# Part II — Grids and Rydberg One of the great successes of quantum mechanics is its ability to predict, with remarkable precision, the energy levels of atoms—most notably hydrogen, one of the most abundant elements in the universe. An analogy for these energy levels and transitions can be drawn on a napkin. Imagine a flat rectangular sheet of paper. Draw two straight lines, from edge to edge, crossing in the middle. Together, these lines divide the napkin into a grid of four cells. Increasing the number of lines on both sides of the napkin further subdivides the surface. With two lines drawn in each direction, the surface is divided into $3 \times 3 = 9$ cells. With three lines, into $4 \times 4 = 16$ cells, and so on. In general, $n$ lines drawn in each direction give a grid of $n^2$ cells. Before any lines are drawn, corresponding to level $1$ (or $n = 1$), the surface of the napkin is undivided. We call this the ground state, and we associate with it the first energy level, $E_1$. Without going into further detail, it is not difficult to see that a natural relationship can be established between these successive subdivisions and the observed atomic energy levels, namely $$ E_n = -\frac{E_1}{n^2}. $$ In this picture, increasing $n$ corresponds to increasing the number of global windings on the surface, represented here by the number of cells in the grid. More windings impose more nodes on the same conserved surface. As the number of nodes increases, the characteristic wavelength associated with the configuration becomes shorter, allowing more nodes to fit on the surface. The total energy scale is fixed; what changes from one level to the next is how the surface is subdivided. Each level is characterized by the size of its cells, with $E_n = E_1/n^2$. Transitions between levels are related to the difference in cell sizes between two subdivisions, and require supplying the energy $$ E_1\!\left(\frac{1}{n^2}-\frac{1}{m^2}\right) $$ to move from level $n$ to level $m>n$. In this abstract but constrained way, we recover the Rydberg series—one of the great successes of quantum mechanics—without invoking wavefunctions, operators, mass, or charge, by viewing atomic energy levels as global subdivisions of a conserved surface. # The Torus To visualize a torus—a donut-shaped object—return to the flat napkin. First, identify and glue together one pair of opposite edges of the napkin. After this identification, the flat napkin becomes a tube. Lines that originally ended on one edge now reappear continuously on the opposite edge. Next, take this tube and identify its two circular ends. Gluing these ends together produces a closed surface with no boundary. The result is a donut-shaped surface, known as a torus. Any lines drawn on the original napkin become closed paths on the torus only if they match their own position when crossing an identified edge. This requirement ensures global continuity of the grid and is called a continuity condition. # Charge Consider a standing electromagnetic wave on the surface of a torus in a source-free Maxwell universe. As pictured with the napkin wrapped into a donut, the surface of a torus is equivalent to a rectangular surface with opposite edges identified. Electromagnetic fields on this surface must satisfy periodic (continuity) conditions along the two independent cycles of the torus, analogous to the two orthogonal directions of the napkin. In the absence of sources, Maxwell’s equations read $$ \nabla \cdot \mathbf{E} = 0, \qquad \nabla \cdot \mathbf{B} = 0, $$ together with the evolution equations for $\mathbf{E}$ and $\mathbf{B}$. These equations allow nontrivial solutions on a closed surface, provided the fields form globally consistent standing waves. On a torus, such standing waves can wrap around the two independent cycles of the surface: along the axis of the tube and around its cross-section. Let $m$ and $n$ denote the integer numbers of nodal lines forming the grid on the surface of the torus. These lines can be interpreted as nodal locations, where the electromagnetic fields change sign. A given toroidal configuration can therefore be characterized by the ordered pair $(m,n)$. A source-free universe means that there are no local electric charges producing radial field lines; instead, $$ \nabla \cdot \mathbf{E} = 0 $$ everywhere. However, closed field lines are still allowed for both electric and magnetic fields. Closed field lines may be viewed as loop circulations. Such circulations naturally generate tangential flows on the surface of the torus. At any local patch of the surface, field lines entering and leaving the patch balance so that the net flux vanishes, while tangential circulations form vortices around them. Seen in this way, the pair $(m,n)$ characterizes the flow of energy along the two independent directions of the torus: a measure of how energy circulates along each cycle. If the energy of a shell is distributed over the surface of the torus, and if $m=n$, then each cell carries an energy density proportional to $E/n^2$. The torus encloses a volume. At distances much larger than the size of the torus, the detailed structure of the surface becomes irrelevant, and the energy distribution can be treated as effectively concentrated at a point. At a distance $r$ from this point, the energy density is therefore distributed over a spherical surface of area $4\pi r^2$, leading naturally to a $1/r^2$ radial dependence. In this model, charge quantization emerges naturally from the allowed integer winding numbers. Thus, charge appears not as an added ingredient, but as a topological property of standing electromagnetic waves on a closed surface, completing the geometric picture introduced with grids, energy levels, and the torus.