As we all know, in 4D there are no non-trivial solutions to the Schrödinger equation for the hydrogen atom in 4D (the only solution is the degenerate one where orbital radius is zero, i.e., the atom simply collapses), and so chemistry as we know it in 3D cannot exist in 4D. I've been thinking about this over the years, and recently discovered an alternative system that shows promise as a viable basis for 4D chemistry. Well, actually, it works for all dimensions, so in theory it could work in 3D too, but that's not what we're interested in here.

The basic idea behind this alternative system is symmetry, rather than electromagnetism. The original motivation was derived from our early interest on this forum in symmetrical polytopes, so I imagined a "polytopic universe" in which the primary objects are polytopes, mainly of high symmetry. Over the years, I incrementally revised possible models for this, and recently reached a point where it shows promising signs that it may lead to an interesting, non-trivial system that could serve as a replacement for chemistry in 4D (and beyond). Or even in a hypothetical 3D polyhedral universe.

It is postulated that there is a "symmetry force", that favors highly-symmetrical shapes and disfavors shapes of low symmetry. This force acts to stabilize objects that have high symmetry, and resists changes of shape to forms of lower symmetry.

In a space of n dimensions, it is postulated that there are carriers of this force, which we will call "symmetry quarks" for lack of a better name. Each symmetry quark represents a finite rotation group in n-dimensional space. The way this works is that these symmetry quarks consist of "symmetry subquarks", which are bound particles that have inherent rotation. That is, a subquark will rotate by some fixed inherent angle A in every unit of quantized time. When A is an irrational division of pi, the subquark will destructively interfere with itself and cancel itself out; so the only subquarks that could exist are those whose inherent rotation angle is a rational division of the circle. Thus, each particle will trace out a regular polygon. However, subquarks cannot exist in isolation; they can only exist in bundles of n subquarks, where n is the dimension of the ambient space. These bundles are the symmetry quarks. Subquarks within a symmetry quark will interact with each other; when the resulting rotation group is non-finite, they undergo destructive interference and the symmetry quark will cease to exist. Therefore, the only symmetry quarks that could exist are exactly those which corresponding with the finite rotation groups. In these quarks, constructive interference happens and the configuration is strengthened, taking the shape of a regular polytope.

Now, these symmetry quarks themselves don't directly form matter in the n-dimensional space; they are merely the carriers of the "symmetry force". We assume that there's some other form of matter -- the exact nature of which is unimportant for our purposes -- that can be formed into various shapes. We postulate that the polytopic universe is filled with a "background radiation" of symmetry quarks -- a sea of symmetry force carriers that, when they encounter an object of matching symmetry, will "resonate" with that object. The closer an object is to the inherent symmetry of a particular quark, the stronger the resonance will be. So over time, symmetrical objects will accumulate more quarks of that particular symmetry. These accumulated quarks will "resonate" with each other, strengthening the action of the symmetry force on the object. So for instance, an object in the shape of a cube will accumulate quarks of octahedral symmetry, and an object in the shape of an icosahedron will accumulate quarks of icosahedral symmetry. These quarks confer extra stability to these objects via the symmetry force. Whereas a non-symmetrical object will have less resonance with symmetry quarks, and thereby have less stabilization by the symmetry force. Therefore, highly-symmetrical shapes are favored over less symmetrical ones.

Now, an object in the shape of a cube will accrue not only quarks of octahedral symmetry, but its 6 square faces would also resonate with quarks that have dihedral square symmetry. This is a weaker resonance since the quarks don't correspond with the full octahedral symmetry, but it does confer a residual stabilization. The same is true for non-symmetrical polyhedra that nevertheless have regular polygonal faces -- the regular polygonal faces accrue quarks of dihedral symmetry, and therefore receive stabilization from the symmetry force, as opposed to a shape with irregular faces. Similarly, Catalan solids would have strong resonance with symmetry quarks of their corresponding symmetry, but would lose the extra stabilization that comes from having regular faces.

These subdimensional resonances have far-reaching consequences: suppose there's a bunch of tetrahedra and icosahedra lying around. The former would have lots of resonance with quarks of tetrahedral symmetry, and the latter would have lots of resonance with quarks of icosahedral symmetry. But both would also have common resonance with quarks of triangular symmetry due to their triangular faces. And since these quarks of triangular symmetry will resonate with each other, the closer a tetrahedron is to an icosahedron the stronger this resonance will be. The overall effect will be a weak attractive force: small perturbations of position will sometimes move the tetrahedron marginally closer to the icosahedron, and the symmetry force will confer a stronger stabilization. Therefore, it will also resist perturbations that move the two objects apart. So over time, perturbations that move them closer to each other will accumulate, whereas perturbations that move them apart will be resisted. The end result is that objects of compatible symmetry will self-assemble.

