Disclaimer: I’m just a curious mind, not a physicist or a mathematician. Please tear this apart and call out every mistake. This essay grew out of a long back and forth between me and Gemini, and got too interesting not to share.
To construct a civilization's mathematical and measurement systems from scratch is an opportunity to prioritize pure elegance. By eliminating historical accidents, we can forge a framework where the universe's fundamental symmetries dictate the structure. The design principles are strict: render everything dimensionless where possible, obliterate unitful constants by setting them to 1, treat the natural geometry of rotation as a first-class citizen, and prioritize divisibility in everyday arithmetic. The result is a system that separates human convenience from physical reality, yet maintains an exact, unblemished mapping between the two.
The Unification of Physics: C = 1
Modern measurement fractures the universe into disparate concepts: meters, seconds, kilograms, and kelvins. This idealized civilization recognizes these are mere shadows of the same underlying phenomena. By defining four fundamental constants as exactly 1, the system achieves total mathematical harmony:
- c = 1 (the speed of light)
- ℏ = 1 (the reduced Planck constant)
- k_B = 1 (the Boltzmann constant)
- G = 1 (Newton's gravitational constant) for cosmological frameworks
This singular choice cascades into profound simplifications. Because c = 1, time and length share identical units; one unit of time is simply the duration light requires to traverse one unit of length. Because ℏ = 1, energy, mass, and frequency are directly comparable. The equation E = ℏω collapses effortlessly into E = ω. Because k_B = 1, temperature is revealed as pure energy, collapsing E = k_B T into E = T.
The Mechanics of Setting a Constant to 1
To understand how we can simply "set a constant to 1", we have to realize that a number in physics is never just a fundamental property of the universe. It is a relationship between a physical reality and the specific tool a human built to measure it.
The Analogy of the Racetrack
Imagine a race car driving around a track at exactly 120 miles per hour. The number 120 is not a fundamental property of the car's engine. It is an artifact of the fact that we chose to measure the track in "miles" and the stopwatch in "hours".
If we want to get rid of the cumbersome number 120 in our calculations, we do not need to physically slow down the car. We just change the ruler. We can invent a new unit of distance called the "Lap". If the track is exactly 120 miles long, then 1 Lap equals exactly 120 miles. Suddenly, the car's speed is exactly 1 Lap per hour.
We did not change reality. We just mathematically resized our unit of measurement so that the specific physical phenomenon (the speed of the car) equals exactly 1.
Applying This to the Universe
This is exactly what physicists do with fundamental constants, and it is a very real, doable practice known as "natural units" or "Planck units".
The speed of light (c) is 299,792,458 meters per second. That massive, ugly number only exists because humans arbitrarily decided how long a "meter" is and how long a "second" is before we ever knew light had a speed limit.
To set c = 1, we throw away the meter. Instead, we define our brand new unit of length to be the exact distance light travels in one unit of time. If light travels exactly 1 length unit per 1 time unit, its speed is mathematically 1.
We can do this for any constant. The reduced Planck constant (ℏ) is a messy number because we measure energy in human-scaled Joules. If we invent a new unit of energy that is perfectly sized to match the baseline frequency of a quantum wave, the conversion factor disappears. It becomes 1.
The Mathematical Derivation
We are not assuming these physical constants magically transform. We are treating this as an algebraic puzzle. We need to find the exact, physical size our new rulers, scales, and clocks must be to make the messy numbers disappear.
The speed of light c has dimensions of length (L) over time (T). To force c to equal 1, we write the equation:
1 = L/T
This simply means our new unit of Length must always equal our new unit of Time multiplied by the actual speed of light.
The reduced Planck constant ℏ connects energy to frequency. It has dimensions of mass (M) times length squared (L²) over time (T). To force it to equal 1, we write:
1 = ML²/T
The gravitational constant G relates mass to the geometry of space. It has dimensions of length cubed (L³) over mass (M) times time squared (T²). To force it to equal 1, we write:
1 = L³/(MT²)
The Boltzmann constant k_B connects kinetic energy to temperature (Θ). It has dimensions of mass (M) times length squared (L²) over time squared (T²) times temperature (Θ). To force it to equal 1:
1 = ML²/(T²Θ)
Because we have four equations and four unknown variables (L, T, M, Θ), we can solve for them using basic algebra. We are essentially asking the math: "Exactly how heavy does our base unit of mass need to be so that both G and ℏ equal exactly 1?"
The math provides a literal, physical answer.
Solving for Mass (M): M = √(ℏc/G)
Solving for Time (T): T = √(ℏG/c⁵)
Solving for Length (L): L = √(ℏG/c³)
Solving for Temperature (Θ): Θ = √(ℏc⁵/(Gk_B²))
When a civilization throws away the meter, the kilogram, and the second, and instead physically builds its rulers and scales to exactly match the outputs of these equations, all the chaotic numbers in physics vanish. The constants do not become 1 through magic. They become 1 because the civilization custom-built the baseline scale of its reality to perfectly swallow them.
