Life in 1KD
Years ago I read Flatland and Planiverse, stories set in a two-dimensional universe. To me these are the epitome of “hard science fiction”, wherein one makes one (or a few) key contrary assumptions, and then works out their physical and social consequences. I’ve tried to do similarly in my work on the Age of Em and the Hardscrapple Frontier.
Decades ago I thought: why not flip the dimension axis, and consider life in a thousand spatial dimensions? I wrote up some notes then, and last Thursday I was reminded of Flatland, which inspired me to reconsider the issue. Though I couldn’t find much prior work on what life is like in this universe, I feel like I’ve been able to quickly guess many plausible implications in just a few days.
But rather than work on this in secret for more months or years, perhaps with a few collaborators, I’d rather show everyone what I have now, in the hope of inspiring others to contribute. This seems the sort of project on which we can more easily work together, as we less need to judge the individual quality of contributors; we can instead just ask each to “prove” their claims via citations, sims, or math.
Here is what I have so far. I focus on life made out of atoms, but now in a not-curved unlimited space of dimension D=1024 (=2^10), plus one time dimension. I assume that some combination of a big bang and hot stars once created hot dense plasmas with equal numbers of electrons and protons, and with protons clumped into nuclei of varying sizes. As the universe or star regions expanded and cooled, photons bound nuclei and electrons into atoms, and then atoms into molecules, after which those clumped into liquids or solids. Molecules and compounds first accreted atoms, then merged with each other, and finally perhaps added internal bonds.
A cubic array of atoms of length L with as many surface as interior atoms satisfies (L/(L-2))^D = 2, which for D = 1024 gives L = 2956. Such a cube has (2956)^1024 atoms in total. As I hereby define 2^(2^10) to be “crazy huge” and 2^(-2^10) to be “crazy tiny”, this is a more than crazy huge array. (“Crazy huge” is ~100K times a “centillion”. “Astronomical” numbers are tiny by comparison to these.)
We thus conclude that solids or liquids substantially smaller than crazy huge have almost no interiors; they are almost all surface. If they are coupled strongly enough to a surrounding volume of uniform temperature or pressure, then they also have uniform parameters like that. Thus not-crazy-huge objects can’t have separated pipes or cavities. Stars with differing internal temperatures must also be extra crazy huge.
The volume V(r,D) of a sphere of radius r in D dimensions is V = r^D pi^(D/2) / (D/2)!. For dimensions D = (1,2,3,8,24), the densest packing of spheres of unit radius is known to be respectively (0.5,0.28,0.18,0.063,1) spheres per unit volume. The largest D for which this value is known is 24, where the sphere volume fraction (i.e., fraction of volume occupied by spheres) is V(1,24) ~= 1/518. If we assume that for D=1024 the densest packing is also no more than one unit sphere per unit volume, then the sphere volume fraction there is no more than V(1,1024) = 10^-912. So even when atoms are packed as closely as possible, they fill only a crazy tiny fraction of the volume.
If the mean-free path in a gas of atoms of radius r is the gas volume per atom divided by atom collision cross-section V(2r,D-1), and if the maximum packing density for D=1024 is one atom of unit radius per unit volume, then the mean free path is 10^602.94. It seems that high dimensional gases have basically no internal interactions. I worry that this means that the big bang doesn’t actually cause nuclei, atoms, and molecules to form. But I’ll assume they do form as otherwise we have no story to tell.
Higher dimensions allow far more direction and polarization degrees of freedom for photons. The generalized Stefan-Boltzmann law, which says the power is radiated by a black body at temperature T, has product terms T^(D+1), (2pi^0.5)^(D-1), and Gamma(D/2), all of which make atoms couple much more strongly to photons. Thus it seems high D thermal coupling is mainly via photons and phonons, not via gas.
Bonds between atoms result from different ways to cram electrons closer to atomic nuclei. In our world, ionic bonds come from moving electrons from higher energy orbital shells at one atom into lower energy shells at other atoms. This can be worth the cost of giving each atom a net charge, which then pulls the atoms together. Covalent bonds are instead due to electrons finding configurations in the space between two atoms that allow them to simultaneously sit in low shells of both atoms. Metallic bonds are covalent bonds spread across a large regular array of atoms.
