Algebraic theory of brick packing II

Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases

“…Our tools are graphic combinatorics and algebra. We also revisit some previous results of Barnes [11] [12], relevant for signed tilings with complex/rational weights, and explain what they imply for , , n n T T n + even.…”

Packing Boxes with Bricks

“…In some problems our bricks will be ?/-dimensional cubes, and in other problems we allow all d\ orientations of one given brick. Our results are special cases of deeper theorems of Barnes [1,2] and of Katona and Sz?sz [19], who provide necessary and asymptotically sufficient conditions for packings of boxes by sets of given integer bricks. Barnes associates polynomials with bricks and boxes and then applies ideas from algebraic geometry.…”

“…The elegance and power of de Bruijn's method inspired a host of variations and generalizations [1,2,4,19,21,36]. We now give a different proof of one of these generalizations [21].…”

“…In some fortuitous situations, the necessary and sufficient conditions coincide. We avoid the wide array of sophisticated algebraic techniques for solving packing problems [1,2,9,10,30,32,33,35] in favor of more elementary combinatorial methods. 15 10 12 16 (a) (b) Figure 3 (a) P?rtition-and-pack strategy to tile a 37 x 32 rectangle, (b) Any 6 x 1 rect angle encloses a net charge of ?6, 0, or +6…”

Packing 1×2×4 bricks in a cubical box

“…Obviously such encodings can be particularly helpful in packing higher-dimensional boxes, for they transform the physical constraints into purely algebraic ones and thus somewhat mitigate the difficulty of not being able to physically visualize an evolving arrangement of bricks. Among the recent works on packings, Barnes' papers [1,2] In this paper we introduce a polynomial encoding of packings which transforms the packing problem into an exercise in polynomial arithmetic. In Section 2, we show how some packing problems can be elegantly solved by algebraic methods.…”