These self-assemblies will, of course, favor large-scale symmetry constructions; for example, 4 objects that are attracted to each other will favor a tetrahedral configuration, because that will resonate with the tetrahedral quarks more. Once such a formation takes place, it will resist being taken apart -- because then it loses the resonance with the tetrahedral quarks and the extra stabilization by the symmetry force. To break apart this configuration would require energy input to overcome the resistance by the symmetry force.

Now consider a bunch of objects that could be assembled in various ways. Say a bunch of balls that could either form a tetrahedral stack, an octahedral stack, or an icosahedral stack. Suppose, by random chance (brownian motion) they self-assemble into a tetrahedron. Then even though an icosahedral configuration might be more stable (higher-order symmetry), once they have reached a tetrahedral configuration they will resist being reassembled in another configuration. I.e., they are in a local minimum, and it will take energy to push them out of that local minimum in order to get them to a global minimum.

Now consider the phase space of all possible configurations of a bunch of objects. There will have local minima where the configuration is highly symmetrical, due to strong resonance with the symmetry quarks and consequent stabilization by the symmetry force. But between these local minima will be less stable, intermediate configurations. This means that, left to itself, the system will naturally fall into one of the local minima and stay there. To effect a change of state, energy will need to be put into the system to move it out of the local minimum into an intermediate, non-symmetrical (unstable) state. It will then fall into another local minimum (another symmetrical configuration) -- and in the process release the amount of energy that corresponds to the depth of that local minimum.

Here we see the analogy with chemistry: these local minima, which will be configurations with high symmetry, can be considered to be "symmetry compounds" or "symmetry molecules", by analogy with chemical compounds. Breaking them up and reassembling them in a different configuration is analogous to a chemical reaction, that changes the shape of a molecule to a different molecule. The energy required to move the system out of the local minimum is analogous to the activation energy of a chemical reaction. And when the system reaches another local minimum, if the energy released is greater than the activation energy, we have an exothermic reaction. If the energy released is less than the activation energy, we have an endothermic reaction.

Furthermore, suppose there's an ambient amount of energy in the environment that the system can absorb. If this ambient energy level is greater than the activation energy of its current local minimum, then the system will spontaneously move out of its current local minimum and move to another. Thus, if there's a bunch of local minima with lower activation energy than the ambient energy level, the system will spontaneously move between them, until it finds a global minimum that's deeper than this ambient energy level. We may call this the "temperature" of the system: a system that's normally stable may, if heated up, undergo a one-way transformation to a globally more stable state. I.e., when the temperature is high, only more symmetrical shapes will exist. Conversely, if the ambient temperatue is low, then less symmetrical states (shallower local minima) will be stable, so less symmetrical shapes can exist. Now, if the ambient energy level is higher than the global minimum of the system, it will spontaneously transition between all possible states: i.e., it melts into a "symmetry liquid" without any fixed shape -- no shape is stable, regardless of symmetry. When the ambient temperature is lowered again, then it will settle in a local minimum, one closest to its exact state at that point in time. I.e., it "freezes" or "condenses" into a solid state.

The existence of local minima is good news, because it means that even though a bunch of balls may be most stable in icosahedral configuration, the system may nevertheless be metastable as a bunch of tetrahedra instead -- because that's a local minimum and there's not enough energy for the system to find the global minimum. This means that in this polytopic universe, multiple objects of lower symmetry can exist simultaneously -- we won't end up with all the matter in the universe assembling itself into a gigantic icosahedral (or 600-cell) configuration. IOW interesting "chemistry" can happen.

Next steps: work out the specifics of this system, i.e., how to assign energy levels to specific configurations of objects. Probably some kind of counting scheme of symmetry quarks. Then one can work out what the local minima ("symmetry molecules") are, and reaction paths between them. It will not be just a simple bunch of reaction rules; a lot of this will depend on the ambient energy level and what are the neighbouring local minima in the system. IOW, just like real-world chemistry, full of interesting complexities and interactions, and fun to speculate on.