The Illusion of Human Units
To understand how distinct concepts collapse into one, you must first recognize that everyday units (meters, seconds, degrees) are historical accidents. A meter was originally defined by a fraction of the Earth's circumference. A degree is tied to the freezing point of water on a specific planet. Because humans invented these measurements in isolation from one another, we were forced to invent constants to make them interact cleanly in equations.
Fundamental constants like c, ℏ, and k_B are not profound physical objects. They are merely translation factors between arbitrary human scales. Setting them to 1 does not change the universe; it strips away the human translation layer, revealing the native geometry of reality.
Spacetime: Why c=1 Unifies Time and Length
Speed is a ratio of distance to time. The speed of light, c, is the fundamental causal speed limit of the universe. We only think of c as a massive, clumsy number (nearly 300,000,000 meters per second) because we defined our length and time scales long before we understood light.
If you set the universe's ultimate speed limit to exactly 1, the equation for the speed of light becomes:
1 = d/t
Multiply both sides by time, and the result is literal geometric truth:
t = d
Time is distance. We already rely on this intuition when we measure vast cosmological distances in light-years. A light-year uses time to measure space. By making c=1 the mathematical default, the universe is no longer split into space and time. It is a single, unified grid of spacetime where one unit of time is simply the dimension extending exactly as far as light travels in that same interval.
On the road this beauty appears instantly. Instead of "40 mph," your dashboard simply reads **60 nc** (60 nano-c). Highway cruising feels natural at 100 nc (about 67 mph in old units). Walking pace is a gentle 5 nc. Because distance and time share the same unit (the ell), calculating arrival time is pure division: the grocery store is 450 kilo-ells away; at 90 nc you cover the distance in exactly 5 kilo-ells — about 50 minutes on human time. No unit juggling ever again.
The Quantum Wave: Why ℏ=1 Unifies Energy, Mass, and Frequency
In quantum mechanics, fundamental particles are waves. The reduced Planck constant, ℏ, is the historical bridge required to convert a wave's frequency (ω) into human energy units like Joules.
When you set ℏ=1, that artificial bridge vanishes. The equation E = ℏω collapses to:
E = ω
This is not a mathematical trick. It is a physical statement. The energy of a quantum system is its frequency. A faster oscillation is literally what energy is at the microscopic level.
Because we have already established c=1, Einstein's famous translation between mass and energy (E=mc²) also collapses. With c²=1, the result is absolute equivalence:
E = m
Mass is energy, and energy is frequency. A particle's mass is merely the baseline frequency of its quantum wave vibrating in place. They are three distinct words describing the exact same physical property.
Kinetic Reality: Why k_B=1 Unifies Temperature and Energy
Temperature is a statistical illusion. When an object feels hot, you are not feeling a unique fundamental force. You are feeling the microscopic kinetic energy of billions of atoms colliding with your skin.
The Boltzmann constant, k_B, exists solely to convert our biological temperature scales (like Kelvin or Celsius) into actual thermodynamic energy. The equation E = k_B T is nothing more than a unit conversion formula.
By enforcing k_B=1, the concept of the degree is abolished. We stop pretending temperature is a distinct phenomenon. The equation becomes:
E = T
Temperature is directly recognized as the average kinetic energy of the particles in a system. A comfortable living room sits at about 300 micro-ε. Your body runs at 310 micro-ε. Water boils when molecules reach 373 micro-ε. You heat frying oil to 450 micro-ε. No separate scale — heat *is* energy.
The Two Pillars of Measurement
Rather than a sprawling system of base units, the universe is quantified by exactly two fundamental metrics.
The first is the spacetime unit, called the "ell" (ℓ). Length and time are both measured in ℓ. Consequently, velocity is strictly dimensionless. It is expressed merely as a fraction of c. A spacecraft traveling at v = 0.5 is understood intuitively to be moving at half the speed of light. Your family car at 60 nc feels perfectly ordinary.
The second is the energy unit, called the "epsilon" (ε). Mass, frequency, and temperature are all measured in ε. The mass of a particle m or the temperature of a star T are simply a count of ε.
Under this regime, derived quantities shed their complexity. Momentum p shares the same unit as energy. Wavelength possesses the unit ℓ and relates to momentum via p = 1/λ. Entropy and heat capacities become entirely dimensionless properties of a system's state space.
The Duodecimal Foundation: Base 12
The bedrock of daily calculation is base 12. Unlike base 10, which caters merely to the biological accident of our digits, base 12 caters to the mathematics of partitioning. It divides cleanly by 2, 3, 4, and 6. In a base 12 culture, the canonical fractions so common in trade, construction, and daily life (halves, thirds, quarters, sixths) possess finite, clean representations.
