# Chapter ᚁ(B): The actual world *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T3, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Its glossary cards](ch03-cards.md) · [Its references](ch03-refs.md) · [Previous chapter](ch02.md) · [Next chapter](ch04.md) *A footnote such as [r-3.3] is reference entry 3.3 and links to it. Section 3 of 28; chapters are named by their Ogham letter, never by number.* ## The ground beneath us ᚁ – [Beith](ch03-cards.md#g-og-beith): associated with the birch tree, Section 3 ## Chapter ᚁ(B): The actual world > “It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.” > — Richard Feynman, on the fine-structure constant · *QED: The Strange Theory of Light and Matter*, 1985 [r-3.13](ch03-refs.md#r-3.13) Formula: (([Λ](ch03-cards.md#g-Lambda)·[G](ch03-cards.md#g-G)·[ℏ](ch03-cards.md#g-hbar))/([c](ch03-cards.md#g-c)³))·([Ω_Λ](ch03-cards.md#g-OmegaL) + [Ω_m](ch03-cards.md#g-Omegam) + [Ω_r](ch03-cards.md#g-Omegar) + [Ω_k](ch03-cards.md#g-Omegak)) {brace: gravity} · (([k_B](ch03-cards.md#g-kB)·[T_CMB](ch03-cards.md#g-Tcmb))/([m_e](ch03-cards.md#g-me)·[c](ch03-cards.md#g-c)²))·(([b](ch03-cards.md#g-wien)·[x_W](ch03-cards.md#g-wienx)·[k_B](ch03-cards.md#g-kB))/([h](ch03-cards.md#g-h)·[c](ch03-cards.md#g-c))) {brace: heat} · ([N_A](ch03-cards.md#g-NA)·[1 mol](ch03-cards.md#g-mol))·(([R](ch03-cards.md#g-gasR))/([N_A](ch03-cards.md#g-NA)·[k_B](ch03-cards.md#g-kB)))·(([F](ch03-cards.md#g-faraday))/([N_A](ch03-cards.md#g-NA)·[e](ch03-cards.md#g-echarge))) {brace: amount} · (([σ](ch03-cards.md#g-sigma)·[ℏ](ch03-cards.md#g-hbar)³·[c](ch03-cards.md#g-c)²)/([k_B](ch03-cards.md#g-kB)⁴)) {brace: light} · ((([e](ch03-cards.md#g-echarge)²)/(4π·[ε₀](ch03-cards.md#g-eps0)·[ℏ](ch03-cards.md#g-hbar)·[c](ch03-cards.md#g-c))))/([α](ch03-cards.md#g-alpha))·([k_e](ch03-cards.md#g-coulombk)·4π·[ε₀](ch03-cards.md#g-eps0))·((2·[h](ch03-cards.md#g-h)·[R_∞](ch03-cards.md#g-rydberg))/([α](ch03-cards.md#g-alpha)²·[m_e](ch03-cards.md#g-me)·[c](ch03-cards.md#g-c))) {brace: electricity} · ([μ₀](ch03-cards.md#g-mu0)·[ε₀](ch03-cards.md#g-eps0)·[c](ch03-cards.md#g-c)²)·(([e](ch03-cards.md#g-echarge)²·[R_K](ch03-cards.md#g-RK))/([h](ch03-cards.md#g-h)))·(([Z₀](ch03-cards.md#g-Z0))/([μ₀](ch03-cards.md#g-mu0)·[c](ch03-cards.md#g-c)))·([Φ₀](ch03-cards.md#g-fluxq)·[K_J](ch03-cards.md#g-josephson)) {brace: magnetism} · ([m_p/m_e](ch03-cards.md#g-mpme)·[α_s](ch03-cards.md#g-alphas)·[sin²θ_W](ch03-cards.md#g-weakmix)·[a_e](ch03-cards.md#g-ae)·[α_G](ch03-cards.md#g-alphaG)) {brace: particle physics} {brace: [physical constants](ch03.md#ch-T3-physical-constants)} + [ds²_light](ch03-cards.md#g-dslight) + [∇·𝐁](ch03-cards.md#g-divB) + [dQ/dt](ch03-cards.md#g-noether) + [(E − mc²)](ch03-cards.md#g-emc2) + [∇_μG^μν](ch03-cards.md#g-bianchi) + [∂_μj^μ](ch03-cards.md#g-continuity) {brace: [physical laws](ch03.md#ch-T3-physical-laws)} #### Type of number Proven transcendental · #### Physical Physics: fixed by definition · Physics: measured · Physics: derived · Key concepts: Physical constants and physical laws. It includes symbols you may not have seen before, such as: [ℏ](ch03-cards.md#g-hbar), [α](ch03-cards.md#g-alpha), and [Λ](ch03-cards.md#g-Lambda). Units cancel into pure numbers and six laws add exact zeros, yet this is the one factor in the equation that stays nonzero, so the actual world can count for something. Welcome to the actual world! Our first steps land us firmly in the physical world. “How far is that?” “How big is it?” To my mind, questions like these are the soil from which numbers grew. This chapter has the one factor in the equation that is *not* zero. I could have arranged for it to vanish too, but I wanted the actual world to be allowed to *“count for something”*. The difficulty here is not the math; it