How It Works

Place a square table on a bumpy (but continuous) floor. Any three legs can rest on the surface, and the tabletop will tilt to match—three points always determine a plane. But the fourth leg has no say in that plane, so it may hover above the floor or try to dig in below it. Define the wobble function:

$$f(\theta) = \bigl(h_{\color{#c25a4e}A}(\theta) + h_{\color{#c49a3a}C}(\theta)\bigr) - \bigl(h_{\color{#4a9a8f}B}(\theta) + h_{\color{#7b6fa0}D}(\theta)\bigr)$$

where $h_i(\theta)$ is the floor height at leg $i$'s position when the table is rotated by angle $\theta$. For a square table, rotating by $90°$ swaps the diagonal pairs $(\color{#c25a4e}A\color{#333},\color{#c49a3a}C\color{#333})$ and $(\color{#4a9a8f}B\color{#333},\color{#7b6fa0}D\color{#333})$, so:

Since $f(\theta)$ is continuous (the floor is continuous) and changes sign over $[0°, 90°]$, the Intermediate Value Theorem guarantees some $\theta^*$ where $f(\theta^*) = 0$—all four legs touch the ground.

Where the Theorem Gets Wobbly

The proof leans on a square footprint—equal diagonals ensure that a 90° rotation swaps the diagonal pairs. Give the table a rectangular base and that symmetry tips over; balance is no longer guaranteed. Steep slopes trip it up too: the full theorem needs a Lipschitz condition (floor slope ≤ ~35°) and legs long enough to keep contact through the rotation. Push past those bounds and the argument loses its footing.

There's also a quietly absurd physical assumption: the legs must be long enough to clear every bump the table rotates over. A dinner table on gently uneven tiles? Fine. That same table straddling a boulder? The math still says "just rotate it," but the legs and tabletop will plunge into the rock before you reach the magic angle. The theorem is about contact points, not collisions.

On the sturdy side: a tripod (three legs) never wobbles at all—three points always define a plane, no theorem required. And the result really is about the geometry of the contact points, not the object itself—any four-footed thing with a square base (like a chair) qualifies.

Reference: Baritompa, Löwen, Polster & Ross, "Mathematical Table Turning Revisited" (2005).

Eames chair 3D model: "Vitra Eames Plastic Chair" by furnny, licensed under CC BY 4.0.