A FEASIBLE ALGORITHM FOR CHECKING n-SCISSORS CONGRUENCE OF POLYHEDRA IN R

While in R2, every two polygons of the same area are scissors congruent (i.e., they can be both decomposed into the same finite number of pair-wise congruent polygonal pieces), in R3, there are polyhedra P and P ′ of the same volume which are not scissors-congruent. It is therefore necessary, given two polyhedra, to check whether they are scissorscongruent (and if yes – to find the corresponding decompositions). It is known that while there are algorithms for performing this checking-and-finding task, no such algorithm can be feasible – their worst-case computation time grows (at least) exponentially, so even for reasonable size inputs, the computation time exceeds the lifetime of the Universe. It is therefore desirable to find cases when feasible algorithms are possible. In this paper, we show that for each dimension d, a feasible algorithm is possible if we fix some integer n and look for n-scissors-congruence in Rd – i.e., for possibility to represent P and P ′ as a union of ≤ n simplexes. 1 Formulation of the Problem Scissors congruence: brief reminder. In a plane, every two polygons P and P ′ of equal area A(P ) = A(P ′) are scissors congruent (equidecomposable) – i.e., they can be both decomposed into the same finite number of pair-wise congruent polygonal pieces: P = P1 ∪ . . . ∪ Pp, P ′ = P ′ 1 ∪ . . . ∪ P ′ p, and Pi ∼ P ′ i . In one of the 23 problems that D. Hilbert formulated in 1900 as a challenge to the 20 century mathematics, namely, in Problem No. 3, Hilbert asked whether every two polyhedra P and P ′ with the same volume V (P ) = V (P ′) are scissors congruent[8]. This problem was the first to be solved: already in 1900, Dehn proved [3, 4] that there exist a tetrahedron of volume 1 which is not scissors congruent with a unit cube; see, e.g., [1, 5, 6, 12] for a detailed overview. Algorithm for checking scissors congruence. Let us consider polyhedra which can be constructed by geometric constructions. It is well known that for such polyhedra, all vertices have algebraic coordinates (i.e., values which are roots of polynomials with integer coefficients); see, e.g., [2]. In [10], we described an algorithm for checking whether two polyhedra with algebraic coordinates in R (or in R) are scissor congruent. When the polyhedra are scissor congruent, this algorithm also enables us to find the corresponding scissor decomposition Pi and P ′ i . In general, the task of checking scissors congruence and – if yes – finding the corresponding decompositions is not feasible. In [9], we have shown that in general, the problem of constructing the corresponding scissor decomposition requires computation time t which grows exponentially with the size s of the input: t ≥ c for some c > 1. In theoretical computer science, such algorithms are known as not feasible, since already for reasonable sizes s, the time c exceeds the lifetime of the Universe – and thus, it is not possible to perform these computations. Only algorithms whose computation time is bounded by a polynomial of the size s of the input are considered ro be feasible; see, e.g., [11, 13]. A natural question. Since the general problem is not feasible, it is desirable to find feasible cases. What we do in this paper. In this paper, we show that for every number n, there is a feasible algorithm for checking n-scissors congruence – – possibility to represent P and P ′ as a union of ≤ n simplexes.