---
license: cc-by-4.0
language:
- en
tags:
- physics
- dimensional-analysis
- buckingham-pi
- vaschy-buckingham
- symbolic-regression
- science
pretty_name: "Vashy: Dimensionally-Consistent Physics Equations"
size_categories:
- 1M<n<10M
configs:
- config_name: default
  data_files: circuits.parquet
---

# 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.

---

## 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**.

---

## 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.

---

## 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.

---

## 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 | — | ✅ |

---

## 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 |

---

## Loading

```python
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:

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

---

## 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

```bibtex
@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

$$S = k_0 \cdot q \cdot \Phi \qquad L_a = k_0 \cdot q \cdot \Phi$$

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:

$$q\Phi/\hbar$$ est une phase → l'**effet Aharonov–Bohm** ;

$$L_a = q\Phi$$: le champ électromagnétique...

$$q\Phi = n\hbar$$ → la **quantification du flux**...