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Holographic duality or the holographic principle is the notion that everything that occurs in (four-dimensional) spacetime can be described in three dimensions, apparently handling space using two dimensions, assuming one additional dimension is necessary to accommodate time. The idea is that within any volume, you could define a physics that models all the same phenomena using processes going on at the volume's outer boundary. Such correspondences have been found for some areas of physics and it is hypothesized that it might be true of the rest. The word holographic indicates an analogy with a hologram, a photograph of a three-dimensional object somehow encoded on the film's two-dimensional surface.
The fact that physical phenomena that seemingly would need four dimensions can be modeled in three is fascinating, but a practical concern is that physical analysis can sometimes be simplified making use of multiple equivalent-but-mathematically-different approaches: if two sets of physical laws, each with their own mathematics, each model the same phenomena, then a physicist can choose the one with the easier mathematics, and sometimes can switch between the two within the course of the analysis (perhaps repeatedly) to find the easiest path to a solution. To the extent that a holographic principle holds, it offers such opportunities.
The current holographic duality of high interest is the AdS/CFT correspondence, such a correspondence between conformal field theory and anti-de Sitter space, the former using four dimensions, and the latter modeling the same physical phenomena using only three.