### Asymptotically Optimal Box Packing Theorems

#### Abstract

Given a protoset of $d$-dimensional polyominoes, we ask which boxes can be packed by the protoset. In some cases, it may be too difficult to give a complete answer to this question, so we ask the easier question about determining all sufficiently large boxes that can be packed. (We say that a box is "sufficiently large" if all edge lengths are ${} \ge C$ for some large $C$.) We give numerous examples (mostly $2$-dimensional) where we can answer this easier question. The various techniques involved are: checkerboard-type colorings/numberings (tile homology), the boundary word method of Conway and Lagarias (tile homotopy), ad hoc geometric arguments, and a very nice theorem of Barnes. Barnes' Theorem asserts that all necessary conditions for a box to be packable can be given in a certain form, and these conditions are also sufficient for large boxes.

Barnes' Theorem has not received the appreciation it deserves. We give a new, purely combinatorial proof of this important result. (Barnes' original proof uses techniques of algebraic geometry.) In the special case that all the prototiles are boxes themselves, we show how to determine all sufficiently large boxes that they pack. We prove a theorem based on Barnes' result that reduces this to a straightforward calculation.