点上的周期欧几里德图
Periodic Euclidean Graphs on Integer Points
摘要：
均匀离散欧氏图是嵌入在欧几里得空间中的图，使得不同顶点之间的距离最小。如果嵌入在$n$维空间中的这样一个图在$n$线性无关平移下保持不变，则它是“$n$ -周期”的，因为它的对称群的商群除以它的对称群的平移子群是有限的。我们提出了一个比伯巴赫定理的改进：给定嵌入在对称群$\bbbS$的$n$维欧几里得空间中的$n$ -周期一致离散欧氏图，存在另一个$n$ -周期一致离散欧氏图嵌入在同一空间中，其顶点是整数点（可能模仿射变换）并且其对称群具有同构于$\bbbS$的（不一定是真的）子群。最后，我们讨论的应用程序的计算机生成的“水晶网”。
英文摘要：
A uniformly discrete Euclidean graph is a graph embedded in a Euclidean space so that there is a minimum distance between distinct vertices. If such a graph embedded in an $n$-dimensional space is preserved under $n$ linearly independent translations, it is "$n$-periodic" in the sense that the quotient group of its symmetry group divided by the translational subgroup of its symmetry group is finite. We present a refinement of a theorem of Bieberbach: given a $n$-periodic uniformly discrete Euclidean graph embedded in a $n$-dimensional Euclidean space of symmetry group $\bbbS$, there is another $n$-periodic uniformly discrete Euclidean graph embedded in the same space whose vertices are integer points (possibly modulo an affine transformation) and whose symmetry group has a (not necessarily proper) subgroup isomorphic to $\bbbS$. We conclude with a discussion of an application to the computer generation of "crystal nets".
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