I suspect it is impossible to split a (any) 3d solid into two, such that each of the pieces is identical in shape (but not volume) to the original. How can I prove this?
Can we slice an object into two pieces similar to the original?
5
$\begingroup$
geometry
-
1On a related note: Kimmo Eriksson proves in [The American Mathematical Monthly Vol. 103, No. 5 (May, 1996), pp. 393-400] that a convex polygon is splittable in two properly congruent pieces iff it has rotational symmetry. – 2010-09-07
-
0A couple of questions. 1) What exactly does split mean? 2) Why is this tagged topology? – 2010-09-07
-
1Is this connected to [the Banach-Tarski paradox](http://en.wikipedia.org/wiki/Banach-Tarski)? – 2010-09-07
-
0Was thinking of a 3d analogue to Pythagoras' Theorem. – 2010-09-07
-
0Isaac, I don't think so, because that paradox involves dividing a sphere into non-spherical pieces. But I am also curious if there are any solutions involving pathological shapes or division methods. – 2010-09-08