is that the physical world brings its units with it. A mass arrives in kilograms, a temperature in kelvins, a speed in meters per second. Our formula needs a number that will mean the same thing whichever units we choose. So the first task is to make comparisons: mass with mass, energy with energy, until the units cancel and only a ratio remains. Physicists call such a quantity [dimensionless](ch03-cards.md#g-dimensionless). It still tells us something about the world; it simply no longer depends on the ruler. The chapter holds two kinds of thing: constants and laws. A constant gives a quantity its value. A law tells us how quantities belong together. I wanted both in the picture: the numbers we find in nature and the relationships we have learned to recognize. Field notes ### [physical constants](ch03.md#ch-T3-physical-constants) {#ch-T3-physical-constants} Seven small braces, each a [dimensionless](ch03-cards.md#g-dimensionless) combination. A confession up front: several impressive-looking brackets are elaborate ways of writing 1, kept because the relationships are worth seeing. The fractions let the constants stand together and show us that they already know each other. The colors tell how we know each value. Blue marks constants whose numerical values define our SI units; orange marks measurements; green marks quantities derived from other constants. On [20 May 2019](ch03-cards.md#g-si2019), the SI fixed the numerical values of [h](ch03-cards.md#g-h), [e](ch03-cards.md#g-echarge), [k_B](ch03-cards.md#g-kB), and [N_A](ch03-cards.md#g-NA), using them to define the kilogram, ampere, kelvin, and mole; the [speed of light](ch03-cards.md#g-c), [c](ch03-cards.md#g-c), had already taken that role for the meter in 1983. [r-3.1](ch03-refs.md#r-3.1) Nothing in nature changed that morning. What changed was where we put the ruler. #### The displayed product There is no denominator: the seven braces multiply. gravity [Λ](ch03-cards.md#g-Lambda)·[G](ch03-cards.md#g-G)·[ℏ](ch03-cards.md#g-hbar) / [c](ch03-cards.md#g-c)³ ≈ 3 × 10⁻¹²²; a comparison on a very strange scale, and the doorway to the [cosmological constant problem](ch03-cards.md#g-ccproblem). The [cosmological constant](ch03-cards.md#g-Lambda) is associated with the energy of empty space. Combined with Newton’s [G](ch03-cards.md#g-G), the [reduced Planck constant](ch03-cards.md#g-hbar) ([Planck’s constant](ch03-cards.md#g-h) divided by 2π), and the [speed of light](ch03-cards.md#g-c), its units cancel, and what is left is the tiny 3 × 10⁻¹²² above. A naive estimate using [the Planck scale](ch03-cards.md#g-plancklen) suggests something of order 1 instead, a miss of about 120 orders of magnitude; even after we have learned to compare the quantities, we can be left wondering why the comparison comes out as it does. [r-3.2](ch03-refs.md#r-3.2) [Ω_Λ](ch03-cards.md#g-OmegaL) + [Ω_m](ch03-cards.md#g-Omegam) + [Ω_r](ch03-cards.md#g-Omegar) + [Ω_k](ch03-cards.md#g-Omegak) = 1; the universe’s budget, and the first hint of the confession. [Dark energy](ch03-cards.md#g-OmegaL) (about 0.685), [matter](ch03-cards.md#g-Omegam) (about 0.315), [radiation](ch03-cards.md#g-Omegar) (about 9 × 10⁻⁵), and [curvature](ch03-cards.md#g-Omegak) (fixed to zero in [ΛCDM](ch03-cards.md#g-lcdm), the standard model of a flat, expanding universe) add to 1 when they are taken consistently from that one model. Rounded, the first two already make 1; radiation is too small to show at that rounding. [r-3.2](ch03-refs.md#r-3.2) heat [k_B](ch03-cards.md#g-kB)·[T_CMB](ch03-cards.md#g-Tcmb) / ([m_e](ch03-cards.md#g-me)·[c](ch03-cards.md#g-c)²) ≈ 4.6 × 10⁻¹⁰; an unlikely pairing: the oldest light we can observe and one of matter’s smallest familiar inhabitants. The cosmic microwave background, the