One relatively simple but interesting implication of living in 1KD is that protecting your faces is cheap but encasing them in a rigid structure is crazy expensive. What do I mean by this? Think of a single atom that you are trying to "shield" from possible collisions with other atoms. One idea is to put that atom inside a 1K-dimensional hypercube. But if you do that, you will need 3^1000 atoms since you will need to put the lone protected atom at the core of a 3X3X3X...X3X3X3 hypecube.

But if instead you are satisfied with just blocking possible collisions by adding an atom on each "face" of this hypercube, you could do so with as little as 2X1000 atoms, which might be manageable.

In the case of a 3D space, this corresponds to putting your atom between 6 atoms, two per dimension, making it the center of the "sandwich" along each dimension. If you wanted to complete the cube, you'd need to also add the atoms at the edges and corners of the cube, which amount to 26 in this case. Manageable in 3D, but not in 1KD.

So I'd conclude that protecting the "faces" of your body is cheap, but encasing them inside a rigid boundary is crazy expensive. So while you will likely find evolved systems with protection shields, this protection will never include filling up all the "corners", or you'll run out of atoms really fast.

Same would likely apply to the protection of wire/cables.

Maybe one way to get an intuition about what a high-dimensional universe might be like is to consider that our own universe is, in a certain sense, much more than 1000 dimensional. They're not spatial dimensions, but bear with me...

The quantum wave function is at least 3N-dimensional where N is the number of particles in the universe, since it assigns a complex number to every possible arrangement of every particle. (Actually there are more dimensions for spin and so on.) And what do we observe?

Well, the quantum wave function often "branches", never to reunite, due to high degrees of freedom. (Copenhagen folks call this "collapsing", since they imagine that the other branches disappear.) Also, it is very difficult to determine the dimension (i.e. number of particles). Different places (arrangements of all particles) quickly become inaccessible to each other. Most space (i.e. possible arrangements of particles) is empty (i.e. gets low weight under Born's rule). Agents work in much lower dimensions, due to locality (we call irrelevant degrees of freedom "distant objects"), and modding over symmetries (e.g. "the ball fell down" means "all the particles in the ball moved down", vastly simplifying the cognitive burden).

This maybe tells us something about what universes with many spatial dimensions may be like. (After all, what *exactly* qualifies a degree of freedom as "spatial"?) But it doesn't tell us much about the specific high-dimensional universes Robin is interested in, the ones that have something like molecules and so on, assuming such universes are even logically possible.

Do you have a similar link for helium or other atoms?

I would've thought that spin 1/2 particles such as electrons would no longer be fermions in the sense of "taking up space", basically for the same reason that strings can't form knots in higher dimensions. In two spatial dimensions, besides bosons and fermions you can have lots of other things -- "anyons" they're sometimes called.[1] In high dimensions, my vague intuition is that all particles are bosons regardless of their spin, unless we make some other very big changes.

Without fermions, there would be no matter "taking up space". Everything would behave more like a boson, the force-carrying particles, like light, which happily pass right through each other. To the degree something like an atom is possible, there would only be one electron shell, occupied by all the electrons.

Of course, we're still free to imagine a world that has high dimension and yet has "solid matter" without specifying the mathematics of how this would be possible in detail. I certainly don't mean to be a spoil sport. I'm just saying, my intuition is that that's not the world you get when you turn up the dimension knob and try your best to leave all the other most fundamental aspects of the theory the same.

Higher dimensional space was one of the big themes in the second and third books of the Three Body Problem series. Not super rigorous, but it was a lot of fun.

In 1000 dimension space, there are 499,500 hyper planes of rotation. There's a lot of potential there for particle spin axes, and storage of energy in rotational modes rather than translation. Could anything ever translate without dissipating all its energy into rotational modes?

There's also lots of types of hyper-volumes, and maybe potential for things to oscillate between volume types.

I'm not sure what all these modes would enable in terms of packing particles together either.

See alsohttps://en.wikipedia.org/wi...The number of dimensions seems to be very important to the behavior of the laws of nature (even if those laws are not yet completely known/understood).

Re:"...what have they figured out that helps answer a question like the ones above?" Caveat: this is not my area of expertise in physics. But I am under the impression that you can't simply add an arbitrary number of new dimensions to our present universe and then not suffer consequences that prevent humans (and maybe even atoms) from ever existing. If I'm right about that, our present existence already precludes a universe with 1000 spatial dimensions. Also, bear in mind that even those 10- or 11- dimensional universe models being discussed by cosmologists have a "compact" or "curled up" nature to them. See, for example: https://www.space.com/strin...

At first blush this stuck me as quite counterintuitive (how could increasing the extent of reality by a factor of “uncountable infinity” when moving from dimension N to N+1 ever become difficult to detect?), but on reflection I’m less sure.

I'm not saying they are wrong, just that I haven't seen them speak to the questions I was trying to ask. But if you have seen something tell me; what have they figured out that helps answer a question like the ones above?

