Push a tumbling doll and it leans, rocks, and comes back. Push it harder and it leans further, rocks longer, and still comes back. There is nothing inside that snaps or winds up: no spring, no pivot, no hidden arm. The whole behaviour comes from where the weight sits. Almost all of the mass is a lump of clay or lead fixed in the bottom of a hollow rounded shell, and that one decision makes exactly one position stable.
The comparison to make is with any ordinary standing object. A bottle stands because its flat base gives it a wide area of support; tip it past the edge of that area and it falls. A rounded doll has no flat base at all, only a small patch where the curve touches the table. It should be far easier to knock over. It is not, because the question is not how wide the base is but what happens to the centre of mass when the shape rolls.
For a curved bottom there is a centre of curvature — the centre of the sphere whose surface the base is a piece of. Roll the doll sideways and the contact point moves along the table, the whole body rotates about that centre, and the heavy lump swings. If the centre of mass lies below the centre of curvature, rolling lifts it. Every degree of tilt raises the weight a little. Gravity, which always pulls that weight down, therefore produces a moment that rolls the doll back toward its upright position. To keep it tilted you would have to hold it up.
Why the weight has to be that low
How low is a practical question, and makers answer it by feel. Put the lump too high and the centre of mass rises above the centre of curvature; tilting then lowers it, and the doll behaves like a bottle — it stays wherever you leave it, or falls all the way over. Put it just below and the restoring moment is weak: the doll returns slowly, wobbles for a long time, and a slight push leaves it leaning for a second before it decides. Put it right at the bottom of the shell and the return is immediate and almost rude. Most dolls sit between the last two, because a toy that snaps back too fast is no fun to rock.
The upper part is deliberately empty. Papier-mâché, thin wood or moulded plastic gives the doll its face and its height at almost no cost in mass. Height still matters, though it matters less than people expect: a tall shell moves the painted head far from the table, which makes the tilt visible, while the weight stays low. This is why the traditional shape is a rounded egg with a small head on top — a shape chosen to show the motion, not to cause it. Strip the paint and the face away and a weighted hemisphere behaves exactly the same.
The radius of the curved base sets the character of the wobble. A tight, small radius returns quickly because a short roll lifts the weight sharply, but the doll feels nervous and easy to tip past its useful range. A broad, flatter curve lifts the weight slowly over a longer roll, so it sways gently and forgives a clumsy push. Makers pick the curve along with the weight, and the two are really one decision: what matters is the vertical distance between the centre of mass and the centre of curvature, not either number on its own.
Where the rocking goes
A doll that were perfectly rigid and perfectly frictionless would rock forever, trading height for speed and back again. Real ones stop after a handful of swings. The energy leaks away in three ordinary places: the shell flexes slightly at the contact patch, the table gives a little, and the lump itself is never bonded so rigidly that nothing inside shifts. Rolling resistance at that tiny patch is small, which is why the wobble lasts as long as it does; the damping is gentle by accident rather than by design. A soft carpet shortens the show noticeably, because the contact patch is no longer a line but a dent the doll has to climb out of each time.
It is worth separating this from the toys that stay up by spinning. A spinning top holds its axis because rotation gives it angular momentum; stop the rotation and it falls at once. A rolling hoop is more precarious still — it has exactly one unstable position and must be corrected continuously by a hook from behind. The tumbling doll needs no motion at all. Its stability is stored in the arrangement of its material, and it is just as stable sitting still on a shelf as it is in the middle of a wobble.
At the other end of the scale, a glass marble is stable in every orientation, so it simply rolls; there is no preferred way up for it to return to. A jianzi sits nearer to the doll: a weighted base and feathers above make one attitude overwhelmingly likely, which is why it lands the same way every kick. The doll is the cleanest version of that idea — it has removed the fall entirely and kept only the return.
This also explains why the toy survives rough handling. There is no adjustment to lose and no part whose alignment matters: as long as the heavy lump stays at the bottom and the base stays round, the geometry does the whole job. A dent in the shell changes how it rocks, not whether it recovers. A crack that lets the lump shift changes the answer, and that is about the only way to break it.
Nothing needs explaining to a child. One push, one slow wobble, one return — the lesson is delivered in about two seconds and can be repeated indefinitely. That is likely why the shape keeps being made: it is a piece of statics you can hold in one hand, with the argument hidden in the base and the conclusion visible on the face.