light left from the early universe, has cooled to about 2.7255 kelvins. Multiply by [Boltzmann’s constant](ch03-cards.md#g-kB) and we have an energy scale; divide by an electron’s [rest energy](ch03-cards.md#g-emc2), 511 keV, and the units disappear. [r-3.3](ch03-refs.md#r-3.3) [b](ch03-cards.md#g-wien)·[x_W](ch03-cards.md#g-wienx)·[k_B](ch03-cards.md#g-kB) / ([h](ch03-cards.md#g-h)·[c](ch03-cards.md#g-c)) = 1; Wien’s law of the brightest glow, hiding a transcendental root. Multiply a hot body’s temperature by the wavelength at which it glows brightest and we always get the same number: [Wien’s constant](ch03-cards.md#g-wien). Planck’s law makes it exactly hc/([x_W](ch03-cards.md#g-wienx)·k_B), where x_W ≈ 4.965 is the nonzero root of x = 5(1 − [e](ch04-cards.md#g-e)^−x), a number proven transcendental, and the reason this chapter’s type of number reads “Proven transcendental.” Multiply it back and the bracket is 1. For the 2.7255-kelvin background above, the brightest glow falls near a millimeter, which is why the oldest light is microwave. [r-3.3](ch03-refs.md#r-3.3) amount [N_A](ch03-cards.md#g-NA)·[1 mol](ch03-cards.md#g-mol) = 6.02214076 × 10²³; a count of particles, and the largest thing in this factor. [Avogadro’s constant](ch03-cards.md#g-NA) is exactly 6.02214076 × 10²³ per mole; [one mole](ch03-cards.md#g-mol) is present only to cancel the “per mole,” leaving a pure count. [R](ch03-cards.md#g-gasR) / ([N_A](ch03-cards.md#g-NA)·[k_B](ch03-cards.md#g-kB)), [F](ch03-cards.md#g-faraday) / ([N_A](ch03-cards.md#g-NA)·[e](ch03-cards.md#g-echarge)) = 1, 1; the gas constant and the Faraday constant, connecting that count to gas and electricity. Since 2019 [the gas constant](ch03-cards.md#g-gasR) is exactly N_A·k_B and [the Faraday constant](ch03-cards.md#g-faraday), the charge of [one mole](ch03-cards.md#g-mol) of electrons, is exactly N_A·[e](ch03-cards.md#g-echarge). [r-3.1](ch03-refs.md#r-3.1) light [σ](ch03-cards.md#g-sigma)·[ℏ](ch03-cards.md#g-hbar)³·[c](ch03-cards.md#g-c)² / [k_B](ch03-cards.md#g-kB)⁴ = π²/60; the energy a hot body radiates, with [π](ch04-cards.md#g-pi) hiding inside. Planck’s law gives the [Stefan–Boltzmann constant](ch03-cards.md#g-sigma) as σ = π²k_B⁴/(60ℏ³c²), so this bracket is secretly π²/60, about 0.1645. [r-3.3](ch03-refs.md#r-3.3) electricity (e² / (4π·[ε₀](ch03-cards.md#g-eps0)·[ℏ](ch03-cards.md#g-hbar)·[c](ch03-cards.md#g-c))) / [α](ch03-cards.md#g-alpha) = 1; the fine-structure constant written out, then divided by itself. The combination brings together [electric charge](ch03-cards.md#g-echarge), [the permittivity of empty space](ch03-cards.md#g-eps0), the quantum scale, and light. Its units cancel to leave α, the [dimensionless](ch03-cards.md#g-dimensionless) strength of electromagnetism, about 1/137, measured and still unexplained. Dividing by α gives 1, because it was α all along. [r-3.4](ch03-refs.md#r-3.4) [k_e](ch03-cards.md#g-coulombk)·4π·ε₀, 2·[h](ch03-cards.md#g-h)·[R_∞](ch03-cards.md#g-rydberg) / (α²·m_e·[c](ch03-cards.md#g-c)) = 1, 1; Coulomb’s 1785 law, and the constant of atomic spectra. [r-3.19](ch03-refs.md#r-3.19) [Coulomb’s constant](ch03-cards.md#g-coulombk) is 1/(4πε₀), so the first is 1 exactly. [The Rydberg constant](ch03-cards.md#g-rydberg), among the most precisely measured constants in physics, equals α²m_e[c](ch03-cards.md#g-c)/(2h), so the second is 1 too. [r-3.4](ch03-refs.md#r-3.4) magnetism [μ₀](ch03-cards.md#g-mu0)·ε₀·c², e²·[R_K](ch03-cards.md#g-RK) / [h](ch03-cards.md#g-h), [Z₀](ch03-cards.md#g-Z0) / (μ₀·[c](ch03-cards.md#g-c)), [Φ₀](ch03-cards.md#g-fluxq)·[K_J](ch03-cards.md#g-josephson) = 1, 1, 1, 1; identities linking empty space, light, resistance, and magnetic flux. [Vacuum permeability](ch03-cards.md#g-mu0) is the magnetic twin