# Information containable in a volume depends on surface area, but a volume of space can contain a volume with a bigger surface area?

The holographic principle has me confused. I'm wondering, if a volume can contain more information if it has a larger surface area, how can it be that a volume with a bigger surface area can be contained in a volume with a smaller surface area?

Example: take the shape of a star symbol and extend it into the third dimension. Think of a box that touches the prongs of the star on each of it's sides. The box has a smaller surface area, but it contains the star.

Does that mean a part of a volume can have the ability to contain more information, the volume it self?

• The entropy (the natural log of the number of microstates compatible with that black hole macrostate, in a theory of quantum gravity) is given by $$Ac^3/4G\hbar$$, where $$A$$ is the area of the event horizon, $$c$$ is the speed of light, $$G$$ is Newton's gravitational constant, and $$\hbar$$ is Planck's constant.
Intuitively, since any attempt to cram more information-carrying stuff into that region will just make the black hole even bigger, this principle implies an upper bound on the amount of information (as quantified by the entropy) that can be encoded inside any spherical region with surface area $$A$$.
Regardless of the precise formulation, the key point here is that in ordinary circumstances (without black holes), the holographic principle only gives an upper bound on how much information can be encoded within a given region. If a region with surface area $$A$$ is contained within a larger-volume region with smaller surface area $$a, as described in the OP, then the amount of information in the smaller-volume region cannot exceed $$ac^3/4G\hbar$$, even though its surface area is larger than $$a$$. This is just like saying if $$x\leq 7$$ and $$x\leq 3$$, then $$x$$ can't be any bigger than $$3$$.