zlacker

O(n^(1/2)) really, since data centers are 2 dimensional, not 3 dimensional.

(Quite aside from the practical "we build on the surface of the earth" consideration, heat dissipation considerations limit you to a 2 dimensional circuit in 3-space.)

>>cperci+(OP)
If you have rows of racks of machines, isn't that 3 dimensions? A machine can be on top of, behind, or next to another that it's directly connected to. And the components inside have their own non-uniform memory access.

Or if you're saying heat dissipation scales with surface area and is 2D, I don't know. Would think that water cooling makes it more about volume, but I'm not an expert on that.

>>frollo+j3
Spatial position has nothing (ok only a little) to do with topology of connections.

>>cperci+(OP)
More fundamentally O(n^(1/2)) due to the holographic principle which states that the maximal amount of information encodable in a given region of space scales wrt its surface area, rather than its volume.

(Even more aside to your practical heat dissipation constraint)

>>frollo+j3
That example would be two dimensions still in the limit computation, since you can keep building outwards (add buildings) but not scale upwards (add floors)

>>manwe1+1h
You can add floors though. Some datacenters are 8 stories with cross-floor network fabrics.

>>mpotea+C6
Hmm, I'll go with that

>>frollo+zo
When you get to, say, 100000 stories, you can't build more stories. At this point your computer costs more than the Earth's GDP for a century, so talking about theoretical scaling laws is irrelevant. Eventually you run out of the sun's power output so you build a Dyson sphere and eventually use all of that power, anyway.

>>immibi+ZO
Oh right, so the height is practically a constant. Square root for sure then.

>>mpotea+C6
Just need to make sure all your computation is done in a volume with infinite surface area and zero volume. Encoding problem solved. Now then, how hyperbolic can we make the geometry of spacetime before things get too weird?

>>frollo+Uy1
All algorithms are O(1) in this case

>>LPisGo+is2
You pick what things are constant and what's variable. If you're scaling a supercomputer to fit a problem, the height is going to max out quickly and can be treated as constant, while the other dimensions are variable.