Mathematics Awareness Week 1995

The Mass-Produced Universe

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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 structure.

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

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