obtained from triangle 1 by performing R first, then F, denoted RF; trian-
gle 6 is likewise obtained by performing F first and then R or FR.
This is essentially the same group as the shuff les of a three-card deck—
one can identify R with the shuff le that simply puts the top card on the
bottom of the deck, and F with the shuff le that leaves the top card alone
but switches the position of the second and third cards. This process of
identifying two apparently different groups with each other is known as
isomorphism—a process that enables mathematicians to translate truths
known about one object to truths known about the other. The proof that
the general quintic has no solution involves a group isomorphic to the
group of symmetries of the regular icosahedron—the regular platonic
solid with twenty faces, all of which are equilateral triangles. Mathemati-
cians nowadays are looking to geometry in the hope that they can dis-
cover things that will translate into problems involving roots of
polynomials.
The French politician Georges Clemenceau once said that war was too
important to be left to the generals. Similarly, group theory was too im-
portant to be left to the mathematicians. Group theory is employed exten-
sively in the sciences, because group theory is the language of symmetries,
and science has discovered that symmetry plays a fundamental role in
many of its laws. I’m not sure whether anyone has written Group Theory
for Anthropologists or Group Theory for Zoologists, but there are books with
similar titles written for biochemists, chemists, engineers—probably the
greater part of the alphabet, and I’d be willing to bet that every letter of
the alphabet is represented when it comes to describing types of groups
(we already observed that the letter s is used for “solvable group”). Notic-
ing patterns, and missing elements of patterns, is often the key to impor-
tant discoveries, and group theory provides an organizing framework that
often points the way to the missing element.
The story of the search for solutions by radicals to polynomial equations
did not end with the discovery that one could not find formulas for the
quintic; rather, it branched off to generate useful and exciting results that
even Cardano and Ferrari, who scaled the summit of what could be done
in this area, would undoubtedly have found every bit as enchanting as
those revealed in Cardano’s Ars Magna.
NOTES- See http:// history1900s .about .com/ od/ 1950s/ a/ hopediamond .htm.
- It would be more accurate to say that polynomials are the only everywhere dif-
ferentiable functions we can calculate. For example, the function f(x), beloved of
The Hope Diamond of Mathematics 99