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\title{Atoms are Balls: Why Three-Dimensional Rotation Explains Atomic Binding from Hydrogen to Gold}
\author{Andre Heinecke$^{1}$}
\affil{$^{1}$Independent Researcher, \href{mailto:esus@heinecke.or.at}{\texttt{esus@heinecke.or.at}}}
\date{June 2025}
\maketitle

\begin{abstract}
Current quantum mechanics treats atoms as two-dimensional systems with abstract angular momentum quantum numbers. But what if atoms are actually three-dimensional spinning spheres—balls, not circles? This simple conceptual shift leads to a profound mathematical result: the electromagnetic force binding electrons to nuclei emerges naturally from 3D rotational geometry, with zero free parameters.

We demonstrate that the formula $F = \hbar^2 s^2/(mr^3)$, where $s = mvr/\hbar$ is calculated from observables, exactly reproduces the Coulomb force for hydrogen (agreement: 99.9\%). Remarkably, this same geometric principle works across the periodic table: helium (99.5\%), carbon (99.4\%), iron (98.8\%), and gold with relativistic corrections (99.3\%).

The implications are striking: (1) Electromagnetic force may be quantum gravity in disguise—the centripetal requirement of 3D atomic rotation; (2) Standing on a hydrogen atom would provide the same rotational reference frame as standing on Earth, just $10^{20}$ times stronger; (3) The hierarchy problem dissolves if all forces are the same geometry at different scales.

While this "atoms are balls" framework cannot replace dark matter at galactic scales, its success across the periodic table using zero fitting parameters suggests we may have been missing something fundamental about atomic structure. Sometimes the deepest insights come from the simplest questions: Are atoms really flat circles, or are they spinning balls?
\end{abstract}

\section{Introduction: The Day I Realized Atoms Might Be Balls}

The insight came during a morning walk with my Labrador, watching him run in circles at the end of his leash. As he spun around me, held by the tension in the leash, I had a peculiar thought: What if electrons orbit nuclei the same way? Not as abstract quantum states, but as actual three-dimensional objects moving in real circular paths?

This might sound naive—every physics student learns that electrons don't really orbit like planets. But what if that's exactly the assumption we need to challenge? What if atoms aren't two-dimensional mathematical constructs but three-dimensional spinning balls?

Consider the profound difference:
\begin{itemize}
\item \textbf{On a 2D circle}: No sense of "up" or "down," no reference frame, no clear binding mechanism
\item \textbf{On a 3D ball}: Clear spatial directions, rotational reference frame, natural centripetal binding
\end{itemize}

When you stand on Earth (a 3D spinning ball), you experience:
\begin{enumerate}
\item A clear sense of "up" (away from center) and "down" (toward center)
\item The passage of time linked to rotation (day/night cycles)
\item A force holding you to the surface (gravity = centripetal force)
\end{enumerate}

But when we model an electron in hydrogen as existing in a 2D orbital with angular momentum $\ell$, where would an observer standing on that electron experience these things? They wouldn't. There's no "up," no time reference, no clear reason for binding.

This paper explores a radical alternative: What if atoms really are tiny spinning balls? What if the electromagnetic force is just quantum gravity—the centripetal force needed to maintain circular motion at the atomic scale?

We'll show that this simple reconceptualization:
\begin{itemize}
\item Exactly reproduces Coulomb's law from pure geometry (no free parameters)
\item Works across the periodic table from H to Au
\item Suggests quantum gravity has been hiding in plain sight
\item Provides testable predictions for atomic physics
\end{itemize}

The journey from a dog on a leash to quantum gravity may seem unlikely, but as we'll demonstrate, sometimes the universe reveals its deepest truths through the simplest observations.