[avmich]

I think a good illustration could be provided by the Gauss law. https://en.wikipedia.org/wiki/Gauss%27s_law

It states that if you have a volume of space - say, a cube 1 meter side - containing some electric charges, and you calculate the total flux of the electric field across the boundary of that volume - that is, across the surface of that cube - then you'll find that total charge Q and total flux FF are proportional, FF = Q / epsilon_0 , where epsilon_0 is a fundamental constant. And that ratio doesn't in fact depend on shape or size of that volume of space.

That means Gauss law allows you to go along the boundary surface, calculate total electric flux and calculate the total charge inside the volume within that surface.

Similarly here, "holographically dual" means that you can derive important properties of matter inside some volume from properties which are observable on the boundary surface of that volume. What are those properties is another matter - but this duality principle says that there is a certain relation between them.

[aaaaaaaaaab]

That’s a bad analogy. You can’t deduce the charge distribution within a volume from the flux through its boundary. You can only deduce its magnitude.

[beambot]

Parent correctly referred to "total charge", not its distribution.

[aaaaaaaaaab]

That wasn’t the issue with their analogy.

The issue was that the holographic principle states that the boundary completely determines what’s inside a volume, as opposed to Gauss’s law which only talks about the total amount of charge.

[avmich]

You're right, Gauss law only talks about magnitude, not distribution. I chose this example because it's simpler than explaining reversal of wave fronts, so the formula FF = Q / epsilon_0 is shorter. Note I mentioned "what those properties are is another matter".

The choice of analogy is less precise but hopefully easier to understand. The idea is just that there could be relationship between boundary conditions and internal conditions.