Physics Taken Too FarEpisode 01

Humanity Sugar Cube

Squeeze every person together as tightly as the tiny core of an atom. All of humanity would fit in a cube about the size of a sugar cube. How hard would that tiny cube pull on things nearby?

An imaginary experiment: real matter could not stay compressed like this. Illustration not to scale. Earth’s own pull is not included.

Playing

Squash gauge

Stronger pull than on Earth
On Earth1× Earth gravitya little squashed
Here, 1 m away3.9 times Earth gravityclearly squashed

Cartoon of a soft toy resting on a shelf: flatter just means a stronger pull. It doesn’t predict real crushing. Falling alone doesn’t squash an object.

Humanity at nuclear density, with an observer probeIllustration, not to scale. A glowing cube 1.3 cm wide holds everyone. A measuring point sits 1 m from its centre, where the cube pulls with 3.9 times Earth gravity. A dashed orange ring at 2 m marks where its pull equals Earth gravity.10 cm1 m10 m100 m2 cmSame pull as Earth · 2 m1.3 cm cubeMEASURING POINT1 m away3.9 times Earth gravity
Illustration, not to scale. The moving dots are just for show.
Everyone fits in a cube1.3 cmwide, about the size of a sugar cube
Its pull 1 m away3.9×Earth gravity

8.2 billion is a round example, not an official count.

Measured from the centre of the cube, 2 cm to 100 m.

Assumes 70 kg per person. You can change this under “Assumptions and extra numbers”.

Twice as far, a quarter of the pull

Here (1 m)3.9 times Earth gravity
Twice as far away (2 m)0.98 times Earth gravity

Double the distance and the pull drops to one quarter, however big the cube is.

Assumptions and extra numbersMass per person, full scientific readouts, the inverse-square plot

Assumptions

An assumption applied to everyone, 40 to 100 kg. Default 70 kg.

  • Population is an illustrative input (1 million to 20 billion), not census data.
  • Density: 2.7 × 10¹⁷ kg/m³, a rounded model value for nuclear matter.
  • Gravity is computed for the equivalent uniform sphere with the cube’s volume, not for the cube itself.
  • Only the cube’s pull is shown; Earth’s own gravity is not added.

Scientific readouts

8.2 billion × 70 kg, r = 1.00 m

Total massM = N × m
5.74 × 10¹¹ kg
≈ 574 million tonnes. Compression doesn’t change it.
Volume at nuclear densityV = M / ρ
2.13 cm³
ρ = 2.7 × 10¹⁷ kg/m³ (rounded model value)
Cube edge (equal volume)a = ∛V
1.29 cm
Shape for illustration only.
Equivalent sphere radiusGravity modelR = ∛(3V / 4π)
7.98 mm
All gravity numbers use this uniform sphere.
Acceleration at r = 1.00 mg = GM / r²
38.3 m/s²
3.91 × g₀ · source only, Earth not added
Escape speed from rv = √(2GM / r)
8.75 m/s
≈ 31.5 km/h · Newtonian, source only
Distance where g = g₀r = √(GM / g₀)
1.98 m
From the centre, source alone
Schwarzschild radiusr_s = 2GM / c²
0.853 fm
= 8.53 × 10⁻¹⁶ m · ≈ 1.0× a proton’s charge radius
Not a black hole

R is 9.36 × 10¹² times r_s. Being dense is not enough: a black hole needs R ≤ r_s.

The inverse-square law

g(2r) = g(r) / 4

Acceleration versus distance, log–logA straight line falling two decades in acceleration for every decade in distance, the signature of an inverse-square law. The observer at 1.00 m feels 38.3 metres per second squared from the source alone; each doubling of distance divides that by four.10⁻³10⁻²10⁻¹10⁰10¹10²10³10⁴10⁵2 cm10 cm1 m10 m100 mdistance from source centre, r (log scale)g (m/s², log scale)Earth standard g₀r2r ÷44r ÷168r ÷64
Doubling ladder from the current distance
Distanceg (m/s²)Share of first
r 1.00 m38.31
2r 2.00 m9.581/4
4r 4.00 m2.391/16
8r 8.00 m0.5991/64
How the science worksCaveats, method, constants and sources
  • Impossible on purpose: compression and confinement are assumed.
  • Gravity uses an equivalent sphere, not the cube.
  • Source-only pull; Earth’s own gravity not added.

