"Bookcases line the walls on either side of me, but they seem to be decorative more than anything, because computers occupy the tables in the center of the room, and no one is reading. The Erudite stare at screens with tense eyes, focused."- A description of the Erudite Headquarters by Tris Prior, in Divergent
The Erudite headquarters or compound is where Erudite members study and work in their expertise, Knowledge.
The Erudite headquarters is located somewhere near, or in Chicago's Millennium Park, as stated in the first book when Tris describes a large rusted lima bean sculpture near the compound, which is most likely the famous "Bean" sculpture in the park.
In the Divergent (film), when Tris visits the headquarters to meet Caleb Prior, the Exterior of the compound is actually Chicago's Joe and Rika Mansueto Library, which is no where near Millennium Park.
In the Insurgent (film), Erudite's headquarters changed drastically from the first film, with the whole compound being today's world's Trump International Hotel & Tower in Chicago.
In both films, the compound seems to be white in color, or made of glass, with advanced technology inside.
In the books, the Interior is basically a large library with book shelves covering the walls, but the books looking more decorative in the shelves, since most of the Erudite members sit at tables with computers, their faces focused on the screens. In the center of the room is a reception desk with a huge portrait of the Erudite leader, Jeanine Matthews, with a plaque reading "KNOWLEDGE LEADS TO PROSPERITY"