bedlam - 2:06 pm on Mar 19, 2008 (gmt 0)

Because it's the only shape with equal dimensions when rotated-and as a result, can't fall into the hole it's covering. an equilateral triangle won't fall into the hole...

Actually, that's not quite right.

It's true that an equilateral triangle can't fall edge-first through the exact middle of a congruent triangular hole, it most definitely can fit (assuming the cover isn't ridiculously thick). This can't happen at all with a circular cover.

Picture it this way: the longest point-to-point distance in an equilateral triangle is along any of its sides. If you hold the cover vertically (i.e. so that the faces of the cover are perpendicular to the ground) above and parallel to one of the edges of the hole, and rotated to any angle such that none of the edges of the triangular cover is parallel to the ground, then the edge presented to the hole will be smaller than the edge of the opening. Let go and it'll fall right through.

It's also probably possible (again, limited by the thickness of the cover and the size of the hole), by holding the cover vertically over the hole with one edge perpendicular to the ground, the corner opposite that edge over the corner of the hole, and the whole cover rotated with respect to the hole to almost any angle such that the perpendicular edge of the cover is not positioned over the midpoint of the edge of the hole opposite the already-mentioned corner.

In other words, with one corner of the cover positioned above one corner of the hole, it gets easier to drop the cover through the hole the closer the faces of the cover come to getting parallel with any of the edges of the hole, and the closer any of the cover's edges gets to vertical.

-b

This message brought to you by your possibly neurotic, friendly neighbourhood former CAD draftsman who actually drew pictures to make sure he wasn't talking nonsense :-D

[edited by: bedlam at 2:42 pm (utc) on Mar. 19, 2008]