A toy for one year olds consists of a series of holes and a bunch of objects to push through those holes. The holes come in shapes of circle, square, triangle, etc. And the objects are cylinder/prism versions of the holes. Usually when a child tries to push an object through the hole he or she usually twists the object until it goes through, although each object can be pushed through without twisting if the correct orientation is found.

What if instead of nice cylindrical objects, they were irregularly shaped but still convex. Assume that it is possible to get these irregular objects through the holes by twisting and manipulating them. Is it still possible to find an orientation that will allow the irregular objects through without twisting?

Let the circular hole have a diameter of one unit.

As sphere of unit diameter would pass through no matter what its orientation, however a sphere is not irregular.

Consider a regular tetrahedron with unit distance between vertices as a foundation.

Let each surface and boundary edges be distorted such that all points are unit distance from the fourth vertex which forms their radial centre.

Such an object, though not truly irregular, no matter what its orientation it will always pass through a circular hole of unit diameter.

Consider a Steinmetz solid. This is formed by the intersection of 3 cylinders along XYZ axes.  Even though the cylindrical diameters might be of one unit the diametrically opposing vertices are about 1.2 units apart, as such this object would be invalid.

Brian Smith has not asked us to define an object so it seems that the principle for which he is looking is that no points should be more than the diameter of the hole from each other, as demonstrated by my 'inflated' tetrahedron.  A little more directly, all surface points of the convex object must be within, or on the surface, of sphere that would pass through the circular hole.

Edited on November 9, 2009, 10:38 pm

Posted by brianjn on 2009-11-09 22:09:37