Magritte's well known picture of the pipe has always been a great inspiration to me. The text under the pipe says in French that "this is not a pipe. Indead it isn't, it is a picture of a pipe. At the same time, this organizational dialgram is not an organization, it is a picture of an organization.
An organization is so much more than what can be captu-reed in a diagram. It is also the informal relationships, emotions, driving forces, egos, needs etc. All of what defines the need for managers and leaders, not just administrators.