Crystals of salt are tiny cubes. Sand dunes arrange themselves
in parallel rows. Saturn's rings are circles. The galaxy to
which our solar system belongs is a spiral. And the universe
itself is an expanding ball of space and time. The world, it
seems, is made up of patterns based on mathematics. Why?
A large part of the answer is symmetry. The regularities that we
see in the natural world are clues to deep symmetry principles in
the fundamental laws of nature -- principles that govern the
structure of space, time, and matter. Our notion of symmetry --
including the "symmetry principle" of physics -- is that of a
conceptual tool, a framework that helps us make sense of the
universe and the world around us.
Mathematicians talk not just of "symmetry" in a general sense, but
have given precise definitions in particular settings of what it
means for objects to be "symmetric." Mathematical symmetry is a
powerful concept that is used to analyze and classify structure,
and to provide models to predict as-yet-unobserved physical
To mathematicians, symmetry is both an attribute of an object as
well as a particular transformation of the object. A symmetry of
an object will change the position of individual points that make
up an object, but leave unchanged the structure of the object.
For example, a mathematician will say that a cubical salt crystal
has 4-fold rotation symmetry (an attribute). This is because if
the cube is turned 90 degrees (a quarter turn) about an axis that
pierces through the center of the square top and bottom of the
crystal, the cube's form doesn't change and it will occupy the same
position as before the turn. This quarter turn (a transformation)
is called a symmetry of the cube.
The mathematics of symmetry provides a framework for much of modern
physics. Albert Einstein's theory of relativity is founded on a
symmetry involving both space and time -- the constancy of the
speed of light. Stemming from that work, scientists today
recognize four distinct kinds of force in nature --
electromagnetism, gravity, and the strong and weak nuclear forces.
Symmetry principles have already led to a unified theory of two of
these forces -- electromagnetism and the weak force. Now
mathematicians and physicists are hot on the trail of a unified
theory of all four fundamental forces.
The deep symmetries of the universe tell us that when space, time,
and matter are transformed in certain ways, the laws of nature
remain unchanged. The laws of physics on Earth today are the same
as they were in a distant galaxy a million years ago. Transforming
places, or times, does not alter the laws. Every electron in the
universe is exactly the same as any other electron -- so a
transformation of matter that interchanges two electrons has no
effect on these laws. In fact, we live in a mass-produced
universe, one that uses the same "components" over and over again.
Not all symmetries of the underlying laws can be realized by a
physical system that obeys them. For example, a salt crystal is
made from huge numbers of atoms and chlorine and sodium, and the
crystals form cubes because those atoms are arranged in a regular
cubic lattice. We are amazed at the beautiful symmetry of this
lattice; but actually the lattice has less symmetry than the laws
that create it.
This year, to highlight its importance, Mathematics and Symmetry
has been chosen as the theme of Mathematics Awareness Week, which
is the last week in April. During Mathematics Awareness Week it is
appropriate to remember that mathematics has led to the discovery
that there is a common principle behind many of nature's striking
patterns and forms. That principle is not just symmetry, but the
changes in symmetry that occur in natural processes.
Back to Math Awareness Week 1995