The Constants

What the fundamental constants encode — and what they look like when expressed in nature’s own units

The fundamental constants of physics ($c$, $\hbar$, $G$, $e$, $\alpha$) are usually treated as conversion factors: numbers you plug into a formula to get an answer in the right units. But each constant can be expressed as a product of Planck units and dimensionless ratios — quantities that keep the same value in any system of units and express physical properties of the systems being described. Written this way, the constant’s physical content becomes visible. What looked like an arbitrary numerical value reveals itself as a specific geometric relationship between natural scales.

Each profile in this collection takes one constant and unpacks it: its Planck-unit decomposition, how it enters physical formulas, what its dimensionless ratios tell us about the structure of the physical world, and where the same constant appears across different areas of physics. Every decomposition is numerically verified against standard values.

The mechanical constants

Three constants contain the mechanical Planck units in their dimensional structure. The measured values of $c$, $\hbar$, and $G$ are customarily used to calculate the values of the Planck length, mass, and time — the scale at which quantum mechanics, relativity, and gravity all become simultaneously significant.

Electromagnetic constants

Electromagnetism introduces a fourth Planck unit — the Planck charge $q_\mathrm{P}$ — and a single dimensionless number, the fine-structure constant $\alpha = (e/q_\mathrm{P})^2 \approx 1/137$. Every electromagnetic constant can be expressed as a Planck-unit combination times a power of $\alpha$.

α Fine-Structure Constant
$\alpha = (e/q_\mathrm{P})^2$ — a pure number with no dimensions. The master coupling of electromagnetism.
e Elementary Charge
$e = \sqrt{\alpha} \, q_\mathrm{P}$ — the Planck charge reduced by the electromagnetic coupling. Every power of $e$ contributes one factor of $\sqrt{\alpha}$.
qP Planck Charge
$q_\mathrm{P} = \sqrt{4\pi\varepsilon_0 \hbar c}$ — the natural unit of charge, customarily calculated from the same measured constants that yield the Planck length, mass, and time.
ε0 Electric Constant
$\varepsilon_0 = q_\mathrm{P}^2/(4\pi\hbar c)$ — the vacuum’s permittivity. Contains four Planck units. An alternate route to the Planck scale through the electromagnetic regime.
μ0 Magnetic Constant
$\mu_0 = 4\pi \, m_\mathrm{P} l_\mathrm{P} / q_\mathrm{P}^2$ — the vacuum’s permeability. Contains Planck units in its dimensions, with a geometric factor of $4\pi$.
Z0 Vacuum Impedance
$Z_0 = 4\pi \, m_\mathrm{P} l_\mathrm{P}^2 / (t_\mathrm{P} \, q_\mathrm{P}^2)$ — a Planck-scale impedance independent of $\alpha$. The electromagnetic field’s intrinsic resistance.

Derived constants

Some constants combine multiple Planck units and dimensionless ratios in ways that encode specific physical relationships — a spectroscopic scale, a field-equation coupling, or a composite of mechanical and electromagnetic structure.

Complete reference

The Planck units themselves — the natural scales that every constant contains.