PhysicsBabel — A Complete Catalogue of Dimensionally-Consistent Physics Equations

Vashy is an exhaustive, machine-generated catalogue of every irreducible dimensionless relation that can be formed from a curated pool of 60 physical quantities spanning all of classical physics. It is built directly from the Vaschy–Buckingham π theorem: any physically meaningful equation must be dimensionally homogeneous, so the set of dimensionally-consistent monomials is exactly the set of dimensionless products the theorem describes.

Each of the 2,325,320 rows is one such relation — for example F = k₀ · m·a (Newton's second law), ℓ = k₀ · v⁻¹·ρ⁻¹·μ (the Reynolds number), or m = k₀ · E·c₀⁻² (mass–energy equivalence) — annotated with metrics that let you filter the physically meaningful laws out of the vast space of mere dimensional coincidences.

k₀ denotes the dimensionless constant of order unity that dimensional analysis cannot determine (e.g. the ½ in kinetic energy, or 6π in Stokes' drag). This is an intrinsic limitation of the method, not of the dataset.

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What is in the dataset

The catalogue is complete: it contains every minimal dimensionless relation of size 2 to 8 over the quantity pool. Size 8 is a hard mathematical ceiling — with 7 SI base dimensions, no irreducible relation can involve more than 8 quantities (a relation of k quantities needs rank k−1 ≤ 7).

# quantities	rows	note
2	22	pairs with identical dimensions (e.g. energy vs torque)
3	593	e.g. F = k₀·m·a, E = k₀·m·v²
4	17,325	e.g. Reynolds, Weber, Strouhal numbers
5	280,646
6	813,919
7	1,140,445
8	72,370	must span all 7 base dimensions at once (rare)
Total	2,325,320

Of these, 46,720 are flagged plausible (a conservative gate for genuine physical candidates) and 20 correspond to classically named laws / numbers.

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Provenance & method

Every quantity is represented as a vector of integer exponents over the seven SI base dimensions. A dimensionless relation among a set of quantities is an integer null vector of the matrix whose columns are those dimension vectors.

A row in this dataset is a circuit of that "dimensional matroid": a minimal dependent set, where every quantity is essential (removing any one destroys the dimensionless product) and which is not a product of smaller relations. Formally, a set of n quantities is a circuit iff its dimension matrix has rank n−1 and the (unique) null vector has full support. Each circuit corresponds to exactly one irreducible dimensionless number, and the catalogue enumerates all of them without duplication.

Dimensional analysis fixes only the form of a law up to k₀; it does not prove physical existence. That is why the dataset ships with relevance metrics rather than claiming every row is a real law.

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Dataset structure

A single split (default) backed by data/circuits.parquet (~108 MB, zstd).

column	type	description
size	int	number of quantities in the relation (2–8)
keys	string	sorted, comma-separated quantity keys (the relation's identity)
symbols	string	the quantities' symbols, ·-joined
domains	string	distinct physics domains involved, comma-separated
pi	string	the dimensionless number (Π group), e.g. ℓ·v·ρ·μ⁻¹
equation	string	the relation solved for one quantity, e.g. ℓ = k₀ · v⁻¹·ρ⁻¹·μ
name	string | null	recognized name of the law / dimensionless number, if any
n_const	int	number of universal constants (c₀, G, h, k_B, e, N_A) involved
n_domains	int	number of distinct physics domains bridged
max_exp	int	largest absolute exponent in the monomial
exp_l1	int	sum of absolute exponents (monomial "weight")
complexity	float	composite complexity score (lower = simpler)
score	float	composite physicality score (higher = more plausible)
plausible	bool	passes the conservative physicality gate (see below)
exponents	string (JSON)	{quantity_key: integer_exponent} for the dimensionless monomial

The plausible gate

A row is flagged plausible = true when it satisfies all of:

• size ≤ 5 — real laws rarely couple more than five quantities;
• n_const ≤ 1 — a genuine law uses at most one universal constant;
• n_domains ≤ 2 — at most one cross-domain bridge;
• max_exp ≤ 3 and exp_l1 ≤ 8 — small integer exponents.