So I'm not too sure about this, but wouldn't things work out to be much lower temperature for a given level of energy by the equipartition theorem, and therefore you get many more bose einstein condensates going around? All around, quantum behaviour should be pretty common.

And I guess everything is way closer to equillibrium, as you point out. So I suppose life has to be much more malthusian if it is going to survive? And just generally more efficient. So I suppose they'd be running of the equivalent of advanced nanotech? Plus, if the temperature is so low, then I'd expect reversible life to be more common. And things like navigation via coupling atoms in yourself to magnetic fields is going to be easier. And maybe life is going to be more slow because of how little energy is available? Long range communication plausibly relies more on entanglement than vibrations.

Given how weakly coupled things seem to be, I'd expect collapse to be less frequent. So we get life that explicitly exploits QM effects. I don't think improved search speed would really matter that much, but I guess intelligent life would rely less on fast memory? Not sure. Maybe things like writing down large stores of information, which should probably be more like sculptures than little 2D sheets, would be easier.

Vision is going to be massively more important if matter is sparse and radiation common. So I guess they rely on that way more. Spatial world modelling gets far more compute, given the lack of data available they need to be highly sample efficient, as well as how many ways things could move. So I suppose geometric reasoning should be prized, and perhaps their mathematics is more oriented towards that? And I guess if they have art, it should be much more visual.

Wait, what's the lagrangian here? Do we have crazy large symmetry groups with titanic amounts of fields? Everything above is assuming something like SU(1)XSU(2)XSU(3).

But... they do attempt to speak to reality and to accept constraints imposed by observed reality. I didn't think you were interested in purely-fictional universes. After all, you said "In principle our standard theories already contain the answers, if only we can think them through."

This is very much in the spirit of what Greg Egan does in Orthogonal (though with a much smaller tweak to physics).

He has posted extensive supplementary materials on his site, including articles that work out E&M, general relativity, thermodynamics, and quantum mechanics in the "Riemannian" universe. I don't see as much on chemistry and biology (I suspect things get too difficult to predict here) but the books themselves include some (less precise, more speculative) extrapolations there:

Wouldn't the main effect of this be that lots of inverse-square-law phenomena (e.g. gravity, electrostatics) become inverse-power-of-1023-laws and so drop off way faster with distance? That might be a problem for having atoms and stars.

Negative curvature combines the long-distance phenomena of high dimensions with the short-distance phenomena of low dimensions. It might be worth considering first.

One relatively simple but interesting implication of living in 1KD is that protecting your faces is cheap but encasing them in a rigid structure is crazy expensive. What do I mean by this? Think of a single atom that you are trying to "shield" from possible collisions with other atoms. One idea is to put that atom inside a 1K-dimensional hypercube. But if you do that, you will need 3^1000 atoms since you will need to put the lone protected atom at the core of a 3X3X3X...X3X3X3 hypecube.

But if instead you are satisfied with just blocking possible collisions by adding an atom on each "face" of this hypercube, you could do so with as little as 2X1000 atoms, which might be manageable.

In the case of a 3D space, this corresponds to putting your atom between 6 atoms, two per dimension, making it the center of the "sandwich" along each dimension. If you wanted to complete the cube, you'd need to also add the atoms at the edges and corners of the cube, which amount to 26 in this case. Manageable in 3D, but not in 1KD.

So I'd conclude that protecting the "faces" of your body is cheap, but encasing them inside a rigid boundary is crazy expensive. So while you will likely find evolved systems with protection shields, this protection will never include filling up all the "corners", or you'll run out of atoms really fast.

Same would likely apply to the protection of wire/cables.

Maybe one way to get an intuition about what a high-dimensional universe might be like is to consider that our own universe is, in a certain sense, much more than 1000 dimensional. They're not spatial dimensions, but bear with me...

The quantum wave function is at least 3N-dimensional where N is the number of particles in the universe, since it assigns a complex number to every possible arrangement of every particle. (Actually there are more dimensions for spin and so on.) And what do we observe?

Well, the quantum wave function often "branches", never to reunite, due to high degrees of freedom. (Copenhagen folks call this "collapsing", since they imagine that the other branches disappear.) Also, it is very difficult to determine the dimension (i.e. number of particles). Different places (arrangements of all particles) quickly become inaccessible to each other. Most space (i.e. possible arrangements of particles) is empty (i.e. gets low weight under Born's rule). Agents work in much lower dimensions, due to locality (we call irrelevant degrees of freedom "distant objects"), and modding over symmetries (e.g. "the ball fell down" means "all the particles in the ball moved down", vastly simplifying the cognitive burden).

This maybe tells us something about what universes with many spatial dimensions may be like. (After all, what *exactly* qualifies a degree of freedom as "spatial"?) But it doesn't tell us much about the specific high-dimensional universes Robin is interested in, the ones that have something like molecules and so on, assuming such universes are even logically possible.

Do you have a similar link for helium or other atoms?

