or eight decimal places. Numerical analysis is not, as its name would sug-
gest, the analysis of numbers; it is the branch of mathematics that deals
with finding approximate solutions to equations—to an accuracy of three,
five, eight (or whatever) decimal places. Knowing that it may not be pos-
sible to find an exact formula for the solution to an equation, yet realizing
that to build something may require an accurate approximation to that
solution, impelled mathematicians to devise techniques for finding these
approximate solutions and, equally important, knowing how accurate
these solutions are. An inexpensive pocket calculator will give the cube
root of 4 as 1.587401052; but if this number is cubed, the answer will not
be 4—although it will be very close to it. The cube root of 4 as given by
the calculator is accurate to nine decimal places—good enough for build-
ing all mechanical devices and many electronic ones. From a practical
standpoint, numerical analysis can generally determine the roots of poly-
nomials with sufficient accuracy to build anything whose construction
depends upon knowing those roots.
At the moment, though, the quest for solutions of polynomials is go-
ing in new directions. Just as the search for the roots of polynomials
took an abrupt turn at the dawn of the nineteenth century and brought
group theory into the picture, relatively new branches of mathematics
are currently being brought to bear on the problem. Many of the most
widely studied groups are connected with symmetries of objects. For
instance, we have looked at S 3 as the set of all shuff les of a three-card
deck. However, if one imagines an equilateral triangle with vertices A,
B, and C, initially starting with A as the top vertex and B and C as the
bottom left and bottom right vertices, the triangle can be rotated or re-
f lected so that the new position of the triangle corresponds to one of the
shuff les.
Triangle 1 2 3 4 5 6
Top Ver te x A C B A C B
Bottom B C A B C A C B B A A CWe can actually see how the group structure arises in this example—
there are two fundamentally different operations from which the others
are constructed. These are a counterclockwise rotation of 120 degrees,
which we could denote by R. Triangle 2 is obtained from triangle 1 by do-
ing R. The other basic operation is to leave the top vertex unchanged but
f lip the bottom two; we denote this by F. Triangle 4 is obtained from tri-
angle 1 by doing F. Similarly, triangle 3 is obtained from triangle 1 by
performing R twice; this operation is denoted RR, or R^2. Triangle 5 is
98 How Math Explains the World