It is possible that there are extensions with other dimensions. Spaces of this kind, however, cannot stand in connection with those of a quite different constitution. Accordingly, such spaces would not belong to our world, but must form separate worlds. ~ German philosopher Immanuel Kant
Our sensory experience is of 3 spatial dimensions (3D). Classical physics assumed the 3D spatial universe a closed (self-contained) physical system. Modern physics proved that assumption wrong. Relativity and quantum mechanics both necessitate extra spatial dimensions (ED).
Matter tells space how to curve. Space tells matter how to move. ~ American physicist John Wheeler
The conceptual unification of spacetime arose through Einstein’s theories of cosmological dynamics. According to general relativity, mass begets gravity, which distorts spacetime, the medium in which matter swims. It is an entangled relation, as matter defines spacetime, just as spacetime is necessary for matter to manifest.
To take effect, gravity’s distortion requires at least 1 extra dimension beyond the 3 observable spatial dimensions. Analogously, a flat (planar) 2D piece of paper becomes 3D when crumpled.
Quantum mechanics faces a similar situation in requiring extra-dimensionality for forces to interact with subatomic particles. This renders 4-dimensional spacetime (4D) as existing within a higher-dimensional (HD) envelope.
HD = 4D + ED
Holistic dimensionality highlights that phenomena are only a fraction of Nature. Other dimensions (ED) exist, which may be inferred by their impact on the 4 dimensions (4D) of spacetime which we perceive. Among other indicators, nonlocality is compelling evidence of 4D–ED interactivity.
Extra dimensions really exist. They’re part of Nature. ~ American physicist Edward Witten