Atoms seem to be possible in higher dimensions. Electrons can have more degrees of spin, and there are far more orbitals all at the lowest energy level around nuclei. Thus nuclei would need to have very large numbers of protons to fill up all the lowest energy levels. I assume that nuclei are smaller than this limit. Thus different types of atoms become much more similar to each other than they are in our D=3 universe. There isn’t a higher shell one can empty out to make an ionic bond, and all of the covalent bonds have the same simple spatial form.
The number of covalent bonds possible per atom should be < ~3*D, and B < ~D-10 creates a huge space of possible relative rotations of bonds. Also, in high dimensions the angles between random vectors are nearly right angles. Furthermore, irregularly-shaped mostly-surface materials don’t seem to have much scope for metallic bonds. Thus in high dimensions most atom bonding comes from nearly right angle covalent bonds. Which if they form via random accretion creates molecules in the shape of spatial random walks of bonds in 1024 dimensions.
It is hard to imagine making life and complex machines without making rigid structures. But rigid structures require short loops in the network of bonds, and for high D these seem unlikely to form due to random meetings of atoms in a gas or liquid; other random atoms would bond at a site long before nearby connected atoms got around to trying.
If a network of molecular bonds between N atoms has no loops, then it is a tree, and thus has N-1 bonds, giving less than two bonds per atom on average. But for P>>2, this requires almost all potential bonds to be unrealized. Thus if most atoms in molecules have P>>2 and most potential bonds are realized, those molecules can’t be trees, and so must have many loops. So in this case we can conclude that molecular bond loops are typically quite long. (How long?) Also, the most distinctive types of atoms are those with P =1,2, as enough of these can switch molecules between being small and very large.
Molecules with only long loops allow a lot of wiggling and reshaping along short stretches, and only resist deformations only on relatively large scales. And when many atoms with B < D-2 are close to each other, most neighboring atoms will not be bonded, and can thus slide easily past each other. Thus on the smallest scales natural objects should be liquids, not solids nor metals. And in a uniform density fluid of atoms that randomly forms local bonds as it cools, the connectivity should be global, extending across the entire expanded-and-cooled-together region.
Perhaps short molecular loops might be produced by life-like processes wherein some rare initial loops catalyze the formation of other matching loops. However, as it seems harder to form higher dimensional versions, perhaps life structures are usually low dimensional, and so must struggle to maintain the relative orientation of the “planes” of their different life parts. Life made this way might envy our ease of creating bond loops in low spatial dimensions; did they create our universe as their life utopia?
We have yet to imagine how to construct non-crazy-huge machines and signal processing devices in such a universe. What are simple ways to make wires, gates, levers, joints, rotors, muscles, etc.? Could the very high D space of molecule vibrations be used to good effect? Copying the devices in our universe by extending them in all dimensions is possible but often results in crazy huge objects. Nor do we know what would be the main sources of negentropy. Perhaps gravity clumping, or non-interacting materials that drift out of equilibrium as the universe expands?
The dynamics of a uniformly expanding universe is described by a scale factor a(t), which says how far things have spread apart at each time. For a matter-dominated universe a(t) goes as t^(2/(D-1)), and for a radiation-dominated universe a(t) goes as t^(2D/((D-1)(D+1)). For matter, density goes as a(t)^-D, while for radiation it goes as a(t)^-(D+1). In both cases, we have density falling as t^-2D/(D-1), which is roughly t^-2 for large D. Thus as a high D universe expands, its density falls in time much like it does in low D, but its distances increase far more slowly. There is little expansion-based redshift in high D.
When an expanding region cools enough for molecules to connect across long distances, its further expansion will tend to pull molecular configurations from initially random walks in space more toward long straight lines between key long-loop junctures. This makes it easier for phonons to travel along these molecules, as bond angles are no longer nearly right angles. For the universe, this added tension is not enough to kick it into an exponentially expanding mode; instead the expansion power law changes slightly. Eventually the tension gets large enough to break the atomic bonds, but this takes a long time as widths change only slowly with volumes in high D. (What are typical diameters of the remaining broken molecules?)
As the universe ages, the volume and amount of stuff that one could potentially see from any one vantage point increases very rapidly, like t^(D-1). However, the density or intensity of any emissions that one might intercept also falls very fast as distance d via d^-(D-1), making it hard to see anything very far. In high dimensions it is extremely hard to have a comprehensive view of everything in all directions, and also very hard to see very far in any one direction, even if you focus all of your attention there.