Read before quoting

What this experiment is not

  1. An impossible thought experiment.

    Mass is simply carried over. The model ignores the energy it would take to compress anything this far and any nuclear transformations that would happen along the way.

  2. Not a stable object.

    Ordinary human matter cannot remain a free-standing cube at nuclear density. In neutron stars, immense gravity provides confinement; this much mass is nowhere near enough to self-confine neutron-star matter.

  3. Squeezing adds no pull from far away.

    Compression does not increase total mass or far-field gravity. Outside any spherical arrangement of the same mass, the pull at a given distance is identical; squeezing only lets a probe get closer to the centre.

  4. An idealized Newtonian sphere.

    Gravity is computed for an isolated, uniform sphere of the same volume. The acceleration is from the source alone; Earth’s own gravity is not added. The cube in the picture is for volume only, and point-mass gravity is not exact near a cube’s surface.

  5. Much is left out.

    Tides, surroundings, how materials respond, and relativistic effects where they would matter are all left out. The probe is an abstract test point, not a person; this page makes no claims about survival.

  6. Dense is not the same as a black hole.

    A black hole needs its radius to be inside its Schwarzschild radius. With the defaults, R is roughly ten trillion times larger than rs, so the comparison in the readouts says no.

  7. The animation is decorative.

    Distances are log-mapped onto the picture and the motion is not physical time. Numbers come from the readouts, not from pixels.

  8. The squash gauge is a cartoon.

    It shows a soft toy held against a shelf, which pushes back against the pull. How flat it is drawn is a bounded visual scale of the pull (as a multiple of Earth gravity), not a material model, and it says nothing about when anything would break. Falling alone doesn’t squash an object; only differences in pull across an object (tidal effects, not modelled here) could stretch or squeeze it.

Methods

How the numbers are made

Six lines of algebra, all in SI units. Every number in the experiment comes from the same small, tested function.

  1. Mass

    M = N × m

    N people at an assumed average mass m. Both are inputs: the 8.2 billion default is illustrative, not a census figure, and 70 kg is a stated assumption.

  2. Volume at nuclear density

    V = M / ρ,   ρ = 2.7 × 10¹⁷ kg/m³

    Nuclear matter saturates near n₀ ≈ 0.16 nucleons per fm³. Multiplying by a nucleon mass (≈ 1.67 × 10⁻²⁷ kg) and converting fm³ to m³ gives about 2.68 × 10¹⁷ kg/m³, which we round to an illustrative 2.7 × 10¹⁷ kg/m³. This is a model value, not an exact constant.

  3. Two shapes, two jobs

    a = ∛V  ·  R = ∛(3V / 4π)

    The cube of edge a is the picture and the title. The equivalent uniform sphere of radius R has the same volume and density and is used for every gravity number. Outside a spherically symmetric body, gravity is exactly that of a point mass (Newton’s shell theorem). That is not exactly true near a cube’s faces and corners, so we never apply it to the cube.

  4. Pull and escape

    g = GM / r²  ·  g / g₀  ·  v = √(2GM / r)

    r is measured from the source centre. The model only answers for exterior points, r ≥ R, and the input rejects anything outside 0.02–100 m rather than quietly clamping it. Even at the largest allowed mass (20 billion × 100 kg) R is about 1.2 cm, so 2 cm is always outside the sphere.

  5. Black-hole check

    r_s = 2GM / c²

    The Schwarzschild radius sets the scale where Newtonian gravity fails badly and a horizon would form. We compare it with R. Being dense is not enough; R would have to shrink to rs.

  6. The inverse square

    g(2r) = g(r) / 4

    On a log–log plot, g(r) is a straight line of slope −2. Each doubling of distance divides the pull by four; each factor of ten divides it by a hundred.

References

Sources

  • Constants
    NIST, Fundamental Physical Constants (2022 CODATA recommended values). physics.nist.gov/cuu/Constants · full listing all.pdf. Source of G, c, g₀ and the proton mass and charge radius used here.
  • Density
    C. Drischler, P. G. Giuliani, S. Bezoui, J. Piekarewicz, F. Viens, A Bayesian mixture model approach to quantifying the empirical nuclear saturation point, Phys. Rev. C 110, 044320 (2024). arXiv:2405.02748 · DOE OSTI copy osti.gov/servlets/purl/2473676. Discusses the saturation density of symmetric nuclear matter, n₀ ≈ 0.16 fm⁻³. Our 2.7 × 10¹⁷ kg/m³ is a rounded conversion of that number, used for illustration.