A 4-dimensional periodic table of chemical elements (4D PT) is presented. The 120 elements in the n = 8 system are located on vertices of a 4D-cubic lattice and specified by Cartesian coordinates (n, l, m, s) based on the four quantum numbers. Each quantum number is represented by a vector along a different spatial direction (axes) in 4D Euclidean space. The 4D PT has a fixed topology governed by Euler–Poincare-type equation and the chemical elements have a fixed connectivity with neighboring elements within the 4D PT. Various geometric transformations by rotations and elongations of vectors enable morphing of one PT to another in a continuum while preserving the 4D topology. The 4D structure exhibits principles of complementarity and zero-cyclic sum in chemical elements. Complementarity enables the extension to n = 9 elements and beyond. The 4D PT of elements extends to isotopes and also to molecules and compounds with the same underlying principles.

中文翻译：

---

化学元素的 4D 立方晶格

提出了化学元素的 4 维周期表 (4D PT)。n = 8 系统中的 120 个元素位于 4D 立方晶格的顶点上，并由基于四个量子数的笛卡尔坐标 (n, l, m, s) 指定。每个量子数由沿 4D 欧几里得空间中不同空间方向（轴）的向量表示。4D PT 具有由 Euler-Poincare 型方程控制的固定拓扑结构，并且化学元素与 4D PT 内的相邻元素具有固定的连接性。通过矢量的旋转和伸长进行的各种几何变换使一个 PT 在连续体中变形为另一个，同时保留 4D 拓扑。4D 结构展示了化学元素的互补性和零循环和的原理。互补性可以扩展到 n = 9 个元素及更多元素。