The two men discovered common ground when Mr. Zuckerberg visited Mr. Gehry’s office, which is in a concrete-walled warehouse not far from the Los Angeles airport. He found an unassuming informality, with plywood tables loosely organized and architectural models stacked most of the way to the high, skylighted ceiling. Mr. Gehry, in an interview by telephone, said Mr. Zuckerberg loved it. “He said he wanted one big room, like my office—but 10 acres.” Mr. Gehry agreed to meet a budget said to be modest, but which has not been disclosed. The project has taken a short 40 months.