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Thinkspace is a "place" where Thinkamancers can have fully immersive Thinkagrams. Unlike normal thinkagrams that have both parties appear as see-through holograms, inside Thinkspace both parties appear as if they shared the same space.
Physically, Thinkspace is a black void with only a beam of light in the center, coming down from an unseen source above.
Thinkamancers can alter their appearance inside Thinkspace, but it takes concentration. Maggie could have altered her appearance in the illustrated Thinkspace if she hadn't been multitasking (poorly) to run down a trap corridor. Charlie in particular enjoys constantly altering his appearance, and those of the people he deals with.
Charlie's Archons typically hold incoming calls in a Thinkspace "waiting room" while routing the call to him.
Normal thinkagrams with the hologram can be seen by everyone, Thinkspace thinkagrams may be more private with the downside of being distracting.