Equilateral triangles can fall in. You have three dimensions: the height of the triangle is less than the side length of the triangle, so it fits in near the si

Equilateral triangles can fall in. You have three dimensions: the height of the triangle is less than the side length of the triangle, so it fits in near the sides of the hole. The OP mentions curves of constant width: https://en.wikipedia.org/wiki/Curve_of_constant_width https://en.wikipedia.org/wiki/Curve_of_constant_width Which includes circles and Reuleaux triangles, which are much more difficult to manufacture than circles. I think this can be rounded off to "circles are the only well-known shape that can't fall into a similarly-sized hole".