When two powers have a physical fight in this universe, their main problem seems to be figuring out their relative locations and orientation. It might be easy to send a missile to hit any particular location, and nearly impossible for the target to see such a missile coming or to block its arrival. But any extended object probably does not know very well the locations or orientations of its many parts, nor is it even well informed about most of the other objects which it directly touches. It knows far less about objects even a few atom’s width away in all directions. So learning the locations of enemies could be quite hard.
Finding good ways to learn locations and orientations, and to fill and update maps of what is where, would be major civilization achievements. As would accessing new sources of negentropy. Civilizations should also be able to expand in space at a very rapid t^(D-1) speed.
A high D universe of trivial topology and any decent age encompasses crazy huge volumes and numbers of atoms. The origin of life becomes much less puzzling in such a universe, given the crazy huge number of random trials that can occur. It should also be easy to move a short distance and then quickly encounter many huge things about which one had very little information. One has not seen it nor heard about them via one’s network of news and talk. This creates great scope not only for adventure stories, but also for actual personal adventure.
I’ve only scratched the surface here of all the questions one could ask, and some of my answers are probably wrong. Even so, I hope I’ve whetted your appetite for more. If so, please, figure something out about life in 1KD and tell the rest of us, to help this universe come more sharply into view. In principle our standard theories already contain the answers, if only we can think them through.
Thanks to Anders Sandberg and Daniel Martin for comments.
Added 1Feb: One big source of negentropy for life to consume is all of the potential bonds not made into actual bonds on surface atoms. Life could try to carefully assemble atoms into larger dimensional structures with fewer surface atoms.
Added 2Feb: In low D repulsive forces can be used to control things, but in high D it seems that only attractive forces are of much use.
I think I disagree with a few of your points. What is the significance of your argument about crazy huge numbers of atoms? That is indeed a very large number of atoms, but whether that seems ridiculous is less relevant than the length scale on which structures would actually tend to form, and this doesn't actually seem like an argument that that would be small (relative to the atomic radius). I have some heuristic arguments on this but they relate to another point too so I'll get to them later. It does seem plausible that living things would have to do without enclosed tubes and cavities though.
The paper you link for the stability of atoms doesn't show what you're using it for. They claim, and I'm willing to believe, that there are stable states of hydrogenic atoms in >=5D where the electron wavefunction is localised near the nucleus, but these states are only stable in the absence of other interactions (such as with other atoms or the electromagnetic field). Their energies are positive therefore the electron isn't actually bound to the atom, it can lose energy by ionising. Worse, the energy spectrum is still unbounded below, so the electron will fall into the nucleus if it doesn't leave entirely. To see that the energy is unbounded below, consider a gaussian wavefunction centred on the nucleus. It is not an energy eigenstate, but its expected energy tends to negative infinity as the size tends to zero. By the variational principle, this provides an upper bound on the energy of the ground state, that upper bound being negative infinity, i.e. the ground state does not exist.
Is your claim that the number of bonds possible per atom is <~3*D based on the packing of spheres, or the number of spin states of an electron? I don't know enough to comment on the latter, but in the former case that doesn't seem right. Optimal sphere packing most likely won't be achieved, but a very rough argument suggests the number of spheres that can touch a central sphere grows like 2^D rather than linearly, and from the results about dimensions where it is actually known, it looks like the growth is still exponential but with a base somewhat less than 2. Even for random packings, I'd guess the number of bonds per atom would still be exponential in the dimension. So long as the molecule has few enough bonds to be largely non-rigid, each atom still has opportunities to collide with more atoms and form more bonds. The density of not-already-bonded atoms in the vicinity of an atom in a non-rigid molecule will be at least not much less than in the space in general, so it will continue to accumulate bonds until the structure becomes at least somewhat rigid (or heats up enough from the huge number of bonds forming that the formation of new bonds is balanced by the breaking of old ones). If the attraction between two atoms at a distance sqrt(2) times the equilibrium bond length is non-negligible, the situation would be even more different, since chains would tend to rapidly curl up into dense structures. This probably wouldn't be the case though since effects fall off so quickly with distance.
I'm a little dubious about your argument about mean free path too, although I'm less sure of this one. In the optimally packed lattice itself, the mean free path is 0. As the density goes down from there, why would the mean free path increase so rapidly?