This is intentionally conservative: it favours precision over recall. The remaining ~2.28 M rows are kept so that the catalogue stays complete and auditable, but the overwhelming majority are dimensional coincidences (e.g. m = k₀ · c⁻¹·k_B), not physical laws.

The score

A heuristic that rewards simplicity (few quantities, small exponents, single domain, at most one constant). Sorting plausible rows by score DESC surfaces textbook laws and named dimensionless numbers first.

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Example rows

equation	pi	name	size	score	plausible
m = k₀ · a⁻¹·F	m·a·F⁻¹	Newton 2nd law	3	9.0	✅
m = k₀ · v⁻²·E	m·v²·E⁻¹	kinetic energy	3	7.8	✅
m = k₀ · E·c₀⁻²	m·E⁻¹·c₀²	mass-energy (E=mc²)	3	6.7	✅
ℓ = k₀ · v⁻¹·ρ⁻¹·μ	ℓ·v·ρ·μ⁻¹	Reynolds number	4	8.0	✅
E = k₀ · f·h	E·f⁻¹·h⁻¹	Planck relation (E=hf)	3	—	✅
i = k₀ · U·R⁻¹	i·U⁻¹·R	Ohm's law	3	—	✅

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The quantity pool (60 quantities, 7 base dimensions)

Base dimensions: M (mass), L (length), T (time), Θ (temperature), I (electric current), N (amount of substance), J (luminous intensity).

Mechanics

symbol	quantity	dimensions
m	mass	M
ℓ	length	L
t	time	T
v	velocity	L·T⁻¹
a	acceleration	L·T⁻²
g	gravitational acceleration	L·T⁻²
F	force	M·L·T⁻²
E	energy	M·L²·T⁻²
τ	torque	M·L²·T⁻²
P	power	M·L²·T⁻³
p	momentum	M·L·T⁻¹
Lₐ	angular momentum	M·L²·T⁻¹
S	action	M·L²·T⁻¹
J	moment of inertia	M·L²
P_p	pressure	M·L⁻¹·T⁻²
ρ	mass density	M·L⁻³
μ	dynamic viscosity	M·L⁻¹·T⁻¹
ν	kinematic viscosity	L²·T⁻¹
ω	angular velocity	T⁻¹
f	frequency	T⁻¹
k	spring constant	M·T⁻²
γ	surface tension	M·T⁻²
A	area	L²
V	volume	L³
Q	volumetric flow rate	L³·T⁻¹

Thermodynamics

symbol	quantity	dimensions
Θ	temperature	Θ
S_e	entropy	M·L²·T⁻²·Θ⁻¹
C	heat capacity	M·L²·T⁻²·Θ⁻¹
c	specific heat capacity	L²·T⁻²·Θ⁻¹
κ	thermal conductivity	M·L·T⁻³·Θ⁻¹
α	thermal diffusivity	L²·T⁻¹
β	thermal expansion coefficient	Θ⁻¹

Electromagnetism

symbol	quantity	dimensions
q	electric charge	T·I
i	electric current	I
U	electric potential	M·L²·T⁻³·I⁻¹
R	electric resistance	M·L²·T⁻³·I⁻²
C_e	capacitance	M⁻¹·L⁻²·T⁴·I²
L_e	inductance	M·L²·T⁻²·I⁻²
E_f	electric field	M·L·T⁻³·I⁻¹
B	magnetic flux density	M·T⁻²·I⁻¹
Φ	magnetic flux	M·L²·T⁻²·I⁻¹
ε	permittivity	M⁻¹·L⁻³·T⁴·I²
μ₀	permeability	M·L·T⁻²·I⁻²
σ	electrical conductivity	M⁻¹·L⁻³·T³·I²
ρ_e	electrical resistivity	M·L³·T⁻³·I⁻²

