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This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a cyclic quadrilateral that can be dissected into a triangle with only 2 intermediate pieces. By the WBG theorem, for dissection from any polygon to any other polygon of same area can be done via a finite number of pieces.

Question: Which convex quadrilateral is the hardest to dissect into a triangle - in the sense that the least number of intermediate pieces needed for its dissection into some triangle of the same area is the maximum (the triangle itself could be freely chosen to minimize the number of pieces)?

Remark: Indeed, if a quad or in general, a convex m-gon is to be dissected into some polygon with n sides, the least number of pieces needed appears to have an upper bound that is a function of m and n (iow, the minimum number of intermediate pieces cannot be arbitrarily high for any dissection) - this appears to follow from the proof of WBG theorem given in above wiki page, where the dissection proceeds via an intermediate rectangle that has unit width. Our question asks for a tighter upper bound in the case where n = 3.

A wider algorithmic question could also be asked where a given m-gon is to be dissected into some equal area polygon with only its number of sides n fixed such that the number of intermediate pieces is minimized.

This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a cyclic quadrilateral that can be dissected into a triangle with only 2 intermediate pieces. By the WBG theorem, for dissection from any polygon to any other polygon of same area can be done via a finite number of pieces.

Question: Which convex quadrilateral is the hardest to dissect into a triangle - in the sense that the least number of intermediate pieces needed for dissecting it into some triangle of the same area is the maximum (the triangle itself could be freely chosen to minimize the number of pieces)?

Remark: Indeed, if a quad or in general, a convex m-gon is to be dissected into some polygon with n sides, the least number of pieces needed has an upper bound that is a function of m and n (iow, the minimum number of intermediate pieces cannot be arbitrarily high for any dissection) - this appears to follow from the proof of WBG theorem given in above wiki page, where the dissection proceeds via an intermediate rectangle of unit width. Our question asks for a tighter upper bound in the case where n = 3.

A wider algorithmic question could also be asked where a given m-gon is to be dissected into some equal area polygon with only its number of sides n fixed such that the number of intermediate pieces is minimized - the question above is of course, the case m = 4, n=3.

Guess: The 'minimal intermediate pieces dissection' of any convex m-gon into any convex n-gon is always such that all intermediate pieces are convex.

This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a cyclic quadrilateral that can be dissected into a triangle with only 2 intermediate pieces. By the WBG theorem, for dissection from any polygon to any other polygon of same area can be done via a finite number of pieces.

Question: Which convex quadrilateral is the hardest to dissect into a triangle - in the sense that the least number of intermediate pieces needed for its dissection into some triangle of the same area is the maximum (the triangle itself could be freely chosen to minimize the number of pieces)?

Remark: Indeed, if a quad or in general, a convex m-gon is to be dissected into some polygon with n sides, the least number of pieces needed appears to have an upper bound that is a function of m and n (iow, the minimum number of intermediate pieces cannot be arbitrarily high for any dissection) - this appears to follow from the proof of WBG theorem given in above wiki page, where the dissection proceeds via an intermediate rectangle that has unit width. Our question asks for a tighter upper bound in the case where n = 3.

A wider algorithmic question could also be asked where a given m-gon is to be dissected into some equal area polygon with only number of sides n fixed with the number of intermediate pieces is minimized.

This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a cyclic quadrilateral that can be dissected into a triangle with only 2 intermediate pieces. By the WBG theorem, for dissection from any polygon to any other polygon of same area can be done via a finite number of pieces.

Question: Which convex quadrilateral is the hardest to dissect into a triangle - in the sense that the least number of intermediate pieces needed for its dissection into some triangle of the same area is the maximum (the triangle itself could be freely chosen to minimize the number of pieces)?

Remark: Indeed, if a quad or in general, a convex m-gon is to be dissected into some polygon with n sides, the least number of pieces needed appears to have an upper bound that is a function of m and n (iow, the minimum number of intermediate pieces cannot be arbitrarily high for any dissection) - this appears to follow from the proof of WBG theorem given in above wiki page, where the dissection proceeds via an intermediate rectangle that has unit width. Our question asks for a tighter upper bound in the case where n = 3.

A wider algorithmic question could also be asked where a given m-gon is to be dissected into some equal area polygon with only its number of sides n fixed such that the number of intermediate pieces is minimized.

This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a cyclic quadrilateral that can be dissected into a triangle with only 2 intermediate pieces. By the WBG theorem, for dissection from any polygon to any other polygon of same area can be done via a finite number of pieces.

Question: Given any finite number n, can one construct a quadrilateral (convex, say) that has a dissection into a triangle via n intermediate pieces - and not via a lesser number of pieces? Or is there some upper bound on n - ie if a convex m-gon is to be dissected into a triangle, the least number of pieces needed has an upper bound that is a function of m?

Note: The final triangle only needs to be some triangle with same area as the quad. I couldn't find any literature on algorithms that dissect an n-gon into some triangle via least number of intermediate pieces.

This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a cyclic quadrilateral that can be dissected into a triangle with only 2 intermediate pieces. By the WBG theorem, for dissection from any polygon to any other polygon of same area can be done via a finite number of pieces.

Question: Which convex quadrilateral is the hardest to dissect into a triangle - in the sense that the least number of intermediate pieces needed for its dissection into some triangle of the same area is the maximum (the triangle itself could be freely chosen to minimize the number of pieces)?

Remark: Indeed, if a quad or in general, a convex m-gon is to be dissected into some polygon with n sides, the least number of pieces needed appears to have an upper bound that is a function of m and n (iow, the minimum number of intermediate pieces cannot be arbitrarily high for any dissection) - this appears to follow from the proof of WBG theorem given in above wiki page, where the dissection proceeds via an intermediate rectangle that has unit width. Our question asks for a tighter upper bound in the case where n = 3.

A wider algorithmic question could also be asked where a given m-gon is to be dissected into some equal area polygon with only number of sides n fixed with the number of intermediate pieces is minimized.