Volume IA field guide to what does not change

The numbers the
universe agrees on.

From the oscillation of a cesium atom to the structure of the human psyche — a catalogue of the constants that hold, everywhere, for everyone.

9,192,631,770
Hz · the second
299,792,458
m/s · speed of light
1/137.036
fine-structure
≈ 150
Dunbar's number
00 Preface

Almost everything is a matter of perspective. A short list of things is not.

Hot and cold depend on what is touching your skin. Fast and slow depend on what you are standing on. Even the passage of time, for the observer in motion, stretches and contracts like fabric. The universe is, in this sense, a place of opinions.

But beneath the opinions there is bedrock. A short list of numbers and ratios that every observer — whether on Earth, on a planet orbiting Proxima Centauri, or falling toward a black hole — would write down the same way. These are the constants. They do not change with location. They do not drift with time. They are, in the most literal sense available to science, what is real.

This guide is broader than physics. The first chapters catalogue the constants of matter, energy, light, and gravity. The later chapters turn inward — toward the constants of life, of perception, of mind. Jung argued that the psyche, like the cosmos, has invariants of its own. Those are here too, treated honestly: not measured in joules, but mapped wherever humans have left a record.

Anchor I · The Measurement of Time

A single atom,
in perfect agreement with itself.

9,192,631,770
Hz · oscillations of cesium-133, by definition, equal to one second

For most of human history, a second was an inheritance — from the rotation of the Earth, the swing of a pendulum, the calibration of a chronometer at sea. These were good but mortal definitions. The Earth wobbles. Pendulums drift.

Today, the second has a sturdier home: a single isotope of cesium. The electron at the outermost shell of cesium-133 occupies one of two slightly different energy states. When it transitions between them, it emits a photon at a frequency so stable, and so identical across every cesium-133 atom that has ever existed, that we now define the second as exactly 9,192,631,770 oscillations of that radiation.

An atom in Tokyo, an atom on Mars, an atom in the cold dust of the Oort cloud — all will resonate at this same frequency. International atomic time, maintained as the weighted average of around four hundred cesium clocks, disagrees with itself by less than one second every three hundred million years.

I Branch

Time & Frequency

Time is the dimension along which causality flows. These constants anchor it.

ΔνCs
I · 01Exact
Cesium-133 Hyperfine TransitionDefines the SI second
9,192,631,770
Hz · cycles per second
By international decree since 1967: every oscillation of a cesium-133 atom between its two ground-state hyperfine levels is one nine-billion-one-hundred-ninety-two-million-six-hundred-thirty-one-thousand-seven-hundred-seventieth of a second. The most reliable measurement humanity has ever made.
tP
I · 02
Planck TimeThe atomic unit of time
5.391 247 × 10⁻⁴⁴
seconds
The shortest interval for which our equations remain coherent. Below this scale, the smooth fabric of spacetime is conjectured to break down into something quantum and foamy. Time itself has a floor.
τ
I · 03
Age of the UniverseTime since the Big Bang
13.787 ± 0.020
Gyr · 4.35 × 10¹⁷ seconds
The proper time elapsed since the cosmic microwave background was released, back-projected through the Friedmann equations. Not a constant in the philosophical sense — it grows by one second every second — but a parameter of every cosmological model.
Anchor II · The Cosmic Speed Limit

Nothing travels faster
than light in a vacuum.

299,792,458
m / s · the speed of light, exactly — the meter is defined from it

The number itself is arbitrary; the constraint is not. Light moves at the maximum speed at which any form of information or causality can propagate through the universe. Above that speed, there is no above.

Since 1983, the meter has been defined as the distance light travels in 1/299,792,458 of a second. The speed of light is therefore exact by construction. Every other length on Earth is now measured against the second — itself measured against a cesium atom — in a chain of definitions that begins, ultimately, with the universe and works its way back.

