Lower Bound Problems For Polygon Dissections

An old theorem from the first half of the nineteenth century by Lowry, Wallace, Bolyai and Gerwien asserts that any simple polygon can be dissected into a finite number of pieces and put back together to form any other simple polygon of the same area. An important research problem which attracts much attention is to decide how economically such dissection can be made. In the present paper, we discuss the minimum number of pieces needed to dissect a regular m ?gon into a regular n-gon of the same area using glass-cuts and polygonal cuts. We obtain the following results.Theorem 4.1 Any glass-cut dissection of a square into a convex n-gon of the same area requires at least -1 pieces.Theorem 4.2 Any glass -cut dissection of a regular m-gon into a regular n-gon, for m n, of the same area requires at least [] pieces.Theorem 4.3 The following bound holds for the function g for all sufficiently large non-negative integers m < n :where g(m,n) denotes the minimum number of pieces needed to dissect a regular m-gon into a regular n-gon using glass-cuts.Theorem 4.4 Any polygonal dissection of a convex n-gon into a square of the same area requires at least [] pieces.