A bisector is something that cuts some other thing into two equal pieces. More concretely, assume we are given a reasonably well-behaved (for example, compact) 3D object and we are looking for planes in space that cut this object in two equal volume bits.
For reasonable concepts of well-behavedness we can given a well-behaved object and a line in space find a bisector that contains that line. Often that bisector will be unique.
We are interested in the opposite situation: Given a well-behaved object let us call a line in space a universal bisector if any plane containing the line is a bisector of the object.
For example, the axis of a cylinder or cone is a universal bisector.
Q: Can you find universal bisectors for
- the regular tetrahedron?
- the "L" which is obtained from a cuboid by removing a full height sub-cuboid from one of its corners?
- are they unique?
- can you give an object that does not have a universal bisector?
Note: this is not a maths question, rigorous proofs are not required. There are good intuitive arguments to answer all parts of this puzzle.
Attribution: conceived by myself