, an operation performed on the objects and a
property of the set of objects, we say that we say that the
property has symmetry with respect to the operation if after performing
the operation the property remains unchanged.
For example, consider the letter A. Intuitively we would say that
A is symmetric with respect to an axis going vertically through the
center. To apply the definition we would take that axis and
reflect the letter A on either side of the axis. Then the space
occupied by the letter is symmetric with respect to reflection in the
axis. Similarly, the space occupied by the letter C is symmetric
with respect to reflection in a horizontal axis. What about the
letter S? It does not have either vertical or horizontal
symmetry. Can you think of an operation that will leave S
occupying the same space? If you wrote S on a piece of paper and
turned the paper upside down it would still be the letter S. We
say that S is symmetric with respect to rotation by 180 degrees about
the point in the center of the letter. How many different types
of symmetry can you find for the letter X?
What about the table and lamp? Do you see a symmetry operation?
Suppose we swapped the colors red and blue in the description. We
would have an equivalent description. So we can say that the
description is symmetric with respect to color swapping. At this
point the symmetry of the equation should be apparent. Swapping
the letters x and y in the equation satisfies the exact same condition.
Example from Euclicean Geometry
Suppose we are asked to
prove that the base angles of an isosceles triange are congruent.
We might start by saying that it really is intuitively
obvious. If pressed further, we might use a symmetry
argument. Suppose we flip the triangle around the vertex
angle, like so:
We can now place AB of the original triangle on CB of the flipped
triangle and the same for CB of the original triangle and AB of the
flipped triangle. Since the vertices line up it follows that the two
triangles are congruent and therefore angle A is congruent to angle C.
If pressed still further, we can express this argument in strictly
Euclidean terms. Given three points A, B and C, the order of the
vertices makes a difference, so triangle ABC is different from triangle
CBA. From the given, we have that AB of the first triangle is
congruent to CB of the second and that CB of the
first triangle is congruent to AB of the second. The included
angle B is congurent to itself so triangle ABC is congurent to triangle
CBA by SAS, and therefore by corresponding parts of congruent
triangles, angle A is congruent to angle C.
Three Examples of the Use of Symmetry
We are ready to examine three more involved examples of symmetry.