Abstract
The Rattle theorem states that a baby's rattle, the union of a 2-sphere Σ and its interior, is a topological 3-cell if the marble rattler in its interior touches every point of Σ as it rolls around inside the rattle. Other than rattlers that themselves contain marbles (such as a solid ellipsoid), there are no known substitutes for the marble in this theorem. Examples are given to show that some natural rattler choices among convex polyhedra fail to tame the rattle. However Σ is nearly tame in E 3 if it can be touched at each of its points by the tip of a cone from a family of congruent cones in Σ∪Int Σ with sufficiently large cone angles.
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