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Boot a Universe: the maths

Tegmark's map of space and time dimensions, as a checklist of published results; and one planet orbiting a star with gravity generalised to n space dimensions, integrated step by step (velocity Verlet). Computed live.

Equations

F ∝ 1 / rⁿ⁻¹

Gravity spreads over the surface of an n-dimensional sphere, so it weakens as 1/rⁿ⁻¹.

V_eff(r) = L²/(2r²) − k / rⁿ⁻²

The effective potential for an orbit with angular momentum L. It has a stable minimum only when n < 4: at 4 or more, a nudged circular orbit spirals in or flies off.

Constants

n

2 to 5 (a continuous dial)

space dimensions

Worked examples ✓ checked on every change

n = 3.9: does an orbit survive?	yes	n < 4: a stable minimum exists
n = 4.2: does an orbit survive?	no	n ≥ 4: no stable orbit (Ehrenfest 1917)

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• Newtonian gravity generalised to n dimensions; fractional n is a mathematical dial, not a physical world.
• The boot checks are published results, summarised — not computed here.

Where it breaks

For the checklist: it's a map of arguments (Ehrenfest, Dorling, Tegmark), not a proof that observers need 3 + 1.

Sources: P. Ehrenfest (1917); J. Dorling, Am. J. Phys. 38, 539 (1970); M. Tegmark, Class. Quantum Grav. 14, L69 (1997).