Naskręcki Bartosz, Dauter Zbigniew, Jaskolski Mariusz
Faculty of Mathematics and Computer Science, A. Mickiewicz University, Poznan, Poland.
Macromolecular Crystallography Laboratory, NCI, Argonne National Laboratory, Argonne, USA.
Acta Crystallogr A Found Adv. 2021 Mar 1;77(Pt 2):126-129. doi: 10.1107/S2053273320016186. Epub 2021 Feb 4.
The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V - E + F = 2 or Euler characteristic χ = V - E + F - I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χ = V - E + F - I = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.
著名的三维多面体欧拉公式(V - E + F = 2)或欧拉特征(\chi = V - E + F - I = 1)（其中(V)、(E)、(F)分别是边界顶点、边和面的数量，且(I = 1)表示单个立体本身），当应用于诸如晶体学不对称单元或狄利克雷域等空间填充固体时，会以这样一种方式被修改，即它们的总和比原来的值小一个单位（即分别为(1)或(0)），本文在此证明这种修改具有普遍有效性。本文给出的关于修改后的欧拉特征(\chi = V - E + F - I = 0)的证明分为两部分。首先，通过基于晶格整数索引元素的奇偶群的简单论证，对平移晶格进行了证明。其次，利用关于欧拉特征不变性的怀特黑德定理，将该论证从晶胞扩展到其不对称单元分量。
