Any triangle can be dissected into three mirror-symmetric pieces. If we are limited to just two pieces, is the number of (non-congruent) triangles that admit a dissection (into mirror-symmetric pieces) finite or infinite?
By chance, recently I was looking through some back issues of Journal of Recreational Mathematics, and saw this related problem: >From Journal of Recreational Mathematics, v. 27 (1995) no. 4, p. 302. Problem 2283. Dissection for Coincidence by Heinrich Hemme, Aachen, Germany On page 56 of Charles W. Trigg's Mathematical Quickies (McGraw-Hill Book Co., New York, 1967), there is the following problem: Q 204: Consider two congruent triangles that can be brought into coincidence only by rotation of one of them through a third dimension. How, by cutting the triangles, could coincidence be effected by motion in a plane only? The solution (by Louis R. Chase) to the problem is given on page 172 as follows (Figure 16): The result can be achieved by dissecting one of the triangles into isosceles triangles. A right triangle will be divided by the median upon the hypotenuse. An obtuse triangle can first be divided into right triangles by the altitude upon the longest side, An acute triangle will be divided into three isosceles triangles by the circumradii to the vertices. It can be shown that this solution may be optimized even further. To that end, the problem might be posed as follows: What types of triangles can be transformed into their respective mirror images by dissection into (a) one piece, (b) two pieces, (c) three pieces? OK, that quote is a mouthful, er, a keyboardful. The connection to Veit's problem is that the dissections for (b), the interesting part, come from two-piece dissections into pieces with reflective symmetry. Is it obvious ... provable ... true(?!) that a two piece dissection must be into pieces with reflective symmetry. Hmmm ... not true for isosceles triangles, when a one-piece "dissection" suffices. For scalene triangles, the question still stands. Solutions to the JRM problem by the Proposer (Hemme), Markus Goetz and Helmut Postl, were published in JRM volume 28, (1996-97) no. 4, on pages 313-315. There is no claim that they have found all triangles which admit such a two-piece dissection. I have not located Trigg's book, so I do not know if it has any information about two-piece dissections. I do recall, many years ago, being given the problem of finding such a two-piece dissection for a particular triangle, given by its angles. I'm not certain any more what those angles were, perhaps 50, 55 and 75 degrees. In any event, those angles work for the particular dissection I eventually found, which is one of several the JRM solvers found. Veit, or others, do you have a reference for the two-piece dissection problem? Perhaps Trigg's book has one, or is the origin of it, but I haven't seen the book. Michael Reid
Any triangle can be dissected into three mirror-symmetric pieces. If we are limited to just two pieces, is the number of (non-congruent) triangles that admit a dissection (into mirror-symmetric pieces) finite or infinite?
Veit
On Feb 13, 2012, at 5:42 PM, Veit Elser wrote:
Okay, I'll try again:
Which triangles can be dissected into two pieces, both of which are mirror-symmetric?
On Feb 13, 2012, at 4:09 PM, Dan Asimov wrote:
Veit -- Can the word "just" be omitted without changing the meaning here, or is it in some way limiting the number of mirror-symmetric pieces that such a triangle can be dissected into?
Thanks,
Dan
Veit asked:
<< Which triangles can be dissected into just two mirror-symmetric pieces?