What is the Perisphere?

What's This? The All-Star Squadron's headquarters.

The facts: Roy Thomas being the historical buff he is was no doubt quite attracted to the 1939 New York World's Fair. The Fair itself promoted futurism, which is perfectly in line with the superhero world, but more than that, it HAD prominently featured in Golden Age stories, most notably in the two issues of New York World's Fair Comics which featured several characters in the future DC's stable, well as Centaur's Fantom of the Fair, which would show up in Roy-penned stories in the 80s (see Who's the Phantom of the Fair for more). With its elegant lines that seem of a piece with the Hall of Justice, it was a perfect place to house the All-Star Squadron, which they would soon do (#21) after its first appearance in the book (#18) and on through the rest of the run and Young All-Stars. In the real world, the Perisphere and Trylon were demolished during World War II and cannibalized to help the war effort, apparently as early as December 1940. Let's call it an Earth-2 divergence since the Squadron moved into it in 1942.
How you could have heard of it: At this point, I'm not sure. You would have had to follow All-Star Squadron, and then it would be just part of the mythos. The Perisphere appeared in Sandman Mystery Theatre #41 to 44, but in its pre-ASS days. Its most modern appearance was in the All-Star Comics 80-Page Giant. Or maybe you're just a World's Fair aficionado.
Example story: All-Star Squadron #19 (March 1983) by Roy Thomas and Jerry Ordway

I've picked a story from BEFORE it was the Squadron's HQ, but Roy shows it off for the first time, knowing it will become a big part of the series. Some of it is really about using the RESEARCH, I admit.

But beyond the exposition, he gives the Perisphere its own character (literally) by sending the real-world automaton Elektro, a feature of the World's Fair, to battle our heroes.

Currently under Brain Wave's control, he will become the Squadron's robotic butler Gernsback once the heroes move in. Did Roy ever use Elektro's robot dog Sparko? I don't have any memory of it.

That's really the Perisphere's role here, as the story could have taken place anywhere else. Basically, Brain Wave has captured the JSA and given them visions of their fighting World War II - Roy's retcon to account for all those Nazi-busting stories of the Golden Age after he spelled out in the series why heroes couldn't take the fight to Germany directly - and he's lured Squadron members there to capture them too.

Hey, we're not done with All-Squadron quite yet...

Who's Next? An African-American hero in the '40s.