关于$\mathbb R^3$和$\mathbb R^4$中$k$ -爬行单形的不存在性
On the nonexistence of $k$-reptile simplices in $\mathbb R^3$ and $\mathbb R^4$
摘要：
一个$d$维单形$S$称为$k$ -爬虫形（或$k$ -爬虫形单形），如果它可以被内部不相交的$k$单形平铺，这些单形都与$S$相互全等和相似。对于$d=2$，三角形$k$ -爬行动物存在于所有形式为$a^2, 3a^2$或$a^2 + b^2$的$k$中，它们已经被Snover、Waiveris和Williams完全表征。另一方面，唯一已知的$d \ge 3$的$k$ -爬行单纯形具有$k = m^d$，其中$m$是正整数。我们大大简化了Matoušek和第二作者的证明，对于$d=3$，$k$ -爬行四面体只能存在于$k=m^3$。然后，我们证明了一个较弱的模拟$d=4$的结果表明，四维$k$ -爬行单纯形可以存在，只有$k=m^2$。
英文摘要：
A $d$-dimensional simplex $S$ is called a $k$-reptile (or a $k$-reptile simplex) if it can be tiled by $k$ simplices with disjoint interiors that are all mutually congruent and similar to $S$. For $d=2$, triangular $k$-reptiles exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only $k$-reptile simplices that are known for $d \ge 3$, have $k = m^d$, where $m$ is a positive integer. We substantially simplify the proof by Matoušek and the second author that for $d=3$, $k$-reptile tetrahedra can exist only for $k=m^3$. We then prove a weaker analogue of this result for $d=4$ by showing that four-dimensional $k$-reptile simplices can exist only for $k=m^2$.
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