I would've thought that spin 1/2 particles such as electrons would no longer be fermions in the sense of "taking up space", basically for the same reason that strings can't form knots in higher dimensions. In two spatial dimensions, besides bosons and fermions you can have lots of other things -- "anyons" they're sometimes called.[1] In high dimensions, my vague intuition is that all particles are bosons regardless of their spin, unless we make some other very big changes.

Without fermions, there would be no matter "taking up space". Everything would behave more like a boson, the force-carrying particles, like light, which happily pass right through each other. To the degree something like an atom is possible, there would only be one electron shell, occupied by all the electrons.

Of course, we're still free to imagine a world that has high dimension and yet has "solid matter" without specifying the mathematics of how this would be possible in detail. I certainly don't mean to be a spoil sport. I'm just saying, my intuition is that that's not the world you get when you turn up the dimension knob and try your best to leave all the other most fundamental aspects of the theory the same.

[1] https://www.quantamagazine....

Higher dimensional space was one of the big themes in the second and third books of the Three Body Problem series. Not super rigorous, but it was a lot of fun.

In 1000 dimension space, there are 499,500 hyper planes of rotation. There's a lot of potential there for particle spin axes, and storage of energy in rotational modes rather than translation. Could anything ever translate without dissipating all its energy into rotational modes?

There's also lots of types of hyper-volumes, and maybe potential for things to oscillate between volume types.

I'm not sure what all these modes would enable in terms of packing particles together either.

See alsohttps://en.wikipedia.org/wi...The number of dimensions seems to be very important to the behavior of the laws of nature (even if those laws are not yet completely known/understood).

Re:"...what have they figured out that helps answer a question like the ones above?" Caveat: this is not my area of expertise in physics. But I am under the impression that you can't simply add an arbitrary number of new dimensions to our present universe and then not suffer consequences that prevent humans (and maybe even atoms) from ever existing. If I'm right about that, our present existence already precludes a universe with 1000 spatial dimensions. Also, bear in mind that even those 10- or 11- dimensional universe models being discussed by cosmologists have a "compact" or "curled up" nature to them. See, for example: https://www.space.com/strin...

At first blush this stuck me as quite counterintuitive (how could increasing the extent of reality by a factor of “uncountable infinity” when moving from dimension N to N+1 ever become difficult to detect?), but on reflection I’m less sure.

Atoms are in fact possible: https://arxiv.org/abs/1205..... I don't assume that stars are possible.

I'm not saying they are wrong, just that I haven't seen them speak to the questions I was trying to ask. But if you have seen something tell me; what have they figured out that helps answer a question like the ones above?

So I'm not too sure about this, but wouldn't things work out to be much lower temperature for a given level of energy by the equipartition theorem, and therefore you get many more bose einstein condensates going around? All around, quantum behaviour should be pretty common.

And I guess everything is way closer to equillibrium, as you point out. So I suppose life has to be much more malthusian if it is going to survive? And just generally more efficient. So I suppose they'd be running of the equivalent of advanced nanotech? Plus, if the temperature is so low, then I'd expect reversible life to be more common. And things like navigation via coupling atoms in yourself to magnetic fields is going to be easier. And maybe life is going to be more slow because of how little energy is available? Long range communication plausibly relies more on entanglement than vibrations.

Given how weakly coupled things seem to be, I'd expect collapse to be less frequent. So we get life that explicitly exploits QM effects. I don't think improved search speed would really matter that much, but I guess intelligent life would rely less on fast memory? Not sure. Maybe things like writing down large stores of information, which should probably be more like sculptures than little 2D sheets, would be easier.

Vision is going to be massively more important if matter is sparse and radiation common. So I guess they rely on that way more. Spatial world modelling gets far more compute, given the lack of data available they need to be highly sample efficient, as well as how many ways things could move. So I suppose geometric reasoning should be prized, and perhaps their mathematics is more oriented towards that? And I guess if they have art, it should be much more visual.

Wait, what's the lagrangian here? Do we have crazy large symmetry groups with titanic amounts of fields? Everything above is assuming something like SU(1)XSU(2)XSU(3).

But... they do attempt to speak to reality and to accept constraints imposed by observed reality. I didn't think you were interested in purely-fictional universes. After all, you said "In principle our standard theories already contain the answers, if only we can think them through."

This is very much in the spirit of what Greg Egan does in Orthogonal (though with a much smaller tweak to physics).

He has posted extensive supplementary materials on his site, including articles that work out E&M, general relativity, thermodynamics, and quantum mechanics in the "Riemannian" universe. I don't see as much on chemistry and biology (I suspect things get too difficult to predict here) but the books themselves include some (less precise, more speculative) extrapolations there:

https://www.gregegan.net/OR...

Wouldn't the main effect of this be that lots of inverse-square-law phenomena (e.g. gravity, electrostatics) become inverse-power-of-1023-laws and so drop off way faster with distance? That might be a problem for having atoms and stars.

Negative curvature combines the long-distance phenomena of high dimensions with the short-distance phenomena of low dimensions. It might be worth considering first.

This is a great post - the most imagination-inspiring thing I read in weeks! A lot here to digest, but really fun!