It all began in 1905 when Albert Einstein asked whether light could exert a push (the “recoil” of a photon cannon) and what that implied for conservation of energy and momentum. The radical answer was: mass is stored energy. Here’s why.
🎼 Musical fact: Metastasis by Iannis Xenakis (1954) applies mathematical formulas and architectural structures to music—an art–science fusion.
These pillars lead to new expressions for momentum and energy.
Want to dive deeper into this fundamental invariant? See The spacetime invariant: from Minkowski to 𝐸² = 𝑝²𝑐² + 𝑚²𝑐⁴.
$$ \mathbf{p} = \gamma\,m\,\mathbf{v}, \quad \text{where} \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}. $$
$$ E^2 = p^2c^2 + m^2c^4. $$
If \(v = 0\) ⇒ \(p = 0\), then $$ E_0 = mc^2. $$ Voilà! The rest energy is bottled up in mass.
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.
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.
Electrons need a delicate balance between Coulomb attraction and zero-point kinetic energy. Tweaking the exponent in the Coulomb term destroys that balance.
