[math-fun] turtleneck sweater for octopi
http://home.btconnect.com/GavinTheobald/HTML/Octagon.html#Hexadecagon I don't recall curved cuts in a polygonal dissection. Can't the circular segments be replaced by segments of hexadecagons? --rwg Are curves essential to any known minimal dissection?
Many of these prompt the question "how on earth were they constructed?" Which the author does discuss, under http://home.btconnect.com/GavinTheobald/HTML/Methods.html WFL On 4/14/16, Bill Gosper <billgosper@gmail.com> wrote:
http://home.btconnect.com/GavinTheobald/HTML/Octagon.html#Hexadecagon I don't recall curved cuts in a polygonal dissection. Can't the circular segments be replaced by segments of hexadecagons? --rwg Are curves essential to any known minimal dissection? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
rwg> Are curves essential to any known minimal dissection? I think (no proof) that you can force a required curve: Consider a big, asymmetric object, perhaps a scalene triangle. Punch a smallish asymmetric hole in it, perhaps a second scalene triangle. This is the source polygon. The target is the same object, with the internal hole rotated. The rotated hole must overlap the original; the rotation amount must be an irrational multiple of pi. This has a two piece dissection, based on cutting a circle around the hole, and rotating the internal piece. I believe I've specified enough restrictions to make polygonal solutions suboptimal. The spoiled ideas include simply cutting a new hole and filling the old hole; flipping either inner or outer triangle; and cutting portions of either triangle and refitting around the edge to do an ersatz rotation. Rich
That reminds me of one of the oldest unsolved problems in geometry: show that 4 pieces are minimal for a scissors dissection of a square to an equilateral triangle. The classic dissection (believed to be optimal) is here: https://oeis.org/A110312/a110312_3v.jpg and this "4" is the conjectured value of a(4) in https://oeis.org/A110000 (one of my favorite sequences, but regrettably only one term, a(3)=1, is known to be correct) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Apr 14, 2016 at 11:43 AM, <rcs@xmission.com> wrote:
rwg> Are curves essential to any known minimal dissection?
I think (no proof) that you can force a required curve: Consider a big, asymmetric object, perhaps a scalene triangle. Punch a smallish asymmetric hole in it, perhaps a second scalene triangle. This is the source polygon. The target is the same object, with the internal hole rotated. The rotated hole must overlap the original; the rotation amount must be an irrational multiple of pi.
This has a two piece dissection, based on cutting a circle around the hole, and rotating the internal piece.
I believe I've specified enough restrictions to make polygonal solutions suboptimal. The spoiled ideas include simply cutting a new hole and filling the old hole; flipping either inner or outer triangle; and cutting portions of either triangle and refitting around the edge to do an ersatz rotation.
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This is part of a letter by Morris W. Hirsch (who was my thesis advisor) to the Notices of the AMS (2014): ----- The term "scissor congruent," introduced in [1], refers to a pair of planar convex bodies A = A_1 u ··· u A_n, B = B_1 u ··· u B_n, where A_i and B_i are congruent compact 2-cells, andfor i,j the interiors of A_i and A_j (respectively, B_i and B_j) are disjoint. The main result is that A and B are scissor congruent iff they have the same area and their boundaries have partitions ∂A = I_1 u ··· u I_m, ∂B = J_1 u ··· u J_m, where I_k and J_k are congruent 1-cells. It follows that a square and a circle are not scissor congruent, andscissor congruent ellipses are congruent. No analog in higher dimensions is known. The term "scissor congruent" was introduced by Dubins et al. in [1] to describe a pair of convex planar bodies A, B having cell decompositions such that there is a bijection between the two-cells of A and those of B with corresponding two-cells congruent through rigid motions. The main result in [1] is that A and B are scissor congruent iff they have the same area and their boundaries have partitions ∂A = I_1 u ... u I_m, ∂B = J_1 u ... u J_m with I_k and J_k congruent arcs. It follows that a square is not scissor congruent to any strictly convex body, and if two ellipses are scissor congruent, then they are congruent. References [1] L. Dubins, M. Hirsch, and J. Karush, Scissor congruence, Israel J. Math. 1 (1963), 239–247. [2] S. Gold, The sets that are scissor congruent to an unbounded con- vex subset of the plane Trans. Amer. Math. Soc. 215 (1976), 99–117. [3] R. Gardner, A problem of Sallee on equidecomposable convex bod- ies, Proc. Amer. Math. Soc. 94 (1985), 329–332. [4] C. Richter, Affine congruence by dissection of discs—appropriate groups and optimal dissections, J. Geom. 84 (2005), 117–132. [5] ________, Squaring the circle via af- fine congruence by dissection with smooth pieces, Beiträge Algebra Geom. 48 (2007), no. 2, 423–434. ----- —Dan
On Apr 14, 2016, at 5:18 AM, Bill Gosper <billgosper@gmail.com> wrote:
http://home.btconnect.com/GavinTheobald/HTML/Octagon.html#Hexadecagon I don't recall curved cuts in a polygonal dissection. Can't the circular segments be replaced by segments of hexadecagons?