Splitting a pizza into thirds? Mark the knife at exactly 0.4. Cutting lumber for a bookshelf into quarters? Measure 0.3 of the plank. Halving a recipe or splitting rent four ways becomes effortless mental arithmetic. While base 60 offers even greater divisibility, base 12 strikes the perfect equilibrium between engineering utility and cognitive load.
The Geometry of Rotation
In Euclidean geometry, radians serve as the default cultural angle, stripping away the arbitrary nature of the 360-degree circle. Furthermore, the circle constant is defined intrinsically by the full rotation: τ (tau) = 2π.
With τ as the standard, formulas speak the language of rotation natively:
- A full turn is exactly τ. - A half turn is τ/2. - A quarter turn is τ/4.
Today we still say "turn 90 degrees" at an intersection and then mentally convert to radians for any real calculation. In this system that conversion disappears forever. Turning your steering wheel a perfect right angle is simply τ/4:
sin(τ/4) = 1, cos(τ/4) = 0
Reimagining Time on a Human Scale: Base 12
While natural units are theoretically pristine, they operate on scales entirely detached from the human experience. To bridge this gap, the civilization introduces a secondary "human layer" using base 12 exponential scaling.
Time provides the most crucial test for this system. If time is merely length (c=1), measuring a human day in universal spacetime units (ℓ) yields numbers too vast to comprehend. Instead, the civilization defines a human-scale baseline called the "Pulse".
The Pulse is defined as a specific base 12 power of ℓ (for instance, 12³² ℓ), chosen to perfectly align the distance light travels with the rhythmic duration of a resting human heartbeat. From this biological and physical anchor, all other timepieces scale elegantly via base 12 multipliers — yet they still feel instantly familiar because they line up with the round numbers we already love in base 10:
- The Pulse: The base unit of human time (roughly one second).
- The Span: 12 Pulses (10 on our scale) — the duration of a deep breath.
- The Phase: 12³ Pulses (1,000 on our scale) — roughly 30 minutes, a standard block for focused labor.
- The Cycle: 12⁴ Pulses (10,000 on our scale) — roughly 10 hours, optimized to map cleanly to a standardized planetary day, syncing biological rhythms with the planet's rotation.
- The Orbit: 12⁶ Pulses (a million on our scale) — a seasonal year, scaled for agricultural and astronomical tracking.
By chaining time to base 12 powers of ℓ, a person checking their calendar is still, fundamentally, measuring the distance light travels through the cosmos. Your morning commute might be "18 Phases" — yet you are always tracking light's journey.
The Elegance in Practice
To preserve readability across all quantities, the civilization standardizes human-scale prefixes based on powers of 12 (12³, 12⁶, 12⁹ for macro scales, and 12⁻³, 12⁻⁶, 12⁻⁹ for micro scales). An architect might design a structural beam measuring 1.2 mega-ells, while a climatologist documents a comfortable room temperature as precisely 300 micro-epsilons (mE). A cook measures exactly 0.6 of a mixing bowl of flour.
The true triumph of this system is witnessed in its mathematics. Across all scientific domains, visual and cognitive clutter vanishes.
In relativistic mechanics, instead of the old E = mc² we calculate with today (cluttered with huge conversion factors and massive numbers), the equivalence of mass and energy is stark and undeniable:
E = m
In quantum mechanics, instead of equations riddled with ℏ factors and extra constants we juggle today, the Schrödinger and Dirac equations shed their cumbersome constants. Momentum and wave vector now unify cleanly as:
p = k
In thermodynamics, temperature is treated natively as energy, rendering entropy a dimensionless geometry of probability. Many logarithmic terms become dramatically cleaner without extraneous Boltzmann factors weighing down the integrals.
If a specialized domain requires a "circle-first" approach to length, a conventional unit u can be introduced such that a circle of radius 1u has a circumference of exactly 2u. This implies that u naturally scales the radius by π. While circumference calculations become trivial, the area adapts cleanly to A = r²/π, maintaining the civilization's core philosophical stance: equations should reflect reality's underlying geometry, never its arbitrary conversion artifacts.
In Closing
To adopt this system is not merely to change how we calculate; it is to change how we perceive reality itself. Our current measurements are artifacts of a fragmented past, forcing the universe to translate its elegant truths into our arbitrary languages of meters, seconds, and degrees. By stripping away these historical accidents and building upon the natural geometry of rotation, the innate divisibility of base twelve, and the absolute unifications of spacetime and energy, we stop imposing our scales on the cosmos. Instead, we finally learn to speak its native tongue. In this framework, mathematics ceases to be a clumsy tool of conversion. It becomes a transparent window into the fundamental architecture of the universe.