关于k-爬行四面体的不存在性
On the nonexistence of k-reptile tetrahedra
摘要：
一个d维单形S称为k-爬行形，如果它可以被单形S_1，S_2，. S_k都是全等且相似的。当d=2时，k-爬行单纯形（三角形）对于k的许多值都存在，它们已经被Snover，Waiveris和Williams完全刻画。另一方面，当d > 2时，只有一种k-爬行单形的构造是已知的，即希尔单形，它只提供了m^d，m= 2，3，.的形式的k。我们证明了当d=3时，k-爬行单形（四面体）只存在于k=m^3时。这部分证实了赫特尔的一个猜想，他断言唯一的k-爬行四面体是希尔四面体。我们的研究一直受到理论计算机科学中的概率包标记问题的启发，该问题由Adler于2002年提出。
英文摘要：
A d-dimensional simplex S is called a k-reptile if it can be tiled without overlaps by simplices S_1,S_2,...,S_k that are all congruent and similar to S. For d=2, k-reptile simplices (triangles) exist for many values of k and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, for d > 2, only one construction of k-reptile simplices is known, the Hill simplices, and it provides only k of the form m^d, m=2,3,.... We prove that for d=3, k-reptile simplices (tetrahedra) exist only for k=m^3. This partially confirms a conjecture of Hertel, asserting that the only k-reptile tetrahedra are the Hill tetrahedra. Our research has been motivated by the problem of probabilistic packet marking in theoretical computer science, introduced by Adler in 2002.
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