[math-fun] scissor congruence
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 http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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. 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 http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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On Apr 15, 2016, at 10:11 AM, 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'm not sure I see the picture you're thinking of here.
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. 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.
The original phrase "scissor" [sic] congruence appears in that paper L. Dubins, M. Hirsch, and J. Karush, Scissor congruence, Israel J. Math. 1 (1963), 239–247. They allowed any continuous curve for the cuts (and required further that the pieces be topological disks. At least most recently, the term "scissors" [sic] congruence" is often used to mean straight cuts only, re dissections of polygons (and polyhedra). Previously, the word for equivalent the Dehn-Sydler kinds of dissection was "equidecomposable" and did not include the word "congruent". There's nice book by Boltyanskii, Equivalent and Equidescomposable Figures (English version 1963), and a paper by Rich Schwartz at https://www.math.brown.edu/~res/Papers/dehn_sydler.pdf <https://www.math.brown.edu/~res/Papers/dehn_sydler.pdf>. It just so happens (?) that there's a long and somewhat technical but (from the first few pages) apparently readable survey article on scissors congruence in the April 2016 Bulletin of the AMS. This article takes a long view, allowing all manner of decompositions that make sense in this context, as well as considering the higher dimensional cases. Re higher dimensions, I knew from Bolyanskii that in 3D, Dehn had invented and shown his Dehn invariant to be necessary for equidecomposability via isometries (1903). And much later (1965), Sydler showed it to also be sufficient. Actually, Bricard (1896) had come up with a more elementary version of the Dehn invariant, but it didn't quite prove what Dehn was able to show. What I learned from that Bulletin article is that the Dehn-Sydler theorem has been extended to 4 dimensions as well. But so far, no higher dimensions. —Dan
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Dan Asimov -
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Warren D Smith