Another Overflow Question - copied
Am copying a question recorded on January 30th, 2022 at https://mathoverflow.net/questions/414979/to-choose-a-set-of-n-triangles-which-together-form-largest-number-of-triangles
It had got deleted by a bot. Not sure if the efforts to get it undeleted will succeed so am copying the contents here:
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Question: Given an integer n, to form a set of m triangles (where m is to be minimized; no constraints on these m triangles) such that these m triangles can all together be arranged without gaps or overlaps to form n different triangles.
Remarks:
The question is trivial for n =2. One takes say, two 30-60-90 right triangles which can together form either an equilateral triangle or a 30-30-120 isosceles triangle.
As given in the above linked post, this question can seen as a dissection problem. Indeed, that for any n, there do exist sets of triangles which can all together form n equal area but non-congruent triangles can be shown from the arguments given in the proof of the Wallace Bolyai Gerwein theorem (https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem) but then the number of triangular pieces could be very large and here, we are looking at minimizing the number of such pieces.
A possible variant is for the set of m pieces allowed to contain polygons other than triangles. A constrained version of the question would be to insist that the dissection be vertex-to-vertex (when the intersection of any two pieces in the layout is either a common side, a common vertex, or empty).
Further question: Given integer n, to find a set of n convex regions such that by putting them all together we can form the largest number of different convex polygonal regions.
Remark: A regular (n-1)-gon with unit side and n-1 very flat and mutually non-congruent triangles all having longest side of unit length can together form (n-2)! different convex polygons. Not sure if this is the best one can do.
Note: 3D and Hyperbolic geometry versions of these questions could also be of interest.
A further thought(17th Feb, 2022): The basic question seems to stay nontrivial even if we allow gaps or overlaps (maybe not both together) to exist between triangles as they form the layouts and only insist that the final layouts have a proper triangular outline.
It had got deleted by a bot. Not sure if the efforts to get it undeleted will succeed so am copying the contents here:
--------------------------
Question: Given an integer n, to form a set of m triangles (where m is to be minimized; no constraints on these m triangles) such that these m triangles can all together be arranged without gaps or overlaps to form n different triangles.
Remarks:
The question is trivial for n =2. One takes say, two 30-60-90 right triangles which can together form either an equilateral triangle or a 30-30-120 isosceles triangle.
As given in the above linked post, this question can seen as a dissection problem. Indeed, that for any n, there do exist sets of triangles which can all together form n equal area but non-congruent triangles can be shown from the arguments given in the proof of the Wallace Bolyai Gerwein theorem (https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem) but then the number of triangular pieces could be very large and here, we are looking at minimizing the number of such pieces.
A possible variant is for the set of m pieces allowed to contain polygons other than triangles. A constrained version of the question would be to insist that the dissection be vertex-to-vertex (when the intersection of any two pieces in the layout is either a common side, a common vertex, or empty).
Further question: Given integer n, to find a set of n convex regions such that by putting them all together we can form the largest number of different convex polygonal regions.
Remark: A regular (n-1)-gon with unit side and n-1 very flat and mutually non-congruent triangles all having longest side of unit length can together form (n-2)! different convex polygons. Not sure if this is the best one can do.
Note: 3D and Hyperbolic geometry versions of these questions could also be of interest.
A further thought(17th Feb, 2022): The basic question seems to stay nontrivial even if we allow gaps or overlaps (maybe not both together) to exist between triangles as they form the layouts and only insist that the final layouts have a proper triangular outline.
posted by R.Nandakumar @ 12:48 PM
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