The Imitation Game is an exciting crowd-pleaser about Turing, a socially awkward misfit whom most everyone finds weird, arrogant, and off-putting. But he’s also a cosmic genius. It’s Turing who cracks the so-called Enigma code, which encrypted all of the Nazi military’s wartime messages. He’s ably assisted by a team of world-class cryptanalysts—“We’re like the Avengers,” jokes Cumberbatch—that includes Joan Clarke, a brilliant mathematician whom Turing champions despite the military’s sexist objections, and with whom he becomes intimate. She’s played with resourceful intelligence and warmth by Knightley, who became mates with Cumberbatch one summer when they shared a commune-like farmhouse while filming Atonement, and discovered a mutual pleasure in cooking and live music. They have a striking chemistry. “Because we really are friends,” Cumberbatch says, “we share this great shorthand. It made doing our scenes together a joy.”

Although The Imitation Game is set 70 years ago, it still feels relevant. For one thing, the team’s code-breaking equipment, known as the Turing Machine, was the prototype of today’s modern computer. “The algorithms Alan used during the war,” Cumberbatch says, “are still used in Google’s search engine.” Yet what gives the movie its sting is the heartbreaking fate of this lonely, decent man. Because his wartime work was marked top secret, the public never knew of his heroic achievements. More tragically, because he was totally unabashed about his homosexuality at a time when it was still illegal in Britain—Turing is today a gay icon—he was a target of police harassment.

“I felt a responsibility to show him properly,” says Cumberbatch. “Alan’s face should be on the back of banknotes like Darwin and Newton. It should be on the front of history textbooks and science books.”

With his big-brain forehead yielding to a softer, more labile mouth and chin, the actor has no peer in showing us the humanity of those who might otherwise seem like intellectual freaks (Sherlock, Stephen Hawking, Julian Assange). “I wanted Benedict for the role even before his name became so big,” says the film’s director, Morten Tyldum. “He’s a perfectionist—we spent a lot of time just finding the right voice—and I knew he could get to the core of Alan Turing, the fragility and the arrogance.”

In person, you grasp what a fine actor Cumberbatch is: He’s not remotely arrogant or fragile. Younger-looking than on-screen, he comes across as sweetly boyish, a quality that may belie his sophistication as an actor but helps seal the deal with his female fans. His words come in enthusiastic flurries—“God, I talk quick,” he says when I play back a moment of tape to make sure the recorder’s working—and you never quite know where his conversation may swirl. He exudes an aura of innocent, almost starstruck pleasure in his good fortune, as when he talks about the “bliss” of hanging out with Depp: “We rolled cigarettes and sat around and talked and talked and talked. He’s a friend now. Which is an amazing thing to think about.”

Benedict Cumberbatch at the 2014 Met Gala

“People kept saying he was the Next Big Thing,” recalls _Sherlock’_s cocreator Steven Moffat, “and I think Benedict was getting impatient for that moment to actually happen.”