On 2016-04-15 10:11, rcs@xmission.com wrote:
My note about a circular dissection is discussing constructions with a minimal number of pieces. If you allow extra pieces, then every piece can be a polygon.
I've seen the theorem about polygon interdissectability (any two polygons with the same area are interdissectable) attributed to Hilbert, likely in either a Gardner column or in the intro to a Dissections book.
You can adjust a side of a triangle to any desired length via gosper.org/A=f(bh).gif . If this is the sidelength of another triangle with the same area, then they will have the same height, and repeating the trick finishes the proof for triangles. gosper.org/shardway.PNG --rwg
I was amazed that Hilbert would be concerned with something so light. Later I learned about his posing the same problem for 3D, and Dehn's negative answer. And much later I realized that this is relevant to the existence of alternative measures for area & volume. I don't know the details, but perhaps some dissections are constraints on alternate measures.
I have a query on the intended meaning of "scissors dissection": Does this mean simply that the pieces must have straight edges? Or that the dissection can be created using straight cuts entirely across the pieces (and more cuts across the resulting pieces: the 4 piece triangle->square would qualify). Other possible (dis)allowances are cuts that terminate without completely crossing the piece, but meet up with other cuts to separate pieces; cuts which turn at vertices (hard to actually cut), and cuts that make holes, which must be initiated by piercing the piece (such as my example with a triangular hole). And maybe cuts that aren't entirely used, that extend into a piece without separating it.
Rich
----- Quoting Warren D Smith <warren.wds@gmail.com>:
Any polygon in Euclidean plane is scissor-congruent to any other of the same area, and this is accomplishable entirely with straight cuts (no curves ever needed). I think that was first shown by Bolyai.
So if you want an example where curved cuts are needed, then you need the two shapes to include at least one with a curved boundary. And then it is trivial.
Make shape #1 be a square. Shape #2 is got by cutting square in two via an S-curve, rotate pieces, glue along common straight line. Obviously a curved cut is needed to get from 1 to 2.
-- Warren D. Smith