Outer Limits of Reason (58 page)

Note these formulas use addition, subtraction, multiplication, division, squares, and square roots.

What about “cubic” equations such as

ax
³
+
bx
²
+
cx
+
d
= 0?

Are there standard formulas to solve such equations? In the sixteenth century Gerolamo Cardano
⁹
showed that there are three solutions and that they are given with fairly complicated formulas.
¹⁰
The solutions employ the usual operations, including the square root and cube root.

Pressing on, we can ask about a “quartic” equation:

ax
⁴
+
bx
³
+
cx
²
+
dx
+
e
= 0.

Lodovico Ferrari (1522–1565) and Niccolò Fontana Tartaglia (1499–1557) found solutions for such problems. Many readers would be anxious if we actually wrote down the four formulas for the four possible solutions. Rather than write them, let's just say that the “quartic formula” uses the usual operations, square roots, cube roots, and fourth roots.

What about a “quintic” equation?

ax
⁵
+
bx
⁴
+
cx
³
+
dx
²
+
ex
+
f
= 0.

Here things get more interesting. One would imagine that there are “quintic formulas” composed of the usual operations and all root operations up to the fifth power. This is not true! There are no such formulas! In the early nineteenth century, Paolo Ruffini (1765–1822) and Niels Henrik Abel (1802–1829)
¹¹
proved that no such general formulas utilizing the usual operations and root operations exist. This means that there will never be a simple formula that provides the solutions for every single
a, b, c, d, e,
and
f
in a quintic equation. This is another clear example of a limitation of mathematics.

In general, the problem is unsolvable. Nevertheless, there are solutions for certain quintic equations that are easy to find. For example,

x
⁵
– 1 = 0

has a solution at
x
=
1.

This is the heart of Galois's work. He was able to use the coefficients of a given quintic equation to determine if the equation would be solvable with the usual operations. To this end, Galois introduced the notion of a
group
. A group is a mathematical structure that models the idea of symmetry. Galois showed how to associate a group with every equation. With these symmetries, he was able to determine if the given quintic equation would be solvable by means of the normal operations. Once his work for solving quintics was understood, it was used in many other areas of mathematics and science.

The idea of describing symmetries involved a major revolution in modern mathematics, chemistry, and physics. Much of modern mathematics and the sciences studies different forms of symmetry and hence different types of groups. It is from this point of view that we can finally understand Weyl's statement about the importance of Galois's letter: modern mathematics and science extensively use concepts that Galois introduced.

We would be in way over our heads if we were to get into the nitty-gritty of how
Galois theory
actually works. Suffice it to say that first the symmetries of a mathematical or physical system are described. With this in place, researchers make sure that the symmetries are preserved with different operations or physical laws. The fact that a system cannot violate its symmetries can be seen as limitations of the system.

The classical unsolvable problems of constructions with straightedge and compass that we met in the last section can all be proved unsolvable using Galois theory. An additional problem we have not discussed asks whether a polygon for which each edge has the same length—called a
regular polygon
—can be constructed. An equilateral triangle and a square can be constructed. What about a pentagon? Or an arbitrary
n
-sided regular polygon? Galois theory tells us exactly which
n
-sided regular polygons can be constructed using a straightedge and a compass. So if

n
= 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, . . ., 257, . . . or 65,537, . . .

then an
n
-sided regular polygon is constructible. By contrast, if

n
= 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, or 25, . . .

then an
n
-sided regular polygon cannot be constructed.

A fun place to see the limitations that Galois theory describes is in the classic children's fifteen-piece puzzle. This well-known puzzle has fifteen little pieces on a four-by-four grid. One is permitted to move the pieces one at a time. The goal is to get the pieces in order, as in the right-hand diagram in
figure 9.6
. However, there are some starting configurations that cannot lead to the ordered configuration. When the 14 and 15 are simply swapped, there is no way to get to the ordered configuration. There is also no way to get from the ordered configuration back to the original.

In fact, there are fifteen factorial (15!) possible configurations. Exactly half of them are called “even permutations,” while the other half are called “odd permutations.” The usual movements of pieces have a certain symmetry that takes even permutations to even permutations and odd permutations to odd permutations. These are the symmetries of the system. The fact that one of the configurations in
figure 9.6
is an even permutation and the other is an odd permutation ensures that no number of legal moves will get us from one configuration to the other.

Another fun place to observe an impossibility related to Galois theory involves Rubik's Cube. Take a completed Rubik's Cube and (unnaturally) twist one of the corners. Mix it up more and then let a poor unsuspecting friend (who has not read this book) try to solve the puzzle. It cannot be done. One twist makes it impossible to solve, no matter how many moves are performed.

In summary, the Galois theory of equations shows the inherent limitations of the usual operations of multiplication, division, exponentiation, and roots in solving equations. Mathematicians have, over the years, developed other techniques that use calculus and infinitary methods to solve some of these problems. So Galois theory demonstrates that some problems cannot be solved
using certain methods
. Similarly, all
n
-sided regular polygons can be constructed if you are permitted to measure exact sizes with the straightedge. The fifteen-piece puzzle can be easily solved by removing the pieces from the puzzle and putting them back correctly. A Rubik's Cube can always be solved by cheating—that is, by taking it apart and putting it back in order. These are all simple tricks for getting around the mathematical limitations described by Galois theory.

9.3  Harder Than Halting

Imagine you land a job working for a construction contractor helping customers design their future dream kitchens. Everything is going fine until the wife of a millionaire walks in and wants to change her kitchen floor. She does not want the usual pattern of squares. She wants something totally different. She wants to only use circles. You show her that circles will not work since this will leave gaps that cannot be filled (as in
figure 9.7
).

What about pentagons (
figure 9.8
)?

Pentagons will also not work, but you try to sell her on hexagons (
figure 9.9
).

There are no empty spaces. Hexagons can be used to tile a room. Some shapes work, and some do not work.
Figure 9.10
has two other tilings that use a single shape.
¹²

There are obviously many more different shapes that can tile without leaving any gaps. The famous Dutch artist M. C. Escher (1898–1972) made some great etchings of strange shapes that fit together perfectly to tile without gaps.

Consider the weird shape in
figure 9.11
called the Myers shape.