II Branch

Light & Electromagnetism

The maximum speed of causality, and the parameters governing every electric and magnetic phenomenon in the cosmos.

c
II · 01Exact
Speed of Light in Vacuum
299,792,458
m / s
Appears in E = mc², in relativistic time dilation, in the Maxwell equations. Since 1983, defined as exact — the meter is now derived from it. No structure in the universe moves faster.
ε₀
II · 02
Vacuum Permittivity
8.854 187 8128 × 10⁻¹²
F / m · farads per meter
The capacity of empty space to permit an electric field. Sets the strength of Coulomb's law and, with μ₀, determines the speed of light through the relation c² = 1/(ε₀μ₀).
μ₀
II · 03
Vacuum Permeability
1.256 637 062 × 10⁻⁶
N / A² · newtons per ampere²
The magnetic counterpart of permittivity — how readily empty space supports a magnetic field. Together with ε₀ defines the impedance of free space.
e
II · 04Exact
Elementary Charge
1.602 176 634 × 10⁻¹⁹
C · coulombs
The charge carried by a single proton, equal and opposite to that of an electron. Since 2019, defined as exact; the ampere is now derived from it. Every charged particle in the universe is an integer multiple.
α
II · 05
Fine-Structure Constant"One of the greatest damn mysteries in physics" — Feynman
7.297 352 5693 × 10⁻³
dimensionless · ≈ 1/137.036
A pure number — no units — that governs the strength of the electromagnetic interaction. Why is it 1/137? No theory yet derives it from first principles. Were it slightly different, atoms could not form.
III Branch

Gravity & Curvature

By far the weakest of the four fundamental forces — yet the architect of every galaxy, every star, every world.

G
III · 01
Newtonian Gravitational Constant
6.674 30 × 10⁻¹¹
m³ kg⁻¹ s⁻²
The strength of gravitational attraction between any two masses. Despite being the first constant of modern physics — Newton's Principia, 1687 — it remains the least precisely measured to this day, owing to gravity's weakness compared to electromagnetism (a factor of 10³⁶).
g
III · 02Exact
Standard GravityAt Earth's surface, by definition
9.806 65
m / s²
Treated as a defined value for engineering. Drop a coin and after one second it is travelling at 9.806 65 meters per second. Not universal — depends on Earth's specific mass and radius — but a fixed standard.
Λ
III · 03
Cosmological ConstantThe dark energy parameter
1.089 × 10⁻⁵²
m⁻²
Einstein added it to keep the universe static, then retracted it as his "greatest blunder." We have since reinstated it to account for the accelerating expansion of space itself. The smallest non-zero number in physics; one of its deepest mysteries.
IV Branch

The Quantum

The granular constants that determine the discrete, lumpy nature of matter and energy at the smallest scales.

h
IV · 01Exact
Planck ConstantThe fundamental quantum of action
6.626 070 15 × 10⁻³⁴
J · s
Energy comes in packets — quanta — of size hν for radiation of frequency ν. Since 2019, defined as exact; the kilogram is now derived from it. The constant that broke classical physics and inaugurated the quantum era.
IV · 02
Reduced Planck Constantℏ = h / 2π
1.054 571 817 × 10⁻³⁴
J · s
More natural in equations with angular frequency. Appears in Schrödinger's equation, the uncertainty principle (Δx · Δp ≥ ℏ/2), and the spin of every particle.
kB
IV · 03Exact
Boltzmann ConstantS = k log W
1.380 649 × 10⁻²³
J / K
The bridge between the microscopic and the thermodynamic — converts between temperature and the average kinetic energy of particles. Since 2019, defined as exact; the kelvin is now derived from it. Engraved on Boltzmann's tombstone.
NA
IV · 04Exact
Avogadro's Number
6.022 140 76 × 10²³
mol⁻¹
The number of particles in one mole — twelve grams of carbon-12. The conversion factor between the atomic and the human-sized. Since 2019, defined as exact.
me
IV · 05
Electron Rest Mass
9.109 383 7015 × 10⁻³¹
kg · 0.511 MeV/c²
Identical across every electron that has ever existed. Were it slightly larger, atoms would be smaller — and chemistry, perhaps, impossible.
mp
IV · 06
Proton Rest Mass
1.672 621 924 × 10⁻²⁷
kg · 938.272 MeV/c²
Roughly 1,836 times the mass of the electron. The ratio m_p / m_e appears to be a fundamental dimensionless parameter — one no theory yet derives.
mn
IV · 07
Neutron Rest Mass
1.674 927 498 × 10⁻²⁷
kg · 939.565 MeV/c²
Slightly heavier than the proton — by 1.293 MeV/c². If reversed, free protons would decay, and hydrogen could not exist.
V Branch

The Cosmos

Parameters describing the universe as a whole — its expansion, its temperature, its density.

