The notched cube tiles Rn

✍ Scribed by Sherman Stein

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 191 KB
• Volume: 80
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(90)90252-d

✦ Synopsis

Herb Holden once observed that packing material consisting of copies of a cube of side 2 inches from which a unit corner cube is removed tiles R3 by translates. In [l] Conlan generalized this fact, proving that a polytope obtained from a unit cube in R3 by deleting a corner box of which two dimensions are rational and one dimension is a unit fraction also tiles R3 by translates. We generalize this result by replacing R3 by R" and removing any restriction on the dimensions of the deleted box.

Our main result is the following. Theorem 1. Let al, u2, . . . , a,, be n real numbers, 0 < Ui < 1, 1 G i s II. Let K be the polytope obtained from the unit cube {(x1, x2, . . . , x,): 0 <xi c l} by deleting the rectangular box {(x1, x2, . . . , x,): 1 -ai <xi 6 1). Then R" cun be tiled by translates of K.