1. Setting the Stage

Imagine rolling cosmic dice before the Big Bang: could the universe have emerged with four, five, or even ten spatial dimensions? We’ll survey the key physical “tests” any dimensionality \(d\) must pass to host chemistry, stars, and talking primates. Starting with plain-language intuition and then backing it up with MathJax formulas so you can see the numbers at work.

If you’re curious how this special choice influences fundamental laws, see Why 𝐸 = 𝑚𝑐².

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2. Gravity, Electrostatics, and Stable Orbits

2.1 Qualitative Overview

• Too few dimensions (\(d \le 2\)): the force decays too slowly—particles spiral into the center.
• Too many dimensions (\(d \ge 4\)): the force decays too quickly—orbits fall apart.
• Exactly \(d = 3\): the familiar \(1/r^2\) law supports closed, repeating paths—Kepler would approve.

2.2 Quantitative Analysis

In \(d\) dimensions Gauss’s law gives   $$ F(r)\propto\frac{1}{r^{d-1}} \tag{1} $$   and the corresponding potential   $$ V(r)\propto\frac{1}{r^{d-2}}\,. \tag{2} $$   Small perturbations about a circular orbit remain bounded only if the effective radial potential has a minimum—this occurs precisely when   $$ d = 3\,. \tag{3} $$   Substituting \(d=4\) flips the sign of the restoring term, so orbits either collapse or escape. The same analysis applies to the hydrogen electron cloud.

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3. Quantum Atoms and Chemistry

3.1 Intuitive Picture

Electrons need a delicate balance between Coulomb attraction and zero-point kinetic energy. Tweaking the exponent in the Coulomb term destroys that balance.