
%keplerr.tex: ``Book Review of ``Kepler's Conjecture'' by George G. Szpiro''
%
%a Plain TeX file by Doron Zeilberger (2 pages)
 
%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\Tilde{\char126\relax}
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
 0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
 height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\parindent=0pt
\overfullrule=0in
%end macros
 
\bf
\centerline
{
``Book Review of ``Kepler's Conjecture`` by George G. Szpiro``
}
\rm
\bigskip
\centerline{ {\it 
Doron ZEILBERGER
}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg@math.rutgers.edu} \hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/   .}
First version: July 17, 2003.  This version: July 17, 2001.
} 
}
 
The most difficult open problems in mathematics are often very easy to state.
One famous example is the Four Color Conjecture, proved in 1976 by
Kenneth Appel and Wolfgang Haken (and described beautifully
in Robin Wilson's wonderful book: ``Four Colors Suffice``).
Another, equally famous, and even older, open problem was the
Kepler conjecture, posed in 1611 by Johannes Kepler, in his correspondence
with Thomas Hariot, who was wondering about the best way to stack
cannon balls. Kepler conjectured that 
the density of a packing of congruent spheres in three dimensions
is never greater than $\pi/\sqrt(18)$. It is achieved
with the so-called {\it face-centered cubic} packing, which is just
the {\it grocer's method} of arranging oranges on a fruit-stand.

Unlike the Four Color Problem, that for many years was mainly
attempted by amateurs, the Kepler conjecture received the attention
of some of the greatest mathematicians, including Newton and
Gauss. It even made it to the Pantheon of open problems
of 1900, Hilbert's famous list of 23 open problems.

Like the Four Color Problem, the ultimate solution of
Kepler's conjecture, by Thomas Hales, was computer-aided,
and the computational and programming effort even exceeded that
of the Four Color Theorem.
Indeed such is the complexity of Hales's proof
that the paper that was submitted
in 1998, is still under review, and its official status is
``submitted''. Currently Thomas Hales and his students are preparing
a ``second-generation'' proof that they hope will be more streamlined
and easier to check.

The substance of the statements of the Four Color Theorem and the
Kepler-Hales Theorems are very different, but the {\it methodologies}
of proof are strikingly similar. Reduce an a priori infinite problem
to checking a finite, albeit huge, number of cases. This {\it idea}
of reduction from the infinite to the finite was made by humans
(H. Heesch for Four-Color and L. Fejes-T\'oth for Kepler), 
as was the strategy of reducing it from {\it finite in principle}
to {\it feasible} (Appel-Haken and Hales respectively).
But the {\it bottom line} required very extensive computer-checking.
In the case of Hales's proof ``it takes over 270 pages
of mathematical text, extensive computer resources, including
3 giga-bytes of data, well over 40 thousand lines of code, about
100000 linear programs each involving perhaps 200 variables
and 1000 constraints''. It is all made mathematically rigorous
by using `interval-arithmetic' rather than the much faster, but
non-rigorous, floating-point arithmetic.

Most mathematicians dislike and mistrust computer-assisted proofs,
since they seem to violate their Platonic a priorism, and `contaminates'
their field with  the contingencies
of experimental science. But the reason
mathematicians were able to manage, so far,
with their tiny computer
between their shoulders, is that they only proved (relatively) trivial
results. What makes the proofs of the Four Color and the Kepler Conjectures
even more significant than their already immense face-values
(as proofs of long-standing extremely difficult problems), is that
they are {\it harbingers} and {\it iconic examples} of future mathematics,
that will be all computer-assisted, and eventually computer-generated.

George Szpiro's excellent book on Kepler's Conjecture
describes very vividly, and extremely lucidly, the fascinating
history of this tantalizing problem, and its ultimate proof
by Thomas Hales. Along the way there are exciting digressions 
into many other topics, including the Four Color Problem, and
philosophical discussions about computer proofs. 
The main text is entirely non-technical, and even the appendices
require very little background. Yet everything is mathematically
accurate, and the author's previous life as a professional mathematician
is clear throughout.

Analogously to the Kepler and Four Color Problems,
the importance of Szpiro's book transcends its 
already very great {\it face value}
of telling the thrilling story of a major math problem.
It is a {\it paradigm} of mathematical popularization, hopefully
to be emulated in the future, thereby making math reach wider and
wider audiences.

\bye