H₀
V · 01Disputed
Hubble ConstantRate of cosmic expansion
67.4 — 73.3
km / s / Mpc
Measurements from the early-universe CMB and from local supernovae disagree by about 9% — the Hubble tension, the largest unresolved discrepancy in cosmology today.
TCMB
V · 02
CMB TemperatureThe afterglow of the Big Bang
2.725 48 ± 0.000 57
K · kelvin
The present-day temperature of the cosmic microwave background — the cooled radiation released 380,000 years after the Big Bang. The same in every direction we point our antennas.
ρc
V · 03
Critical DensityFor a flat universe
9.47 × 10⁻²⁷
kg / m³ · ~5.7 protons per m³
The density at which the universe is exactly flat. Measurements suggest the actual density is, remarkably, within 0.5% of this value.
Ω
V · 04
Density Parameter
1.000 ± 0.005
dimensionless
The ratio of the actual density to the critical density. Within measurement error: exactly one. The universe appears to be flat to a precision that demands explanation. The favoured candidate: inflation.
VI Branch

The Mathematical

Constants that arise not from measurement but from pure logic. Unitless. Eternal. The same in any universe that contains arithmetic.

π
VI · 01
PiCircumference / diameter
3.141 592 653 589 793 …
transcendental · irrational
Appears throughout physics wherever there is rotation, oscillation, or rotational symmetry — in the wave equation, in the period of a pendulum, in the normal distribution.
e
VI · 02
Euler's NumberBase of the natural logarithm
2.718 281 828 459 045 …
transcendental
The number whose derivative is itself. Underlies all growth, decay, and compounding processes. Joined with π and i in Euler's identity: e^iπ + 1 = 0.
φ
VI · 03
Golden Ratio(1 + √5) / 2
1.618 033 988 749 894 …
algebraic
The limiting ratio of consecutive Fibonacci numbers. Found in the spiral of sunflower seeds, in pinecones, in nautilus shells — wherever growth proceeds by adding to itself.
γ
VI · 04
Euler–Mascheroni Constant
0.577 215 664 901 532 …
unknown if rational
The limiting difference between the harmonic series and the natural logarithm. After centuries of effort, it is still unknown whether γ is rational. One of the great open questions of number theory.
δ
VI · 05
Feigenbaum ConstantThe rhythm of chaos
4.669 201 609 102 990 …
dimensionless
The ratio at which period-doubling bifurcations crowd together as a system approaches chaos. Universal — the same number governs the route to chaos in dripping faucets, populations, electric circuits, neurons.
G
VI · 06
Catalan's Constant
0.915 965 594 177 219 …
rationality unknown
Appears in combinatorics, hyperbolic geometry, and certain definite integrals. Like γ, no one knows if it is rational or irrational.
ζ(3)
VI · 07
Apéry's Constant
1.202 056 903 159 594 …
irrational
The sum of 1/n³ for all positive n. Proven irrational by Roger Apéry in 1978 — a result that stunned the mathematical world. Whether ζ(5), ζ(7), … are irrational remains open.
K
VI · 08
Khinchin's Constant
2.685 452 001 065 306 …
dimensionless
The geometric mean of the continued-fraction coefficients of almost every real number is the same number. Almost no number is exempt. Why this universal constant exists at all is, in a precise sense, miraculous.
VII Branch

The Planck Scale

The natural units of the universe — derived from G, c, and ℏ alone. The scale at which the laws of physics, as we know them, cease to apply.