Are curves essential to any known minimal dissection?
My favorite scissors dissection problem is this one.
From a square S=ABCD with center M and vertices labelled clockwise from its northwest corner, cut out the triangle T = CDM.
Next, reattach T to what was the original square by rotating it ninety degrees clockwise about C until DC coincides with BC. Here's picture of the resulting 6 sided figure. Call it H https://www.flickr.com/photos/thane/26339632922/in/dateposted-public/ Let's say we want to cut H into two pieces that can be reassembled to form a new square congruent to S. Of course, one way to do that is simply return the triangle we cut out back to where it came from. The puzzle is to find a different solution. On Thu, Apr 14, 2016 at 9:21 AM, Dan Asimov <asimov@msri.org> wrote:
This is part of a letter by Morris W. Hirsch (who was my thesis advisor) to the Notices of the AMS (2014):
----- The term "scissor congruent," introduced in [1], refers to a pair of planar convex bodies
A = A_1 u ··· u A_n, B = B_1 u ··· u B_n,
where A_i and B_i are congruent compact 2-cells, andfor i,j the interiors of A_i and A_j (respectively, B_i and B_j) are disjoint. The main result is that A and B are scissor congruent iff they have the same area and their boundaries have partitions
∂A = I_1 u ··· u I_m, ∂B = J_1 u ··· u J_m,
where I_k and J_k are congruent 1-cells. It follows that a square and a circle are not scissor congruent, andscissor congruent ellipses are congruent. No analog in higher dimensions is known.
The term "scissor congruent" was introduced by Dubins et al. in [1] to describe a pair of convex planar bodies A, B having cell decompositions such that there is a bijection between the two-cells of A and those of B with corresponding two-cells congruent through rigid motions.
The main result in [1] is that A and B are scissor congruent iff they have the same area and their boundaries have partitions
∂A = I_1 u ... u I_m, ∂B = J_1 u ... u J_m
with I_k and J_k congruent arcs. It follows that a square is not scissor congruent to any strictly convex body, and if two ellipses are scissor congruent, then they are congruent.
References [1] L. Dubins, M. Hirsch, and J. Karush, Scissor congruence, Israel J. Math. 1 (1963), 239–247.
[2] S. Gold, The sets that are scissor congruent to an unbounded con- vex subset of the plane Trans. Amer. Math. Soc. 215 (1976), 99–117.
[3] R. Gardner, A problem of Sallee on equidecomposable convex bod- ies, Proc. Amer. Math. Soc. 94 (1985), 329–332.
[4] C. Richter, Affine congruence by dissection of discs—appropriate groups and optimal dissections, J. Geom. 84 (2005), 117–132.
[5] ________, Squaring the circle via af- fine congruence by dissection with smooth pieces, Beiträge Algebra Geom. 48 (2007), no. 2, 423–434. -----
—Dan
On Apr 14, 2016, at 5:18 AM, Bill Gosper <billgosper@gmail.com> wrote:
http://home.btconnect.com/GavinTheobald/HTML/Octagon.html#Hexadecagon I don't recall curved cuts in a polygonal dissection. Can't the circular segments be replaced by segments of hexadecagons?
Are curves essential to any known minimal dissection?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
participants (6)
-
Bill Gosper -
Dan Asimov -
Fred Lunnon -
Neil Sloane -
rcs＠xmission.com -
Thane Plambeck