An n-shell Lattice Model for 3D Quasicrystals**

** submitted in February 2012 to the Letters section of Nature Preface This manuscript was originally completed in 2011 under the title An n-shell Lattice Model for 3D Quasicrystals. It was submitted in February 2012 to the Letters section of Nature, following the guidelines of Nature Guide to Authors: Summary paragraph for Letters (June 10). The submission system generated the following files: 204364_0_merged_1329190564: full file 204364_0_auth_cover_letter_1612108_lzd5hm: cover letter 204364_0_figure_1610807_lz7jj5: cover artwork 204364_0_supp_1610805_lz7jfb: supplementary artworks These files were created between 19:43 and 20:00 on 13 February 2012, documenting the complete submission process. Although the manuscript was not accepted, the proposed n-shell quasicrystal lattice model represented an original structural theory at that time. All lattice-structure figures in this work were produced by the author himself. Atomic coordinates were first computed using the author’s own Atomic Environment Calculation (AEC) method, and then visualized with a traditional crystallographic drawing program. This workflow itself reflects the close relationship between quasicrystals and crystallography. The original 2011 manuscript marked these figures with “By Li Shichun” to record the originality of both the model and the computations. Fourteen Years of Development in Quasicrystal Research In the following decade, structural studies of quasicrystals continued to advance. A notable development is the magic-number formula proposed by Canestrari (2025), whose shell atom numbers match exactly the values listed in Table 1 of my 2011 manuscript. This agreement is not coincidental; it indicates that shell structure and atomic enumeration have become central topics in quasicrystal research. Unlike Canestrari’s numerical formula, the 2011 model provides a 3D lattice-generation framework applicable to any n-shell quasicrystal, built upon: the shell-generation equation u + v + w = n, the QC cell and rhombohedral lattice as structural units, the Iₕ point group as the symmetry constraint, and twenty coherent rhombohedral cells as the geometric construction. Thus, the 2011 model is not merely a sequence of numbers but a structural theory. From Quasicrystals to Cubic Crystals: A Unified Shell-Structure Perspective In 2026, I further developed the Rubik’s Cluster Shell Model (RCS) for cubic crystals, using generalized Miller indices to generate shell structures. This model shows that: coordination numbers are determined by index degeneracy, shell radii follow √(h² + k² + l²), the shell structures of 1×1×1, 5×5×5, and 7×7×7 Rubik’s Cubes can be systematically generated, and the translational symmetries of space groups No. 222–230 can be reduced to the shell-generation mechanism of point group No. 221. This reveals a unified mathematical logic underlying shell structures in both cubic crystals and quasicrystals. Together, the 2011 and 2026 works form a Unified Shell-Structure Theory, summarized as: Quasicrystals: u + v + w = n Cubic crystals: ⟨hkl⟩ → √(h² + k² + l²) Both systems share: group-theoretical generation, shell enumeration, coordinate computability, structural scalability. Why Publish the 2011 Work Without Modification The decision to publish the 2011 manuscript in its original form is based on the following scientific reasons: The model was more than a decade ahead: The 2011 shell-number sequence was independently confirmed in 2025. The model contains deeper structural insight: It is a lattice-generation theory, not a numerical formula. Historical integrity: Preserving the original text helps document the true trajectory of theoretical development. Foundational role in the unified theory: The 2011 and 2026 models together establish a cross-system structural framework. For these reasons, the 2011 manuscript is presented here unchanged, so that it may be read, understood, and evaluated within today’s scientific context. References Li, S. C. AEC: A New Tool for EET, TFDC and Crystal Formula. Materials Science Forum, Vol. 689, 245–254 (2011). Li, S. C. An n-shell Lattice Model for 3D Quasicrystals. Nature submission system record, 2012. Canestrari, R. Magic-number formula for quasicrystal shells. Nature Communications volume 16, Article number: 1655, 2025. Li, S. C. Rubik’s Cluster Shell Model: A Novel Theory Inspired by the Rubik’s Cube. Zenodo, 2026. Shechtman, D. et al. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953, 1984. Steinhardt, P. J., Levine, D. Quasicrystals: A new class of ordered structures. Phys. Rev. Lett. 53, 2477, 1984. (Additional references may follow those listed in the 2011 manuscript.)