P
VII · 01
Planck Length
1.616 255 × 10⁻³⁵
m
The smallest length with physical meaning in our current theories — 10²⁰ times smaller than a proton. To probe shorter would require energy that forms a black hole. The fabric of space may not be smooth below this scale.
tP
VII · 02
Planck Time
5.391 247 × 10⁻⁴⁴
s
The time light takes to cross one Planck length. The first moment after the Big Bang for which our equations even pretend to apply.
mP
VII · 03
Planck MassAbout a flea's egg
2.176 434 × 10⁻⁸
kg
The only Planck unit on a human scale. But concentrated into one Planck length, it would form a microscopic black hole. The threshold where gravity, quantum mechanics, and a unified theory collide.
TP
VII · 04
Planck Temperature
1.416 784 × 10³²
K
The highest temperature with physical meaning. The temperature of the universe, perhaps, in its first 10⁻⁴³ seconds.
"The most incomprehensible thing about the universe is that it is comprehensible."
— Albert Einstein, 1936
VIII Branch

Chemistry & Atoms

The constants that govern how matter combines, how electrons orbit, how heat radiates.

F
VIII · 01
Faraday ConstantCharge of one mole of electrons
96 485.332 12
C / mol
The bridge between electricity and chemistry. To deposit one mole of any singly-charged ion in electrolysis takes exactly this many coulombs. F = N_A × e.
R
VIII · 02
Molar Gas Constant
8.314 462 618
J / (mol · K)
PV = nRT. The constant of proportionality in the ideal gas law. Equal to N_A × k_B — the macroscopic face of Boltzmann's molecular constant.
R
VIII · 03
Rydberg ConstantFrequency scale of atomic spectra
1.097 373 156 8160 × 10⁷
m⁻¹
Determines the wavelengths of all light absorbed and emitted by hydrogen. The colors of distant stars, the bar codes of the elements — all derive from R∞.
σ
VIII · 04
Stefan–Boltzmann ConstantHow hot things glow
5.670 374 419 × 10⁻⁸
W / (m² · K⁴)
A black body's radiative power is σT⁴. Why a doubled temperature radiates sixteen times brighter. Why stars are so terrifying and so beautiful.
u
VIII · 05
Atomic Mass Unit1/12 of a carbon-12 atom
1.660 539 066 60 × 10⁻²⁷
kg · ≈ 931.494 MeV/c²
The natural mass scale for atoms and molecules. A proton weighs about 1.007 u; the entire periodic table is mapped onto this scale.
a0
VIII · 06
Bohr RadiusSize of a hydrogen atom
5.291 772 109 03 × 10⁻¹¹
m · 52.9 picometers
The most probable distance of the electron from the nucleus in a hydrogen atom in its ground state. The size of an atom is not arbitrary — it falls out of ℏ, m_e, e, and ε₀.
IX Branch

Information & Computation

The newest branch of physics: the laws and limits of information itself. There are constants here, too.

kBT ln 2
IX · 01
Landauer's LimitEnergy cost of erasing one bit
≈ 2.87 × 10⁻²¹
J · at room temperature
Erasing one bit of information requires at least this much heat dissipation. The thermodynamics of computation has a floor. Every laptop, every brain, every CPU on Earth obeys it.
SBH
IX · 02
Bekenstein BoundMaximum information per volume
S ≤ 2πkR·E / ℏc
bits in a sphere
There is a maximum amount of information that can be contained in a finite region of space with a finite amount of energy. Exceed it, and you have built a black hole. The universe limits its own memory.
C
IX · 03
Shannon Channel CapacityC = B log₂(1 + S/N)
log₂(1 + S/N)
bits / second / Hz
Not a single number but a universal law: the maximum rate at which information can be reliably transmitted over a noisy channel. Every modem, every wireless protocol, every fiber-optic cable is bounded by this.
Ω
IX · 04
Chaitin's ConstantThe halting probability
uncomputable
The probability that a randomly chosen program halts. A specific, well-defined real number between 0 and 1 — yet no algorithm can compute its digits beyond a finite few. The shadow of Gödel and Turing rendered as a number.
BB(n)
IX · 05
Busy Beaver FunctionMaximum output of an n-state machine
BB(5) = 47,176,870
BB(6) > 10↑↑15
A function that grows faster than any computable function. After only six states it dwarfs every number that has ever been written down. The mathematical embodiment of "the unknowable."
X Branch

Biology & Life

Every organism on Earth — every microbe, every redwood, every whale — uses the same code. These are its constants.

