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Comment by dvh | Hacker News Reader

Imagine what kind of computers could you make if you could radiate heat into more than one dimension. Or maybe you would get greedy and use up all available dimensions for interconnects and use one remaining dimension for heat dissipation anyway.

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A hypercube's surface area to volume ratio grows without bound with dimension, so radiating heat ought to be easier with more dimensions even if you build a fairly blobby (I'm sure that's the technical term) core, though an n-sphere's peaks at n=7, so you don't want to get too blobby in higher dimensions.

Took me a minute to figure out what you are saying. Then I realized you are referring to the 16pi^3/15 surface area ratio of a 7-sphere. After n>7 it does seems to infinitely move towards zero. But the heat equasion is different in higher dimensional space and governed by the parabolic Harnack inequalities. I'm not sure how surface area makes sense here.

A CPU, a planar object, does so by extending its fins in the three-dimensional direction, but not without limit. Similarly, even if the fourth dimension can be used, it cannot be used without limit. Also, just as there are no two-dimensional objects without thickness in three-dimensional space, there are no three-dimensional objects without thickness in four-dimensional space, and all three-dimensional objects are considered to have size in the fourth-dimensional direction in four-dimensional space. However, if there is such a simple fourth spatial dimension, it should be easily recognizable, so if there is a fourth spatial dimension, it should be closely attached to other known dimensions so that it cannot be found. Finding or creating a usable space out of a closely attached fourth spatial dimension would be a large cost.

Can’t we already radiate heat into 3 dimensions?

I think what he means is that chips are 2D, so heat is radiated orthogonal to the plane of the circuit -- up or down.

Yes. See: radiators

Heat from the chip to the radiator still flows in one direction. So you're effectively limited by the bandwidth of this joint even if you have radiators or liquid-cooling (remote radiators) later in the pipeline.

Wouldn't a 4th dimension allow connection between any two points in a 3D space? I think that's how that works but I'm no mathemagician.

Essentially, but you still have to travel the distance in a parallel dimension. Think about layers on a circuit board or road overpasses. You can connect two things separated by something by going "up" in the third dimension, but you still have to actually go the distance after avoiding the collision (and then go down).

so in current design you travel the distance in the time dimension

Sort of. I don't believe that time is considered a spatial dimension. But from a looser conceptual framework, basically yeah you have to spend some time to travel space.

Also note that even using 3 dimensions to traverse a 1-dimensional space doesn't mean there's guaranteed to be no distance between two points. A string can be super densely coiled in 3D space where there's basically no 3-dimensional distance between two points on the string, but it can also be stretched out or loosely crumpled and still have lots of 3d space between two points on the string.

No, time has nothing to do with this. The point is that two points separated by an object in 3D space (say, a point inside a box and one outside the box) can be connected by something in 4D space, just like two points on a 2D plane on opposite sides of a river can be connected by a bridge in 3D space.

The time dimension is entirely separate. You always travel a distance in the time dimensions, in space the distance just exists, there is no concept of traveling without time. This is also visible in the math of SR - the time dimension is different from the three spatial dimensions (opposite sign in the distance metric).

It is possible by deforming it in the fourth dimension, like folding paper. However, the shapes that can be created by folding are considered limited.

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