Broken glass geometry If topology is called rubber-sheet geometry, would it be accurate to describe the "cut and shuffle" topic of "piecewise isometries" as broken glass geometry ?
Isometry sounds more geometrical than topological. What would be a good name for "piecewise topology" ?  If you take a topological space and break it up and put it back together in a random way, not necessarily in one piece, then what could you say about the topology of the resulting jumble ?
 A: You might be interested in reading about Hilbert's Third Problem. Hilbert asked the question: Given two polytopes of the same volumes, can we cut one up into polytopal pieces and reassemble it to form the other. The answer is NO in dimensions $\geq 3$; there are many additional subtle obstacles. 
A: If the pieces are of given shapes, then this is essentially a jigsaw puzzle, perhaps in a higher dimension than 2.  A mathematical term for problems of this type is "dissection".
If the pieces don't have a specific form and don't have to be put back together "in one piece", I don't think much can be proved about the "resulting jumble".
The  Banach-Tarski paradox gives an example of what is possible, at least if the Axiom of Choice is allowed.  This is a result that proves a ball (solid sphere) can be partitioned into a finite number of subsets that can be recombined to form two balls, each congruent to the original.  (The subsets are not measurable.  It is not a true paradox, but it is so called because it shows the Axiom of Choice has implications that defy common intuition.)
