[frollogaston]

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.

[manwe150]

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)

[frollogaston]

You can add floors though. Some datacenters are 8 stories with cross-floor network fabrics.

[immibis]

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.

[frollogaston]

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

[LPisGood]

All algorithms are O(1) in this case

[frollogaston]

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.

[jvanderbot]

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