高维立方晶格$\mathbb{Z}^n$的仿射二面体子群与准晶体学
Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices $\mathbb{Z}^n$ and Quasicrystallography
AI总结：
该研究通过将二面体群$I_2(h)$的仿射扩张作为仿射$B_n$子群，从高维立方晶格投影视角研究准晶体，推导了其投影规律并验证了低维实例，为$h$重对称准晶体提供了新视角。
摘要：
准晶体可描述为高维立方晶格的投影，而作为仿射$B_n$子群的二面体群$I_2(h)$（阶为$2h$，$h=2n$为Coxeter数）的特定仿射扩张，为$h$重对称准晶体学提供了不同视角。仿射$I_2(h)$构造为仿射$B_n$的子群，后者是立方晶格$\mathbb{Z}^n$的对称性群。具有局部二面体对称性（阶为$2h$）的无限离散群作用于同心$h$边形，这些$h$边形由将$2^n$个顶点的立方晶格Voronoi胞投影到Coxeter平面得到。Voronoi胞面-面拼接，因此导致Coxeter平面上菱形瓦片存在重叠的拼接。需注意，投影后的Voronoi胞是$h$个菱形拼接$h$边形的拷贝经$2π/h$角旋转后的重叠。在对具有仿射对称性$\tilde{B}_n$的晶格$\mathbb{Z}^n$及其仿射二面体子群$\tilde{I}_2(h)$进行一般讨论后，结合实例推导了其到Coxeter平面的投影。呈现了具有仿射对称性$\tilde{B}_n$（$n=1,2,3,4,5$）的立方晶格，结果显示晶格$B_3$的投影得到六边形晶格，$B_4$的投影描述具有8重局部对称性的Ammann-Beenker准晶体晶格，$B_5$的投影描述具有厚、薄菱形的10重局部对称性准晶体结构。由此可直接证明，更高维立方晶格到Coxeter平面的投影可能产生具有12重、18重等对称性的准晶体结构。
英文摘要：
Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extensions of the dihedral group $I_2(h)$ of order $2h$, $h=2n$ being the Coxeter number, as a subgroup of affine $B_n$ offers a different perspective to $h$-fold symmetric quasicrystallography. Affine $I_2(h)$ is constructed as the subgroup of the affine $B_n$, the symmetry of the cubic lattice $\mathbb{Z}^n$. The infinite discrete group with local dihedral symmetry of order $2h$ operates on the concentric h-gons obtained by projecting the Voronoi cell of the cubic lattice with $2^n$ vertices onto the Coxeter plane. Voronoi cells tile the space facet to facet, consequently, leading to the tilings of the Coxeter plane with some overlaps of the rhombic tiles. It is noted that the projected Voronoi cell is the overlap of $h$ copy of the $h$-gons tiled with some rhombi and rotated by the angle $2π/h$. After a general discussion on the lattice $\mathbb{Z}^n$ with the affine symmetry $\tilde B_n$ and its affine dihedral subgroup $\tilde I_2(h)$ its projection onto the Coxeter plane has been worked out with some examples. The cubic lattices with affine symmetry $\tilde B_n$ $(n=1,2,3,4,5)$ have been presented and shown that the projection of the lattice $B_3$ leads to the hexagonal lattice, the projection of the lattice $B_4$ describes the Ammann-Beenker quasicrystal lattice with 8-fold local symmetry and the projection of the lattice $B_5$ describes a quasicrystal structure with local 10-fold symmetry with thick and thin rhombi. It is then straight forward to show that the projections of the cubic lattices with even higher dimensions onto the Coxeter plane may lead to the quasicrystal structures with 12-fold, 18-fold symmetries and so on.
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