A Question on Dissecting Triangles
The Wallace-Bolyai-Gerwein theorem answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by translations and rotations. The answer: this can be done if and only if two polygons have the same area.
It is easy to see this: An equilateral triangle can be cut into 2 identical 30-60-90 degrees right triangles which can be patched together to form a 30-30-120 degrees triangle. So, via 2 intermediate pieces, we can go from an equilateral triangle to the 30-30-120 triangle of the same area.
Question: Find two triangles equal area T1 and T2 such that T1 can be cut into 3 (and not less than 3) pieces which can be reassembled into T2. Not sure if such a T1-T2 pair exists.
One can readily generalize and ask for equal area {T1-T2} pairs which can be dissected into each other via n (and not less than n) intermediate pieces. Note that we are not asking about the general triangle to any other equal area triangle dissection but for specific triangle pairs.
Remark: n cannot be arbitrarily large as per the W-B-G theorem. The method based on Montucla's dissection of one rectangle to another equal area rectangle with specified length (as given in Greg Frederickson's 'Dissections - Plane and Fancy') might give a large (but bounded) number of intermediate pieces.
Further question: What can one say about equal area {T1-T2} pairs which are the worst for mutual dissection - ie for which the number of intermediate pieces is the highest possible? A guess is that a triangle pair with equal area and equal perimeter is bad for mutual dissection.
Further Note - Going to quadrilaterals from triangles: we first note the problem is trivial for rectangles; indeed, for any rectangle R and any n, there is a different rectangle with same area as R and n times as long which can be got by cutting R into n equal, length-parallel strips and attaching them end to end. For general quadrilaterals, things seem less trivial. For any cyclic quad Q (each pair of opposite vertices add to 180 degrees), it appears that we can cut Q into 2 pieces and patch these to get another quad Q' different from Q. But for non-cyclic Q's, one is not sure of finding such a Q' even for n =2. For general n, things appear harder.
Here is a discussion page on these questions:
https://mathoverflow.net/questions/350735/on-dissecting-a-triangle-into-another-triangle
It is easy to see this: An equilateral triangle can be cut into 2 identical 30-60-90 degrees right triangles which can be patched together to form a 30-30-120 degrees triangle. So, via 2 intermediate pieces, we can go from an equilateral triangle to the 30-30-120 triangle of the same area.
Question: Find two triangles equal area T1 and T2 such that T1 can be cut into 3 (and not less than 3) pieces which can be reassembled into T2. Not sure if such a T1-T2 pair exists.
One can readily generalize and ask for equal area {T1-T2} pairs which can be dissected into each other via n (and not less than n) intermediate pieces. Note that we are not asking about the general triangle to any other equal area triangle dissection but for specific triangle pairs.
Remark: n cannot be arbitrarily large as per the W-B-G theorem. The method based on Montucla's dissection of one rectangle to another equal area rectangle with specified length (as given in Greg Frederickson's 'Dissections - Plane and Fancy') might give a large (but bounded) number of intermediate pieces.
Further question: What can one say about equal area {T1-T2} pairs which are the worst for mutual dissection - ie for which the number of intermediate pieces is the highest possible? A guess is that a triangle pair with equal area and equal perimeter is bad for mutual dissection.
Further Note - Going to quadrilaterals from triangles: we first note the problem is trivial for rectangles; indeed, for any rectangle R and any n, there is a different rectangle with same area as R and n times as long which can be got by cutting R into n equal, length-parallel strips and attaching them end to end. For general quadrilaterals, things seem less trivial. For any cyclic quad Q (each pair of opposite vertices add to 180 degrees), it appears that we can cut Q into 2 pieces and patch these to get another quad Q' different from Q. But for non-cyclic Q's, one is not sure of finding such a Q' even for n =2. For general n, things appear harder.
Here is a discussion page on these questions:
https://mathoverflow.net/questions/350735/on-dissecting-a-triangle-into-another-triangle
posted by R.Nandakumar @ 5:06 AM
0 Comments:
Post a Comment
<< Home