Organisational and technological systems are generally structured by linear, procedural and additive modular methods. The new degrees of freedom available at a logical level are rarely acknowledged. Unexplored, abstract logical spaces of literal higher-dimensionality also exist.
Higher dimensions are not magical, metaphysical mystery tours in some “other” spatial place or additional bolt-on to reality; they consist of the abstract relationships and recombinatory, recursive patterns of information inside the world with which we are already familiar.
The naturally-occurring patterns and symmetries in (and of) optimally-concise methods for information encoding are a pointer to what to expect. It is not what and how you add which inflates dimensionality so much as what and how you remove or substract that opens this door.
Language (and cognition) is poorly-structured to make this intelligible, but it is not an impossible task.