Tony Bott was 12 when he first ventured up to his father's third-floor study in Newton and burst through the door without knocking, only to find he was the one in for a surprise. A large lacquered spool -- the kind used for cable lines -- hung from the ceiling next to the desk, acting as a decorative shelf. On that day, his father was standing on the spool, swinging back and forth.
''I'm just testing the strength of the chains and the length of the swing," Raoul Bott told his son.
A curious sight, perhaps, but not unusual. Dr. Bott spent a prestigious academic career thinking deeply about how to view surfaces and spaces. His field was topology and geometry, and as a mathematician he had few peers at the end of the 20th century.
Dr. Bott, 82, professor emeritus at Harvard University, died of cancer Dec. 20 at his home in Carlsbad, Calif.
On the history wall in the Mathematica room at Boston's Museum of Science, his name appears twice -- once with one of the exhibit's shortest entries: ''1959 Bott periodicity theorem." The brevity of the entry, which falls on the wall a few centuries to right of historical figures such as Isaac Newton and Copernicus, belies its significance.
''He had a tremendous influence in the development of modern geometry and topology," said Clifford Taubes, chairman of Harvard's math department. ''I would say that his contributions to this were as great as any one person."
Among the mathematics awards Dr. Bott received were the National Medal of Science in 1987, the Wolf Prize in Israel in 2000, and two from the American Mathematical Society -- the Oswald Veblen Prize in 1964 and the Steele Prize for lifetime achievement in 1990.
At 6-foot-2 with blue-gray eyes, a thick Hungarian accent, and an even thicker silver beard -- he used to nibble on the edge that drooped near his mouth when contemplating a vexing question -- ''his physical presence alone was hard to miss," said his daughter Jocelyn Scott of Rancho Santa Fe, Calif.
Dr. Bott's disposition was no less forgettable.
''His theorems were fantastic, but there are people with fantastic theorems who are not loved the way he was loved. Everyone considered him a father figure," Taubes said. ''He was just such a gentleman and gregarious. He loved to laugh, he loved life. He taught us to look for beauty and art in everything."
The art form many consider a close relative of mathematics was also a passion.
''Music was his love," said Dr. Bott's wife, Phyllis. He took piano lessons as an adult at the Longy School of Music in Cambridge, and among his favorite composers was Johann Sebastian Bach.
''I think of him as a mathematician's composer," Phyllis Bott said. ''Many mathematicians are very fond of Bach."
In a 2000 interview, Dr. Bott likened his choice of math problems to pursue to his choice of musical compositions to learn. ''As in music, one falls in love with different things at different times," he said. ''Right now for me it's the Fourth Partita of Bach. What brings on these impulses is hard to say."
Born in Budapest, Dr. Bott left Europe in 1938, when he was 15, with World War II on the horizon. After a year in England, he moved with his family to Canada, where he graduated from McGill University in Montreal with a bachelor's and a master's in engineering. He changed his focus to mathematics and moved to Pittsburgh, where he received a doctorate from Carnegie Institute of Technology.
Two years at the Institute for Advanced Study at Princeton followed, during the time Albert Einstein was in residence. Dr. Bott then taught the University of Michigan and accepted a professorship at Harvard in 1959, where he remained until retiring to emeritus status in 1999.
Allyn Jackson, an editor for Notices of the AMS, conducted a lengthy interview that the American Mathematics Society journal published in 2000. She remembered Dr. Bott as ''a total charmer."
''He looks like a bit of a heartbreaker, and he has the personality to match," Jackson said. ''He was not at all a nerdy mathematician-type whatsoever."
She added, ''He used to like to say that there are dumb mathematicians in the world and smart mathematicians. And he said he was one of the dumb ones, which obviously wasn't true, but he liked to understand things on a basic level."
Taubes, the mathematics chairman at Harvard, said Dr. Bott ''worked very hard to say it right, to say it as cleanly as possible. His papers are gems -- not incompressible, jargon-filled. They were works of art."
Before becoming a colleague, Taubes was a student of Dr. Bott's and said the professor's manner put at ease students who entered the classroom intimidated by being in the presence of a famous mathematician.
''It was delightful -- it was just wonderful to be in his class, to see how this beautiful mathematics flowed," Taubes said. ''Mathematics wasn't impenetrable: It was beautiful, it was so seductive. I, for one, would have been a physicist if I had not been in his class, but I was seduced by mathematics."
Outside the classroom, he found beauty in nature, particularly while walking the beach on Martha's Vineyard, near the home he and his wife purchased in Chilmark, where he would delight in spending time with his family and, sometimes, sketching math problems in the sand.
''He would be on the beach every day that it wasn't pouring rain -- and even sometimes when it rained," Phyllis Bott said. ''Everyone knew him. They called him the mayor of the beach."
Taubes said Dr. Bott was the kind of person who made colleagues think, ''I'll try to be more like Raoul Bott -- not just as a mathematician, but how to live one's life."
Along with his wife, his son, Tony, of Harwich, and daughter, Dr. Bott leaves twin daughters, Renee of Berkeley, Calif., and Candace of Cambridge; and nine grandchildren.
A memorial service has been planned for 1:30 p.m. Jan. 29 at the Memorial Church at Harvard.