The
symmetery…
Is
symmetry a force, like gravity or magnetism?
One of the books I quoted in my previous post is titled The Force of Symmetry. A literal reading of the title implies that
symmetry is a force, like the force of gravity.
But the title should be read impressionistically rather than literally,
maybe as The Influence of Symmetry in
Physics, for instance. The influence of symmetry in physics nowadays is
huge, because symmetry in its mathematically most abstract form is considered
to be the explanation for the existence of forces.
What
is a force? That’s easy to answer. A force is a push or a pull. Your body is being pulled toward the center
of Earth by gravity right now, and the chair you’re sitting in is pushing up on
you against the pull of gravity. In
terms of Newton’s third law, these aren’t the action-reaction forces,
however. The reaction force of the
Earth’s gravity pulling on your body is your body’s gravity pulling back on Earth. The action-reaction forces in the case of the
chair you’re sitting in are the “contact forces” of the chair pushing up on you
and your butt pushing back on it. Well,
partly your thighs pushing back also.
The
contact forces are electromagnetic forces in the atoms of your body and of the
chair. The action-reaction pair in the
case of you pulling up on Earth’s mass and Earth pulling down on your mass is
due to the gravitational force.
What
causes forces? Physics has a pretty easy
answer for that, too. Forces are caused
by fields. Even the contact forces of two objects in apparent direct contact with each other are caused by the interaction of the electromagnetic fields of the objects. But when you ask “what is a field?” well, now you’re getting into
the land of abstraction. Now you’re getting closer to seeing the influence of
symmetry in physics.
Before
getting into that, however, let’s not forget the simple symmetries we see all
around us. We see, for instance,
numerous examples of squares, rectangles, circles and other such geometric
figures. We see the human body’s
bilateral symmetry. In all these
examples, if you perform some transformation or operation such as rotating the
object around a certain axis by a certain angle, you find that in some cases
the object is unchanged by the transformation.
Draw a square on a piece of paper and then rotate the paper by 90°. The
square looks just the same as it did before you rotated it. Of
course, there are things you can do to prevent the square from looking the same
after you rotate it by 90°. You can draw
each side of the square in a different color.
Or if you use a rectangular sheet of paper, you will have an external
reference that tells you your square has been rotated by 90° (but once you
rotate it by 180° the external reference itself looks the same as it originally
did).
The
square can be said to have a four-fold symmetry. If you start rotating it around an axis
through its center, you find there are four angles at which symmetry clicks in
and the square looks like it originally did, no matter what the original
position of the square was. A hexagon has a six-fold symmetry, as its name implies. Snowflakes also have sixfold symmetry.
How many “folds” of symmetry does a circle have? We’re getting into the physicist’s idealized
notion of symmetry here. The circle has
“perfect” two-dimensional symmetry, or an infinite number of choices of
rotations about its center that leave it unchanged. Squares, hexagons, etc. have discrete or
countable degrees of symmetry. The
circle has continuous 2-dimensional symmetry.
Likewise, a sphere has continuous or complete 3-dimensional symmetry. Another
way of putting this is that you can crawl around on the surface of a perfect
sphere as much as you want, and no place on it will appear different from any
other place. On a cube or any other polyhedron, you can crawl around and find
its sides and corners. You won't be able to identify any corner or any side as being different from the others, but you can observe that corners are different from sides, and both of these are different from the "vertices" where more than two sides meet.
