Mathematics and the Real World (60 page)

We now ask, does this also apply to spaces of higher dimensions, for example, the boundary of a ball in four-dimensional space? In section 32 we discussed the fact that physicists claim that physical space has more than three dimensions, but our senses cannot perceive them. It is difficult to develop a feeling or intuition about multidimensional spaces, but they can be defined quite simply in mathematical terms. If we follow Descartes and describe normal three-dimensional space by using coordinates (
x
,
y
,
z
), and the boundary of the ball is given by the equation
x
²
+
y
²
+
z
²
= 1, four-dimensional space will be described by four coordinates, (
w
,
x
,
y
,
z
), and the boundary of a four-dimensional ball will be given by the equation
w
²
+
x
²
+
y
²
+
z
²
= 1, and likewise for spaces with more dimensions. The boundary of a ball in four-dimensional space will have three dimensions, just as the boundary of a ball in three-dimensional space has two dimensions (think of small squares or rectangles on the surface; each has two dimensions).

Poincaré asked: Is the boundary of a ball in a space with more than three dimensions, where the boundary of the ball has at least three dimensions, characterized by the property of the shrinkability of a loop on it? He himself did not suggest the possibility that the answer was positive. The intuitive reaction that this property of shrinkability is characteristic of the boundary of a ball emerged only after years of attempts to discover whether it was true. Indeed, it was proved, among other things, that the
boundary of a ball in more than six dimensions is characterized by the property of a loop being shrinkable. It was Steve Smale who came up with the proof, for which he was awarded the Fields Medal. It was also proven that the property also applied if the boundary of the ball had four dimensions (the ball has then five dimensions). That was proven by Michael Freedman, for which he received the Fields Medal in 1986.

The question of the boundary of a three-dimensional ball (a ball in four dimensions) remained unanswered, despite many attempts to answer it. Richard Hamilton made a notable contribution by outlining a method for solving the problem, and in 2002 and 2003 Grigori Perelman, from St. Petersburg, wrote three papers that were considered preliminary papers, that is, before they had received confirmation from a professional journal that it would publish them, after they had been refereed (a process that can take years). Today there are Internet sites that publish papers at that stage, and that is accepted practice, especially if the author wishes to press ahead and preserve his rights as being the first to publish his findings (“first dibs”). The only reservation is that those papers have not passed the stage of being approved by referees. In his papers Perelman explained his solution of the Poincaré conjecture, basing it on the method outlined by Hamilton.

At this point we revert to the question of what is a proof. Two students of the Chinese American mathematician Shing-Tung Yau of Harvard University, named Zhu Xiping and Huai-Dong Cao, published a paper (after it had been approved by referees) setting forth a complete proof of the Poincaré conjecture. Their proof, as they themselves stated, was based on the Hamilton-Perelman method. Specifically, they claimed that Perelman's proof was incomplete and that the missing part was not trivial. Their claim was supported by their tutor, Yau, a famous mathematician who had won the Fields Medal in 1982 and the Wolf Prize in 2010.

The dispute was a professional one; that is, it centered on the question of what is a complete proof, although it is difficult to ignore the aspects of prestige and money involved in being the first ones to solve the problem. As one of the Clay Institute's millennium problems, the first solution of the problem would entitle its discoverer to one million dollars. The dispute was heated and intense, with accusations leveled in both directions. Perelman,
who in an understatement could be described as a sensitive person, decided to abandon mathematics in reaction to the incident. Others did the work for him, completing what was missing from his proof, noting the fact that the missing parts were not significant. The mathematics community recognized Perelman's achievement, and eventually his Chinese adversaries did also. In 2006 it was decided to award him the Fields Medal, but he refused it. The Clay Mathematics Institute also recognized the partial publication and subsequent complementary sections published by other mathematicians as a whole publication and decided to award Perelman a million dollars, as promised. Once again Perelman refused to accept the prize and withdrew to his city of St. Petersburg, where he is currently absent from the scene of research in mathematics.

64. PURE MATHEMATICS VIS-À-VIS APPLIED MATHEMATICS

The title of this section appears to reflect a distinction between two types of mathematics, and, indeed, this distinction is acceptable to some mathematicians. I would like to put forward and justify the claim that this differentiation is artificial. I will also argue against the use of the term
pure mathematics
itself: Is mathematics that is not pure really impure or contaminated? When people use the term
pure mathematics
, they mean mathematics for its own sake, that is, mathematics that is not motivated by the applications for which it is intended. We will see that even when a mathematician undertaking research does not relate to the possible applications for his results, his findings are likely to be very useful. Here are some examples.

We have mentioned perfect Platonic bodies previously. The fact that only five such bodies existed was known already at the time of Plato. As far as we can reconstruct, the research that led to the discovery of perfect bodies and to the proof that there were only five such bodies was mathematical research for its own sake. Very soon, however, the result became extremely useful. The five perfect bodies became the basis of one of the
first models of the structure of the world. Another attempt to use this “pure” mathematics result was made by Kepler, as we saw above (see mainly section 17). In the course of time, these uses lost their value as better mathematical models were discovered to describe the world. Conceptually, however, these were applications of mathematics just like the other uses made of mathematics to describe the world.

