In The Three-Body Problem by Liu Cixin, one of the most imaginative and haunting ideas is the dual-vector foil, a hypothetical weapon that collapses three-dimensional space into two dimensions. When activated in the solar system, the author describes the resulting 2D space as resembling The Starry Night by Vincent van Gogh—beautiful, surreal, and terrifying.
This concept invites serious thought experiments grounded in physics. What would really happen if our solar system were flattened into a two-dimensional plane? Here are some speculations.
Gravity
In three dimensions, Newtonian gravity follows the familiar inverse-square law:
$$ F(r) = G \frac{m_1 m_2}{r^2}. $$
But in a 2D universe, the mathematical form of gravity changes. The gravitational force becomes:
$$ F(r) \propto \frac{1}{r}. $$
and the Poisson equation leads to a logarithmic potential:
$$ \Phi(r) \propto \ln r, $$
This has several dramatic implications:
- Gravitational wells become deeper and more confining.
- Escape velocity becomes ill-defined, since the potential never truly levels off.
- Orbital motion becomes unstable; orbits that would be closed ellipses in 3D now form precessing “rosette” patterns.
According to Bertrand’s Theorem, only a $1/r^2$ (gravity in 3D) or linear $r$ (harmonic oscillator) force law produces stable, closed orbits. In 2D, with a $1/r$ force, circular orbits are only marginally stable, and most motion would eventually degrade into collision or ejection.
Dimensional Freedom
In 2D, area scales as $r^2$ rather than volume as $r^3$. There is less “space” for matter to occupy. The Sun, as the most massive body in the solar system, would likely expand dramatically in a 2D environment.
With only two spatial dimensions, there is less room to maneuver. Combined with unstable gravitational dynamics, this would make collisions between celestial bodies far more likely. Planets, asteroids, and satellites would spiral into one another unless finely tuned—an improbable scenario in such a dynamically volatile system.
Gas and Pressure
Gravitational self-attraction weakens at small radii in 2D due to the softer force law. At the same time, hydrostatic equilibrium—the balance between pressure and gravity—also behaves differently. For a 2D gas disk, the pressure gradient satisfies:
$$ \frac{dP}{dr} \sim -\rho \frac{G M(r)}{r}. $$
This suggests that:
- Gas giants would become much more extended and diffuse, lacking the gravitational compression that gives them their current shapes.
- Terrestrial planets, composed of rigid matter, might remain relatively compact.
The result would be a solar system of bloated gas envelopes and compact rocky remnants.
The Sun’s own internal balance between pressure and gravity would also collapse. Nuclear fusion could slow dramatically or stop altogether.
Optics
In 2D, light would also be confined to a plane. This has consequences for visibility and optics:
- Opaque objects would only be visible along their edges—think of coins seen from the side.
- Terrestrial planets would appear as thin luminous rings when viewed at perpendicular angles, not as fully illuminated disks.