Chemistry

symbol	quantity	dimensions
n	amount of substance	N
M_m	molar mass	M·N⁻¹
c_n	molar concentration	L⁻³·N
E_m	molar energy	M·L²·T⁻²·N⁻¹
R	molar gas constant	M·L²·T⁻²·Θ⁻¹·N⁻¹
z	catalytic activity	T⁻¹·N

Photometry

symbol	quantity	dimensions
I_v	luminous intensity	J
E_v	illuminance	L⁻²·J
L_v	luminance	L⁻²·J

Universal constants

symbol	quantity	dimensions
c₀	speed of light	L·T⁻¹
G	gravitational constant	M⁻¹·L³·T⁻²
h	Planck constant	M·L²·T⁻¹
k_B	Boltzmann constant	M·L²·T⁻²·Θ⁻¹
e	elementary charge	T·I
N_A	Avogadro constant	N⁻¹

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Loading

import pandas as pd
df = pd.read_parquet("data/circuits.parquet")

# the physically-plausible laws, best first
laws = df[df.plausible].sort_values("score", ascending=False)

# every relation involving viscosity, within mechanics
df[df["keys"].str.contains("viscosity") & (df.domains == "mechanics")]

Or with the 🤗 datasets library:

from datasets import load_dataset
ds = load_dataset("<user>/vashy", split="train")

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Intended uses

• Symbolic regression & scientific ML: a dimensionally-valid hypothesis space / prior over candidate equations.
• Physics education: browsing dimensionless numbers and their structure.
• Benchmarking: testing whether models can recover known laws from dimensional constraints.

Limitations & caveats

• Dimensional consistency ≠ physical truth. Most rows are coincidences; use plausible and score, and validate against physics.
• The constant k₀ is undetermined by dimensional analysis.
• Only single dimensionless numbers are represented. When a phenomenon needs two or more independent Π groups, the true law is Π₁ = f(Π₂, …) with an arbitrary function f; the dataset lists the individual irreducible Π groups (the circuits), not the functional relation between them.
• Pool-dependent. The catalogue is exhaustive for this 60-quantity pool; a different pool yields a different catalogue.
• Deliberate dimensional collisions in the pool (energy vs torque, action vs angular momentum, entropy vs heat capacity, acceleration vs gravity) are real and produce legitimate size-2 identities.

License

Released under CC-BY-4.0. The underlying facts are mathematical consequences of SI dimensional definitions.

Citation

@misc{r&d_mediation_2026,
    author       = { R&D Mediation },
    title        = { PhysicsBabel (Revision 963a461) },
    year         = 2026,
    url          = { https://huggingface.co/datasets/RANDMEDIATION/PhysicsBabel },
    doi          = { 10.57967/hf/9544 },
    publisher    = { Hugging Face }
}

Discovering equations by AI

For instance, charge × flux = angular momentum

$$S=k_0\cdot q\cdot \mathrm{\Phi }{L}_a=k_0\cdot q\cdot \mathrm{\Phi }S\; =\; k\_0\; \backslash cdot\; q\; \backslash cdot\; \backslash Phi\; \backslash qquad\; L\_a\; =\; k\_0\; \backslash cdot\; q\; \backslash cdot\; \backslash Phi$$

The product of charge × magnetic flux has the dimensions of an action and of an angular momentum: [C]·[Wb] = [J·s]. This isn't a coincidence — it's the backbone of three chapters of quantum physics:

$$q\mathrm{\Phi }\mathrm{/}\mathrm{\hslash }q\backslash Phi/\backslash hbar$$ est une phase → l'effet Aharonov–Bohm ;

$$L_a=q\mathrm{\Phi }L\_a\; =\; q\backslash Phi$$: le champ électromagnétique...

$$q\mathrm{\Phi }=n\mathrm{\hslash }q\backslash Phi\; =\; n\backslash hbar$$ → la quantification du flux...