The research on the structure of bodies in a space has continued from the days of Plato until today, and among other things researchers in geometry tried to classify bodies that were semi-Platonic, that is, bodies whose boundaries consisted of parts of Platonic bodies. The research was completed some decades ago by Victor Zalgaller of St. Petersburg, who currently has affiliations with the Weizmann Institute in Israel. He proved that there are exactly ninety-two such bodies, and he found them all. Victor is an active mathematician in the fields of geometry and optimization. In December 2010 we celebrated his ninetieth birthday at the Weizmann Institute. He is also another example of the claim relating to age that we put forward in the previous section: in the last two years, he has published several original papers, including a number as sole author. This research of his, like the research that led to the discovery of all the perfect bodies, was carried out for its own sake, but I expect that it will also be found to have uses and applications.

Another example of research in mathematics for its own sake that turned out to have uses relates to tiling an area. A floor, say, can be tiled with square or rectangular tiles, used in most buildings, or diamond shaped (rhomboids) or triangular tiles, and so on. In older, historical buildings, such as palaces built by Islamic rulers in the Iberian Peninsula, the tiles are not even convex (i.e., the tiles may have, say, the shape of the letter L), yet they serve as flooring that is very pleasing to the eye, and the regular repeat of the pattern follows interesting rules of symmetry. Here too it was proved that the number of symmetrical patterns is finite, and examples of all these can be seen in the floors of palaces, such as the famous Alhambra in Granada, Spain.

Now here is a mathematical question asked for its own sake in the 1960s: Can a non-repeated tiling pattern be found with a small number
of tile shapes? Non-repeated, or non-periodic, tiling means that there is no possibility of moving the tiled space in a certain direction so that the tiling after the move will merge with the tiling before the move. Roger Penrose of the University of Oxford, winner of the Wolf Prize in 1988, gave an interesting answer in the mid-1970s. He showed that it was possible to tile the area in a non-repeating pattern with only two kinds of tiles, both of diamond shape. Penrose's construction was mathematics, per se, although some places, including Texas A&M University, actually used the tiling proposed by Penrose in one of their halls.

In 1982 the crystallographer and computer scientist Alan Mackay of the University of London performed a computerized experiment to discover what the light diffraction pattern would look like if the atoms of a crystal were ordered in the Penrose tiling pattern. Such patterns are the main instrument used in the identification of crystallographic structure of crystals. Mackay's mathematical experiment can also be seen as mathematics for its own sake, as no one thought or believed that atoms in nature can arrange themselves in non-repeated patterns, as was written in all the textbooks.

However, at the same time, in 1982, the Israeli Dan Shechtman of the Technion, the Israel Institute of Technology, who was then on sabbatical at the United States National Bureau of Standards in Washington, DC, performed an experiment on the diffraction of light. He discovered, in contradiction to deep-rooted belief, a system that did not conform to the periodic structure. In the absence of a mathematical model the results of the experiment would have remained just a description of the findings, without an understanding or an explanation of the discovery. Therefore, Shechtman and a colleague from the Technion, Ilan Blech, published a model that explained the result. Within a short time two physicists who were then at the University of Pennsylvania, Paul Steinhardt and Dov Levine, found that the structure revealed by Shechtman was perfectly consistent with the results of Mackay's theoretical experiment, that is, Penrose's mathematical tiling pattern. Mathematics provided the explanation and the confirmation of the existence of what had been discovered. Researchers in crystallography laboratories searched for, and found, other instances of the phenomenon, but the leading theoretical crystallographers still persisted in expressing doubts and
claiming that there might have been an error in Shechtman's experiment. They clung to the old mathematics that described repeated patterns and considered that to be the only mathematics that described crystals in nature. When many other crystals were discovered that did not have repeated patterns, the crystallography “establishment” also agreed to adopt the mathematics that described crystals in nature. Thus a door was opened to one of the most useful areas in material science. His breakthrough earned Shechtman the Nobel Prize in Chemistry in 2011.

In section 31 we referred to another example of applied mathematics: how group theory is a major tool in understanding the structure of elementary particles. The link between mathematics and particle physics was discovered in the 1960s, but group theory itself existed much earlier. The theory is formulated in textbooks in an abstract form. Here is its full version.

A group is a collection, which we will denote by
G
, of elements, which we will denote by the letters
a
,
b
,
c
, and so on. An operation takes place between the members of the group that we will indicate by the + sign (like the plus sign in the addition of numbers, although in our case we are not referring to addition, as the members of the collection are not necessarily numbers). The result of the operation
a
+
b
is a new member of the group
G
. The operation has certain properties:

1. Associativity
   :
   a
   + (
   b
   +
   c
   ) = (
   a
   +
   b
   ) +
   c
   .
   
2. There is a member called the
   zero
   , or the
   neutral
   ,
   member
   , which will denote 0, that fulfills
   a
   + 0 =
   a
   for all members of the group.
   
3. For each member, there exists its
   inverse member
   , denoted by –
   a
   , that satisfies the equality
   a
   + (–
   a
   ) = 0.
   

That is all. We reemphasize that although we use the symbol for addition, which we are familiar with from its use with numbers, and we use the symbol 0 to indicate the “neutral” member, here we are not dealing with numbers. These are abstract operations, and the usual symbols are used to help the brain absorb the abstract structure. Care must be taken not to confuse the dual use of the usual symbols. For example, the properties that we have listed
do not
imply that the equality
a
+
b
=
b
+
a
holds for all two members of the group. In some groups, this equality holds. A group in which that does apply is known as a commutative group.