4
X · 01
DNA Nucleotide BasesA, T, G, C (and U in RNA)
4
letters in the genetic alphabet
Every gene on Earth, from the oldest bacterium to the most recent human child, is written in four letters. Three-letter words (codons) encode amino acids; 4³ = 64 codons map to 20 amino acids plus stop signals.
20
X · 02
Standard Amino AcidsBuilding blocks of proteins
20
+ 2 rare additions (selenocysteine, pyrrolysine)
Every protein in every cell in every organism is built from these twenty molecules. The chemistry of life is, in this sense, a very small alphabet.
3.4 nm
X · 03
DNA Helix PitchOne complete turn of the double helix
3.4 × 10⁻⁹
m · 10 base pairs per turn
The double helix turns once every 3.4 nanometers along its length. A human cell's DNA, unwound, would stretch about two meters. The total in your body: 30 billion kilometers — beyond Pluto.
≈ 50
X · 04
Hayflick LimitMaximum cell divisions
≈ 40 — 60
divisions before senescence
Most human somatic cells can divide only about fifty times before their telomeres erode and division halts. A biological constant of cellular mortality. Cancer cells, alone, learn to bypass it.
−70
X · 05
Neuronal Resting PotentialVoltage across the membrane
−70 ± 10
mV · millivolts
Every resting neuron in every brain on Earth holds its interior approximately 70 millivolts more negative than its exterior. The substrate of every thought you have ever had.
≈ 30
X · 06
ATP per GlucoseYield of cellular respiration
≈ 30 — 32
ATP molecules
A single molecule of glucose, fully oxidized, yields roughly thirty molecules of ATP — the universal energy currency of cells. Every step you take, every breath, is paid for in ATP.
XI Branch

Perception & Cognition

The constants of the human interface — the resolution, bandwidth, and bottlenecks of the only mind you can directly study.

ΔI / I
XI · 01
Weber FractionJust-noticeable difference
≈ 0.02 — 0.10
fractional · sense-dependent
The smallest change in stimulus you can detect is a fixed fraction of the original. Double the weight in your hand, and you must double the difference to feel it. Perception is logarithmic.
7 ± 2
XI · 02
Miller's Magical NumberShort-term memory capacity
7 ± 2
discrete items
Roughly seven discrete chunks of information can be held in working memory at one time. Phone numbers, license plates, lists — all bounded by this. Modern research suggests the number may be closer to four; the principle survives.
≈ 150
XI · 03
Dunbar's NumberStable social relationships
100 — 230
individuals · ≈ 150 median
The maximum number of stable interpersonal relationships a human can maintain — derived from neocortex size across primates. Hunter-gatherer bands, military companies, professional networks, and Christmas-card lists all cluster around this number.
XI · 04
Foveal VisionHigh-acuity visual field
≈ 2
degrees of arc
Only the central two degrees of your visual field are in sharp focus — about the width of your thumbnail at arm's length. The rich, detailed world you seem to see is reconstructed by your brain, three saccades per second.
≈ 50
XI · 05
Flicker Fusion ThresholdWhere motion becomes continuous
16 — 60
Hz · context-dependent
Above roughly 50 frames per second, the human visual system stops seeing individual frames and begins to see motion. The basis of cinema, animation, and every screen you have ever looked at.
≈ 100 ms
XI · 06
Reaction Time FloorThe shortest deliberate response
100 — 250
ms · milliseconds
The minimum time required for a sensory signal to reach the brain, be classified, and trigger a motor response. Below 100 ms, the response is reflex — and disqualifies the runner from the starting blocks.
Anchor III · The Inner Cosmos

Patterns that recur
across every culture.

Carl Jung's claim was that the psyche, like the cosmos, has invariants. He called them archetypes — innate organizing structures of the unconscious that surface, again and again, in the myths, dreams, religions, and art of every people who have ever lived. The Hero. The Shadow. The Wise Old Man. The Mother. The Trickster. They appear in cultures that have never spoken to each other. They appear in the dreams of patients who have never read about them.

These are not constants in the sense of joules or kelvins. They are not measured in instruments. But they recur with a regularity that demands a name. Jung's hypothesis — that the human psyche shares deep structure across the species — remains contested in detail, and increasingly supported in outline by cognitive science and comparative mythology. What follows is a brief catalogue of the most stable of these patterns.

XII Branch

The Psyche

Universal patterns of the inner cosmos — Jung's catalogue of what the mind keeps producing, no matter where in the world it grows up.

