Imagine for a moment that we are at the end of 1900. Physical science is in crisis, as Rayleigh-Jeans law dismally fails to fit the intensity distribution of a blackbody, and the questions of stability of the classical atom of circular orbits of electrons around a positively charged nucleus is a serious anomaly because it throws open the question of stability of all matter. Planck’s solution of the blackbody radiation problem relied on the hypothesis that energy is quantized. His solution to the blackbody radiation problem specifically does not require a detailed theory of how and why energy is quantized but simply that it is. At the same year, Ivars Fredholm resolved a great open problem of nineteenth century mathematics: he showed that a bounded domain in the plane has a discrete set of pure tones or a discrete set of eigenvalues of the Laplacian. Now while the observables-as-operators approach to description of reality by quantum mechanics allows the description of observables by eigenvalues of operators, a direct link had not been made in 1900 between quantization of energy and pure tones of a geometric object: the entire universe.

Spectrum of the Laplacian on noncompact manifolds need not be discrete and even on the euclidean spaces the spectrum of the Laplacian is not discrete. If there be a link between Fredolm’s solution of the discreteness of the spectrum of Laplacian on bounded domains on the plane would be connected to the energy spectrum if the universe could be shown to be compact. If we seek such evidence, we can find it in Penzias and Wilson’s discovery in 1964 of the cosmic background radiation which was used to reinforce the Big Bang cosmology which was given instead as evidence for the expansionary universe models.

We can proceed as follows: first, although the Nobel prize for chemistry was given to Daniel Shechtman in 2011 for ‘the discovery of quasicrystals’, we can note that what Shechtman had clearly discovered from early 1980s is that 5, 8, 10, and 12 fold rotational symmetries are observed in crystalline structures. These can arise as symmetries of four dimensional crystals but not for three dimensional crystals. Shechtman’s discoveries can be used parsimoniously as evidence for existence of four macroscopic spatial dimensions. Of course the actually observed ‘quasi-crystals’ do not have translational invariance because they cannot have translational invariance, but even the standard models of quasicrystals use projections of higher dimensional crystals, which makes the conclusion of the existence of four macroscopic spatial dimensions reasonable.

Next, although the redshift of distant galaxies has been interpreted as a universal Doppler effect, a parsimonious conclusion would be that these result from some mechanism that reduces the energy of the photons uniformly, like frictional drag and thereby lowers the frequency of light towards red.

Now the cosmic background radiation has a uniform lower bound, around 2.7 K. Assuming that diffusion produced the cosmic background radiation, we can use the Gaussian upper bounds established for heat kernels on complete noncompact riemannian manifolds with a lower bound on Ricci curvature to conclude that our actual universe must be compact.

So we are living in a compact four dimensional universe. It has to be a sphere if it is to replicate the quantization of energy for the hydrogen atom. We can deduce that the radius of the universe must be 1/h in length-equivalent where h is Planck’s constant. Ricci curvature for any three dimensional submanifold can be calculated using the second fundamental form as well which gives us a version of the gravitational field equations with the ‘cosmological constant’ filled in: it is h^2, and this matches experimental determination of the cosmological constant.

The model of photons often employed locally is as a superposition of plane waves. If we take seriously the description of the universe above, then we must replace the plane waves by spherical harmonics for a 4-sphere. Recall that one can consider harmonic homogeneous polynomials on 5-dimensional euclidean space. These restricted to the 4-sphere are precisely the eigenfunctions of the spherical Laplacian, the degree k harmonic homogeneous polynomials have eigenvalue -k(k+2).