On June 30, 1962, Sandy Koufax pitched his first no-hitter for the Dodgers. Up against the New York Mets, Koufax struck out 13 batters and walked five to lead his team to victory. It was his first of four career no-hitters—a feat achieved, to this day, fewer than 300 times in the history of Major League Baseball. The following summer, LIFE put Koufax on its cover, calling him “The Mostest Pitcher: Most Wins, Most Shutouts, Most Strike-outs.” Southpaw Sandy, as they called the left-handed pitcher, said that his success in that position stemmed from his failure as a batter. Even as a kid, he knew enough about his own weaknesses to tell his coach, who tried him out at pitcher. It was perhaps no coincidence that the photo that graced the magazine’s cover could have doubled as a menswear ad, had they swapped Koufax’s baseball cap for a fedora. The magazine made no secret of his dapper looks, writing that on top of being “the best pitcher in the past decade,” Koufax was also a hot ticket among female fans. The description reads like a personal ad, save for the mention of his then-girlfriend Linda Kennon: He also stands out as baseball’s most attractive bachelor. Tall and 27, he has poise, a literary mind and a resonant, softly modulated voice. He lives alone in a suburban home with several oil paintings, a well-stocked library and a self-installed stereo system. His car is gold-colored and his girl was runner-up for the title of Miss U.S.A. In addition to his penchant for oil paintings and good books, Koufax made his principles quietly clear. Two years after the LIFE cover story, he would make headlines not for his pitching but for refusing to pitch, when he sat out the first game of the 1965 World Series because it fell on Yom Kippur, one of the most important holidays of the Jewish year. Koufax, for his part, could have done without the attention that went along with his celebrity. “I don’t like to be on display,” he told LIFE, though he obliged when kids mobbed him for his autograph after games. Despite that sentiment, Koufax, now 79, will be forever on display: at 36, he became the youngest player ever inducted to the Baseball Hall of Fame. Mark Kauffman—LIFE Magazine Liz Ronk, who edited this gallery, is the Photo Editor for LIFE.com. Follow her on Twitter @lizabethronk.