Selbst
XII · 01
The SelfTotality of conscious and unconscious
∮ ψ
archetype of wholeness
Not the ego, which is merely the conscious part. The Self is the entire psychic system: known and unknown, light and shadow, integrated. Often imaged as a mandala, a circle squared, a cosmic egg. The goal of individuation is its approximation.
Schatten
XII · 02
The ShadowWhat you refuse to recognise
— ψ
repressed material
The qualities the conscious mind disowns and projects onto others. Everyone has one. The Shadow appears in dreams as a same-sex stranger, an enemy, a pursuer. Confronting it is the first task of individuation — and the hardest.
Anima
Animus
XII · 03
Anima & AnimusContrasexual archetypes
♀ ↔ ♂
inner counterpart
The feminine principle in the masculine psyche (anima); the masculine in the feminine (animus). The figure who appears in dreams as an idealised lover, a mysterious guide, an inner voice. The bridge to the unconscious.
Persona
XII · 04
The PersonaThe social mask
∂ ψ / ∂ society
adaptive façade
From the Latin for actor's mask. The face one presents to the world — necessary, functional, dangerous when confused with the Self. A doctor's white coat, a parent's calm voice, a politician's smile. Useful tools, terrible owners.
∪ ψ
XII · 05
Collective UnconsciousThe shared substrate
trans-personal
structural · pre-individual
Jung's most controversial proposal: a layer of the unconscious common to all humans, inherited rather than learned. Source of the archetypes. The reason a child in Ulaanbaatar and a child in Lima dream of dragons.
→ Selbst
XII · 06
IndividuationThe lifelong work
∫ life
developmental telos
The process by which a person becomes a psychological individual — distinct, integrated, whole. Encounters with the Shadow, the Anima/Animus, and finally the Self. Rarely completed; valuable in any case it is attempted.
XII · 07
The MandalaUniversal symbol of wholeness
○ ⊕ □
geometric archetype
A circle, often containing a square or fourfold division. Appears spontaneously in patients' drawings; in Tibetan thangka; in Christian rose windows; in alchemical diagrams; in Navajo sand paintings. The visual signature of the Self.
XII · 08
SynchronicityAcausal connecting principle
meaning ⊥ cause
phenomenological
Jung's term for meaningful coincidence — events linked not by causality but by significance. Disputed as a metaphysical claim, undeniable as an experience. Whether real or projected, it is a constant of how the mind reads the world.
Coda

Why this is reliable.

Consider what an astonishing thing it is that these numbers exist. We have no obvious right to a universe whose behavior can be summarised in a few dozen parameters. We have no guarantee that the speed of light here is the speed of light in a galaxy a billion light-years away. And yet every check we have ever performed has come back negative. The constants are constant. Not approximately. Not on average. Constant.

Distant light. When we observe a quasar twelve billion light-years away, the photons reaching our telescopes were emitted twelve billion years ago. Their spectral lines fall exactly where laboratory measurements predict, to within parts per million. Atoms in the early universe obeyed the same fine-structure constant they obey today.

The cosmic microwave background. The relic radiation from 380,000 years after the Big Bang has a temperature of 2.725 kelvin in every direction. For that radiation to be so uniform, the physical laws governing matter and light must have been identical in regions of the early universe causally disconnected from each other.

Atomic clocks. If the fundamental constants were drifting, modern frequency standards would have noticed. They have not. The drift, if any, is bounded at less than one part in 10¹⁷ per year.

The constants of mind are softer evidence, by necessity. We cannot put the Shadow under a spectrometer. But here too there is a remarkable invariance: the same patterns appear in cultures that have never spoken, in dreams of patients who have never read the theory, in myths recorded a continent and a millennium apart. Whatever they ultimately are, they are there.

Almost everything in your life is in flux. Relationships, weather, opinions, prices, governments, stars. But somewhere a cesium atom is ringing at 9,192,631,770 cycles a second, and it will keep doing so for as long as there are cesium atoms. Somewhere a child is dreaming of a hero, a shadow, a wise stranger — the same dream a child dreamt in Sumer four thousand years ago. These are the only things you can be sure of. They are also, perhaps, the most beautiful things we have found.