According to Greg Kuperberg
it’s quite striking that real computers are very close to two-dimensional, and yet they are mostly used in a RAM machine mode with an emulation of complete circuit connectivity.
On the other hand, transistors in real computers are not very far away from melting. Even though many computers look 3-dimensional, most of the geometry of a computer is within each chip of the computer, and that geometry is almost completely 2-dimensional. One reason for that is the photolithography used to make the chips. But another reason is that there is no way to carry away the heat from a 3-dimensional block of transistors. Without that problem you could sandwich many chips together in a sort-of 3-dimensional pile. The heat problem effectively limits real computers to the power of 2-dimensional cellular automata. However, this 2-dimensional geometry is mostly used to simulate a RAM machine. It cannot be an efficient simulation, but it is what happens in practice, since most higher-level languages create a RAM machine environment for software. It’s also a pain to design algorithms for a 2D computational grid rather than for a RAM machine.
According to Joe Fitzsimons
The rate at which a region of space can be cooled scales as the surface area, where as the heat produced scales as the number of irreversible gates. For a 2D array these scale in the same way, but for a 3D array the heating scales as the volume (R^3) where as the cooling scales as the surface area of a bounding box (R^2). Clearly you need to balance the rate at which heat is produced with the rate at which it is removed, and hence you have a scaling problem with 3D arrays. This is entirely independent of the cooling mechanism.