Mathematics Points

 

Axioms of Euclidean Plane Geometry

The great advantage of expressing geometry as an axiomatic system was that it no longer was necessary to memorize long lists of independent facts about the nature of the universe--one only had to know a small set of axioms, and by applying to them the rules of inference, one could reconstruct the entire collection of geometric truths.

Noncommutative Algebra

a great many of thc most familiar algebraic relationships originated from real problems, some of them geometric and some from economics and physical science. Parabolas were familiar as conic sections and as the paths of projectiles, and the simple formula for a parabola could easily be graphed using the techniques of analytic geometry. Volume formulas suggested cubic equations, and they were also easy to graph and analyze. But it was not much harder to use the same techniques to analyze polynomials of degree four or five or higher. Few objected that this was not algebra, even though it no longer had a convenient geometric interpretation.

Later, mathematicians began to realize that laws for combining elements in other sorts of systems satisfied most of the axioms of ordinary algebra, and so these systems behaved in large measure like numbers. An important example is the collection of symmetries of a square. We can rotate a square by one-quarter turn in a counterclockwise direction to move all four vertices. Rotating the square again results in a half turn and rotating once again yields a three-quarter turn. It does not matter in which order we make one turn after another. The rotational symmetries of the square form a commutative system.

But the collection of all symmetries of the square is not a commutative system. When we combine such reflections with rotations, the order in which we apply them makes a difference.

If we reflect across a diagonal and then rotate by a quarter turn, the effect is the same as a vertical reflection. On the other hand, if we first rotate by a quarter turn and then reflect across the diagonal, the effect is the same as a horizontal reflection.

But although the collection of symmetries is not commutative, it is algebra, albeit non-commutative algebra.

Another accepted non-commutative system was the algebra of quaternions.

If it was relatively easy for people to accept noncommutative algebra, why was it so difficult for them to accept an alternate geometry, one that satisfied some but not all of the axioms of Euclid?

Reflecting across a diagonal then rotating a quarter turn (top) gives a different effect from rotating a quarter turn then reflecting (bottom).

No one had suspected that there could be a geometry having triangles whose angles summed to greater than or less than 180 degrees, and then these geometries appeared.

Beyond the Third Dimension

                                                                    by
                                                    Thomas F. Banchoff

Originally distributed by Scientific American Library, W. H. Freeman and Co.
All electronic rights reserved, Copyright Thomas F. Banchoff