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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.
Charlie's Archons typically hold incoming calls in a Thinkspace "waiting room" while routing the call to him.
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.
Psychic combat takes place in this realm. Thinkspace uses Signamancy to represent different aspects of the warring combatants' psyches. For example, when Charlie attempts to study Lilith's mind and she resists, the struggle is represented as a flock of crows besieging a castle, and Charlie speculates on whether her Shockamancy attack is a motor control or emotion regulation system.
Normal thinkagrams with the hologram can be seen by everyone, Thinkspace thinkagrams may be more private with the downside of being distracting.