On Visible Tilings

摘要

A tiling of a d-dimensional cube C is a finite collection of d-dimensional cubes, or tiles, whose interiors are pairwise disjoint and whose union is C. A tiling is said to be visible if every tile has a face on the boundary of C. It is easy to see that if {A_1, A_2,...,A_n} is a visible tiling of the unit cube (square) in the case d = 2, and if s_i is the side length of A_i for each i, then Si< 4 and that the sum can be arbitrarily as close to 4 for appropriately chosen tiles. Extrapolating from this we show that if C is the unit cube in dimension d and {A_1,A_2,..., A_n} is a visible tiling, then < 2d and the sum can be made arbitrarily close to 2d.