In Chapter 3 of The Great Gatsby, what is Nick's encounter with Gatsby like?

Nick's first meeting with Gatsby is actually kind of funny. Gatsby is throwing a party -- one of his giant parties that fills his mansion with people from all over. Nick is accorded the special honor of actually being invited -- most of Gatsby's guests just show up. So Gatsby is singling Nick out, even though has never met him. Later, at the party, Nick is a little non-plussed because, among all these people, there is no sign of Gatsby himself. Finally, after meeting up with Jordan Baker, he falls into conversation with a stranger:

“I was still with Jordan Baker. We were sitting at a table with a man of about my age and a rowdy little girl, who gave way upon the slightest provocation to uncontrollable laughter. I was enjoying myself now. I had taken two finger-bowls of champagne, and the scene had changed before my eyes into something significant, elemental, and profound. At a lull in the entertainment the man looked at me and smiled.

“Your face is familiar,” he said, politely. “Weren’t you in the Third Division during the war?”

“Why, yes. I was in the Ninth Machine-gun Battalion.”

“I was in the Seventh Infantry until June nineteen-eighteen. I knew I’d seen you somewhere before.” 

The exchange some more talk; the man invites nick to go with him in his hydroplane in the morning. Then Nick says:

“This is an unusual party for me. I haven’t even seen the host. I live over there ——” I waved my hand at the invisible hedge in the distance, “and this man Gatsby sent over his chauffeur with an invitation.” For a moment he looked at me as if he failed to understand.

“I’m Gatsby,” he said suddenly.

“What!” I exclaimed. “Oh, I beg your pardon.”

“I thought you knew, old sport. I’m afraid I’m not a very good host.” 

Nick is embarassed by the accidental meeting, but Gatsby is completely understanding. He smiles:

“It was one of those rare smiles with a quality of eternal reassurance in it, that you may come across four or five times in life. It faced — or seemed to face — the whole external world for an instant, and then concentrated on you with an irresistible prejudice in your favor. It understood you just so far as you wanted to be understood, believed in you as you would like to believe in yourself, and assured you that it had precisely the impression of you that, at your best, you hoped to convey. Precisely at that point it vanished — and I was looking at an elegant young rough-neck, a year or two over thirty, whose elaborate formality of speech just missed being absurd.” 

It is characteristic of Fitzgerald, I think, that Gatsby's whole persona is captured in this one smile. The smile that "understood you just so far as you wanted to be understood" is like a mirror -- in a way, looking at Gatsby is like seeing your own best image reflected back onto yourself. Once the smile fades, the impression goes, and we see Gatsby, maybe, for what he really is, an "elegant rough-neck." Like Gatsby's books, which are described earlier in the chapter as a masterpiece of fakery (“What thoroughness! What realism! Knew when to stop, too — didn’t cut the pages.”), Gatsby himself is a facade.