The Newtonian paradigm is based upon a shortcut. Newton took the geometric center of the uniformly dense material body as representing its location in order to utilize his calculus for physical problems. This shortcut was meant only as a convenience and was never meant to be taken as “real.” The question of the geometric form of matter, thus, haunts us to this very day.
Early in the 20th century, physicists attempted to come to terms with the form of matter. Quantum theory was based upon the “elemental signatures” that were emitted by heated gases in the form of the series of abnormally intense frequency ranges. These series could be mathematically reproduced through simple formulae (at least in the case of hydrogen gas), and this fact opened the door to the notion that atoms consist essentially of standing waves.
Standing waves are the “distinct solutions” to the patterns of phase variation (amplitude) that can occur within the confines of a bounded space. These solutions depend entirely upon the various symmetries that are possible within the given dimensional context. The one-dimensional solutions are the harmonics of the stringed instruments. The two-dimensional solutions are the various vibrational patterns that the surface of a drum can attain. In three dimensions, we find the spatial patterns that have found application in everthing from acoustics to video game lighting.
Since we live in a 3D universe, these volumetric standing wave patterns should be of extreme interest to any theoretical physicist. In particular, the notion of standing-wave-as-atomic-orbital-shape has been an extraordinary way of understanding the physical foundations of chemistry. Most people do not realize that the configurations of the atomic orbitals are simply the mathematical solutions to an equation that dates back to the time of Kant’s Critiques (the Laplace equation).
Two of the founders of modern quantum theory, de Broglie and Schrodinger, were of the belief that matter ultimately consists of “spatial waves” in these particular configurations. However, there is not much “elegant mathematics” that can be done with this kind of wave theory (that is, the mathematics is exceedingly cumbersome and approximative). Human nature being what it is, the Newtonians took over and interpreted these wave solutions as representing “particle probability densities” (this interpretation was credited to Max Born). But because the “matter wave” description could not be ignored, they settled on the various doctrines that go by the names of uncertainty, complementarity, and correspondence. In the end, though, all that these doctrines mean is: duality.
Because the academic establishment could not stomach the idea of completely abandoning the easily calculable point-mass methodology, duality became the order of the day, and professional physicists eventually went back to their Newtonian ways. Today, the only people who advocate for the atoms-as-standing-wave approach are the various fringe theorists and crackpots that find their homes on the internet. However, the question still remains for any credentialed scientist who wishes to attempt to answer it: What is the geometry of matter?
The fundamental difficulty of the standing wave approach is the boundary problem. That is, the locations of particular standing waves are determined solely by the boundaries that contain it. On the one hand, if every atom is a standing wave that is contained within its own individual “micro sphere,” we are face with the uncomfortable position of having a fundamental interior/exterior duality. That is, there is a nonlocal physics of interior space at the same time that there is a local physics of exterior space.
The only other option is to conceive of atomic standing waves as being defined only by the boundary of the universe. But in this case, every atom will be piled on top of every other atom, because they all share the same boundary. It seems, therefore, that the atomic wave theorist is at an impasse: he needs each atom to be defined by distinct boundaries, and yet he also does not like the philosophical dualities that such a method implies. But there is another option!
Consider the distinction between interior spaces and exterior surfaces. We typically think of surfaces as being the two-dimensional limits of three-dimensional spaces. Geometrically, however, we can generalize this notion to surfaces being the n-1 dimensional limits of n-dimensional spaces. Thus, if we think of the 3d space of the universe as being the surface of a 4d hyperspace, then our problems can be solved.
That is, we can think of each atom as being defined from within the confines of a normal, 3d sphere. But then, we can perform a kind of reverse map projection such that the inside of the sphere “wraps” around the outside of the hypershere. In this way, the entire 2d surface of the defining sphere can be made to meet at a single point at some arbitrary location on the 3d surface of the hypersphere. I know this is difficult to comprehend, but the lower dimensional analogy is very simple: a 2d standing wave that is bounded by a circle can be made to wrap around the surface of a 3d sphere, such that the entire circle is made to meet at a single point (this is the reverse case of the azimuthal equidistant projection, which can be seen on the flag of the United Nations).
In this way, every atom-as-standing-wave has its own unique boundary in the form of a “locatable point” somewhere on the surface of a hypersphere. The interior/exterior duality is avoided, because you cannot go “outside” of a central point-boundary. That is, no matter where you happen to be on the hypersphere, you are always inside of every atom.
The good thing about performing this reverse map projection function is that the local spatial units of the original pre-projection standing waves are compressed/bent the closer that you get to the central boundary point. This is a good thing, because it means that, from the perspective of an independent observer, the amplitudes of the given wave appear to grow larger, in perfect agreement with the inverse square laws of physics. Within this paradigm, we are able to understand the atomic structure of the universe in a theoretically cohesive manner.