A quiet and modest hero of the life of the mind, Dr. Appel (pronounced ap-PELL) had emerged in triumph after plunging into the thorny mathematical thicket represented by the “four-color theorem.” This was a mathematical assertion that, shorn of caveats and qualifications, says that any map can be drawn with only four colors.
Four colors were sufficient, under the theorem, to ensure that no two adjacent elements on a map were drawn in the same color.
For a century, the theorem had resisted the most determined efforts to prove it or to show that it was not so; it was sometimes described as merely a conjecture.
Proof of the theorem was not necessary for politics: not vital to the addition of states to a map, to the demarcation of international boundaries or to the publication of atlases. But it was important in the rarefied world of mathematics.
Seemingly, it was something that anyone could demonstrate satisfactorily at a desk with a stack of paper, a few crayons and a bit of geometrical flair.
But stating the theorem was one thing. It was something entirely different to provide mathematical proof, an argument set out with the implacable logic of the theorems remembered from high school geometry texts.
In the plan of attack followed by Dr. Appel and his colleague Wolfgang Haken, the first step was to reduce to a minimum the universe of all mathematically possible arrangements of shapes on a flat surface. There were hundreds of them. Then it was necessary to show that the four-color theorem worked for every one of them.
Nobody would want to try this without a computer. In fact, it took more than 1,000 hours of computer time to do it.
But on June 21, 1976, Dr. Appel, and his colleague, then both at the University of Illinois at Champaign-Urbana, made their announcement. Q.E.D. They had done it.
Their proof, however, was not one that would cause many non-mathematicians to slap their foreheads and wonder why they had not conceived of it themselves. Even many in the mathematical world found its reliance on computers disturbing.
As practiced for centuries by the great names in the discipline, mathematics had been a celebration of the intellect and imagination, among the purest examples of the human mind at work.
Reliance on the unthinking, robotic efforts of a machine seemed somehow crude and tawdry, a stain on the idea of mathematical creativity.
As time went on, the work of Dr. Appel and his colleague was regarded as a major step toward bringing the computer into the intellectual realm as a full partner in discovery.
Kenneth Ira Appel was born Oct. 8, 1932, in Brooklyn and attended Queens College in New York City and the University of Michigan, from which he received a doctorate in mathematics in 1959.
He served in the Army and worked in Princeton, N.J., for the Institute for Defense Analyses.
He joined the Illinois faculty in 1961 and went to the University of New Hampshire in 1993, retiring in 2002.
Survivors include his wife, Carole Stein Appel, and two sons. A daughter died this year.
In the same way that earlier mathematicians had sought to find logical flaws in efforts to prove the theorem, modern ones challenged the work of Dr. Appel and his colleague.
But in 1989, Dr. Appel and his colleague published a book that appeared to be a milestone in the debate. Its title never seemed to be widely adopted as an American motto or even as a saying among the academic cognoscenti. But it nevertheless bespoke a bold assertiveness.
It was called: “Every Planar Map Is Four-Colorable.”