of [ε₀](ch03-cards.md#g-eps0), with μ₀ε₀c² = 1. The [von Klitzing constant](ch03-cards.md#g-RK), the quantum of resistance, is [h](ch03-cards.md#g-h)/e² (another famous R, and the collision with ᚏ is deliberate). The [impedance of free space](ch03-cards.md#g-Z0), about 376.73 ohms, is μ₀c. [The flux quantum](ch03-cards.md#g-fluxq) [h](ch03-cards.md#g-h)/(2e) and [the Josephson constant](ch03-cards.md#g-josephson) 2e/[h](ch03-cards.md#g-h) are reciprocals. [r-3.1](ch03-refs.md#r-3.1) [r-3.4](ch03-refs.md#r-3.4) particle physics [m_p/m_e](ch03-cards.md#g-mpme) ≈ 1836.15; a comparison we can picture: a [proton](ch28-cards.md#g-mp) against an electron. We could weigh them in kilograms or any other unit of mass; their ratio would be the same. No theory predicts it; like α, it has to be measured. [r-3.7](ch03-refs.md#r-3.7) [α_G](ch03-cards.md#g-alphaG) ≈ 1.75 × 10⁻⁴⁵; gravity’s answer to α, and one of the two reasons this factor is so small. It is Gm_e²/(ℏc), the strength of gravity between two electrons, 10⁴² [times weaker than electromagnetism](ch03-cards.md#g-hierarchy). Gravity between such small things is extraordinarily weak. [r-3.7](ch03-refs.md#r-3.7) [r-3.12](ch03-refs.md#r-3.12) [α_s](ch03-cards.md#g-alphas)·[sin²θ_W](ch03-cards.md#g-weakmix)·[a_e](ch03-cards.md#g-ae) ≈ 0.1180 · 0.22305 · 0.00116; one measured number from each of the three forces the [Standard Model](ch03-cards.md#g-smodel) describes, with gravity’s answer already above. One number for each force: for the strong force, [the strong coupling](ch03-cards.md#g-alphas), the coupling of the [Yang–Mills theory](ch28-cards.md#g-ymgap) we meet in the epilogue once quarks are added; for the weak force, [the weak mixing angle](ch03-cards.md#g-weakmix), a number [the Standard Model](ch03-cards.md#g-smodel) does not predict; and for electromagnetism, [the electron’s anomalous magnetic moment](ch03-cards.md#g-ae), a small departure from the simplest relativistic prediction that [quantum electrodynamics](ch03-cards.md#g-qed) explains with remarkable precision. All three are positive. [r-3.5](ch03-refs.md#r-3.5) Multiply the seven braces together (3 × 10⁻¹²², 4.6 × 10⁻¹⁰, 6.02 × 10²³ and π²/60, eleven brackets equal to 1, and the particle ratios) and the first term comes to about 1.3 × 10⁻¹⁵⁴: tiny, finite (compared to the Spiritual world), and, as the optimist in me humbly chose to arrange it, positive. [r-3.8](ch03-refs.md#r-3.8) **Fine print.** Measured values carry uncertainties, and the product uses the values cited on each card. The density parameters sum to 1 only when taken from one consistent cosmological model. The [elementary charge](ch03-cards.md#g-echarge) e here is not [Euler’s e](ch04-cards.md#g-e), which lives in Chapter ᚂ(L). The factor’s particular size comes from my choice of arrangement; the relationships inside it belong to physics. ### [physical laws](ch03.md#ch-T3-physical-laws) {#ch-T3-physical-laws} How do you put a law into the formula? Every law I chose says that some quantity is exactly zero: [the interval along a light ray](ch03-cards.md#g-dslight), [the divergence of a magnetic field](ch03-cards.md#g-divB), the gap between a resting body’s energy and [mc²](ch03-cards.md#g-emc2). So I write down the quantity the law says must vanish, and let it contribute that zero. [r-3.6](ch03-refs.md#r-3.6) [r-3.11](ch03-refs.md#r-3.11) The zeros speak about different things, an energy here and a spacetime interval there, and physics would never add an energy to an interval. But zero is the one value that reads the same on every ruler: zero joules is zero electronvolts, and zero meters is zero miles. So each law comes home as the plain number 0, and the six are added to the constants rather than multiplied, which leaves the actual world