标题： 凯勒的立方体铺砌猜想在高维中是错误的 显示英文标题

标题： Keller's cube-tiling conjecture is false in high dimensions

摘要： O. H. Keller 在 1930 年提出猜想，认为在任何用单位$n$-立方体对$\Bbb R^n$的铺砌中，都存在两个具有完整面共有的立方体。O. Perron 证明了该猜想对于$n\le 6$成立。我们证明了对于所有$n\ge 10$来说，存在一种用单位$n$-立方体对$\Bbb R^n$的铺砌，使得没有任何两个$n$-立方体具有完整的面共有。 显示英文摘要

摘要： O. H. Keller conjectured in 1930 that in any tiling of $\Bbb R^n$ by unit $n$-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $n\le 6$. We show that for all $n\ge 10$ there exists a tiling of $\Bbb R^n$ by unit $n$-cubes such that no two $n$-cubes have a complete facet in common.