Martin Kruskal, the prolific Princeton mathematician whose work provided a theoretical underpinning for a new form of fiber-optic communications, controlled thermonuclear fusion and the study of black holes, died Dec. 26 at his home in Princeton, N.J., after a series of strokes. He was 81.

His death was announced Thursday by Princeton University, where he spent 38 years.

"We have lost a great man, but he left a great legacy for us to celebrate," said Princeton mathematician Ingrid Daubechies. "Martin Kruskal was an outstanding scientist and mathematician who will be remembered for many seminal contributions. He was also an exceptionally generous, friendly and accessible man."

Kruskal was perhaps most famous for his work on the soliton, an unusual form of wave that is able to maintain its integrity when it encounters a second wave.

Two waves running into each other in a body of water, for example, will tend to cancel each other out.

In contrast, a soliton -- which can exist not only in water, but in a variety of other materials -- passes intact through the second wave. That property makes solitons useful for sending multiple signals through fiber-optic cables without interference and will probably be the basis for the next generation of undersea communications cables.

Solitons were first observed in a canal near Edinburgh in 1834 by the young Scottish scientist John Scott Russell. When the canal boat he was studying suddenly stopped, the bow wave formed into a great heap of water and sped down the canal as a single, solitary wave, passing through other waves it encountered. Russell followed it on his horse for more than two miles before finally losing it in the windings of the canal.

The observation languished in the backwaters of science for more than 130 years until the mid-1960s, when Kruskal and Norman J. Zabusky of Bell Laboratories observed a similar wave in the transport of energy in an atomic crystal.

They named the wave a soliton because of its solitary nature and developed the complicated mathematics needed to describe it. Those equations also turned out to describe Russell's wave, and the two were shown to be mathematically equivalent.

In working out the soliton math, Kruskal and his colleagues for the first time developed a general method for solving so-called nonlinear differential equations -- which describe many natural processes but which were previously thought to be unsolvable.

That feat led to his being awarded in 2006 the American Mathematical Society's prestigious Steele Prize for a Seminal Contribution to Research, one of many such prizes he received. As a result of his work, the citation said, "nonlinearity has undergone a revolution: from a nuisance to be eliminated to a new tool to be exploited."

Martin David Kruskal was born Sept. 28, 1925, in New York City. His father, Joseph, was the prosperous owner of a major fur wholesale business, while his mother, Lillian Oppenheimer, became famous as an expert on origami.

Something about his parents' genes or environment apparently favored logical thinking, because all three of their sons became prominent mathematicians. Martin's older brother, William, is a statistician best known for the Kruskal-Wallis test, which is part of every major statistical computation system.

His younger brother, Joseph Jr., is well known for Kruskal's Algorithm in computer science, the Kruskal Tree Theorem and the formulation of multidimensional scaling.

Martin studied math at the University of Chicago and at New York University, where he received his doctorate in 1952. A year earlier, he had moved to Princeton to join Project Matterhorn, a then-classified effort to develop controlled thermonuclear fusion as an energy source. The project later became the Princeton Plasma Physics Laboratory, and Kruskal, as associate head of the theoretical division, provided much of the theoretical underpinnings for the project.

In 1959, while still holding his post on Project Matterhorn, Kruskal was appointed a lecturer in astronomy at Princeton. A crucial 1960 paper developed what are now called Kruskal Coordinates, used in Einstein's general theory of relativity to explain black holes.

In 1989, Kruskal became a professor emeritus of mathematics and astrophysics at Princeton and moved to the mathematics department at Rutgers University, also in New Jersey.

There he spent much of his time studying the bizarre field of surreal numbers -- a field that includes not only all the previously known types of numbers, but also numbers smaller than the smallest fraction and larger than infinity.

Such numbers have many potential practical applications, such as finding a solution to the unified theory of all forces in the universe.

In his spare time, he also developed the Kruskal Count, a card trick in which a magician "guesses" a card selected by an audience member according to a certain counting procedure.

Kruskal was often described by colleagues as "sartorially challenged." One of the few occasions on which he was seen wearing a tie was when he received the National Medal of Science in 1996.

"The story is told that the tie on that occasion was an outlandish creation purchased at a secondhand store and that President Clinton was much taken with the choice," said mathematician J.C. Eilbeck of Heriot-Watt University in Edinburgh.

Kruskal was also famous for his schedule, working through the night and then sleeping until after noon.

Eilbeck told the story of one young postgraduate student who sought an appointment with Kruskal and was directed to appear at his house at 1 a.m.

"On leaving after an energetic and wide-ranging discussion, he discovered another student at the doorstep with a 2 a.m. appointment," Eilbeck said.

One advantage of that schedule was that it left Kruskal plenty of time for his children.

After teaching in the afternoon, he would go home and spend the evening playing strategy games like chess and backgammon with them.

He wouldn't lose on purpose, said his daughter, Karen, now a lawyer in Brookline, Mass., but he developed elaborate handicap systems to make sure they had a chance to win.

Their living room featured a 4-by-8-foot chalkboard, and his daughter remembered a constant stream of visitors who gathered with Kruskal around the board to discuss their latest findings.

"I was pretty old by the time I realized that blackboards were not in everyone's living room," she said.

Kruskal's wife of 56 years, Laura, is a world-renowned practitioner and teacher of origami, and Kruskal was himself an expert. He also had a great love of hiking and swimming.

In addition to his wife and daughter, Kruskal is survived by two sons, Clyde and Kerry, and five grandchildren.