still counting for something. [r-10.19](ch10-refs.md#r-10.19) #### The displayed sum There is no denominator: six zeros are added. [ds²_light](ch03-cards.md#g-dslight) = 0; light’s path, the most basic zero in relativity. [The spacetime interval between two events on a light ray](ch03-cards.md#g-dslight) is exactly 0, however far the ray travels in space and time. [∇·𝐁](ch03-cards.md#g-divB) = 0; Maxwell’s magnetic equation. In classical electromagnetism, with [no magnetic monopoles](ch03-cards.md#g-divB), the divergence of the magnetic field is 0 everywhere: its field lines never begin or end, so nowhere is there a source or a sink of magnetism. [dQ/dt](ch03-cards.md#g-noether) = 0; Noether’s theorem, the link between symmetry and conservation. Under its assumptions, every continuous symmetry gives a quantity whose rate of change is 0. Physics that works the same today as tomorrow conserves energy; physics that works the same here as over there conserves momentum; the phase symmetry of electromagnetism conserves charge. Q stands for whichever of these we pick. [(E − mc²)](ch03-cards.md#g-emc2) = 0; Einstein’s most famous equation, written as a zero. A body at rest has energy E = mc², so [E − mc²](ch03-cards.md#g-emc2) = 0, and that zero is what the term contributes. [∇_μG^μν](ch03-cards.md#g-bianchi) = 0; the contracted Bianchi identity, the bookkeeping of general relativity. [The Einstein tensor](ch03-cards.md#g-efe), the side of Einstein’s field equations that measures how spacetime curves, has zero divergence whatever the shape of spacetime. Because the equations tie that curvature to matter and energy, energy and momentum must be conserved at every point: geometry leaves them no choice. [∂_μj^μ](ch03-cards.md#g-continuity) = 0; the continuity equation, conservation of charge. [The electric current](ch03-cards.md#g-continuity), counted over space and time together, has zero divergence: charge that leaves a region must flow out through its edge, so charge is never created or lost. The second and final term is 0: six laws, each exactly 0, added together. They leave the factor’s size alone while giving it much of its meaning. **Fine print.** Each law is read in the setting where it applies, and [E − mc²](ch03-cards.md#g-emc2) = 0 is applied to a body at rest. Each zero is added only as the plain number it hands home; the book does not claim that an energy and a spacetime interval can be added as physical quantities. What this chapter adds to the formula: ≈ 1.3 × 10⁻¹⁵⁴ + 0 ≈ 1.3 × 10⁻¹⁵⁴, the one factor that is not zero The constants supply the quantities; the laws show the relationships that make a physical world intelligible. The units have canceled, but the world remains. Our brief stroll has brought us to our garden. We will start with a few varieties you easily recognize, then look among them for some that are less ordinary. There is more growing in our backyard than you might expect. Watch and read - [Fine Structure Constant - Sixty Symbols](https://www.youtube.com/watch?v=WlSLIW0gZtk) Laurence Eaves on why 1/137 is the number physicists worry about, and why the electricity bracket in this chapter is a confession. - [How We’re Redefining the kg](https://www.youtube.com/watch?v=Oo0jm1PPRuo) Veritasium at NIST on the 2019 redefinition, which is the reason some constants in this chapter are exact by decree and others are measured. - [Physical constant](https://en.wikipedia.org/wiki/Physical_constant) Wikipedia: the list, with CODATA values - [2019 revision of the SI](https://en.wikipedia.org/wiki/2019_revision_of_the_SI) Wikipedia: which constants became exact and why - [Lambda-CDM model](https://en.wikipedia.org/wiki/Lambda-CDM_model) Wikipedia: the Planck 2018 numbers behind the cosmology bracket