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PreprintCC BY 4.0Zenodo

The Crystallographic Restriction and the Six-Fold Boundary: Why Six Is Forced, Why Five Is the Baseline, and Why Quasicrystals Are Rendered

  • Paul Rodgers1
  1. 1University of Kentucky
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Abstract

The foundational architecture, geometric intuitions, and motivating philosophy of this framework originated independently with the author prior to and separate from any AI involvement. Subsequent mathematical derivation, connection to established physics literature, computational verification, and error-checking were conducted through extended technical work with AI systems (Anthropic’s Claude and Google’s Gemini). The author directed this process, evaluated and selected among proposed derivations, and takes full responsibility for the accuracy and originality of the final work.

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Cite

BibTeXRIS
BibTeX
@misc{rodgers2026crystallographic,
  title = {{The Crystallographic Restriction and the Six-Fold Boundary: Why Six Is Forced, Why Five Is the Baseline, and Why Quasicrystals Are Rendered}},
  author = {Paul Rodgers},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23539438},
  url = {https://zenodo.org/records/23539438},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Crystallographic Restriction and the Six-Fold Boundary: Why Six Is Forced, Why Five Is the Baseline, and Why Quasicrystals Are Rendered
AU  - Paul Rodgers
PY  - 2026
DA  - 2026/10/21
PB  - Zenodo
DO  - 10.5281/zenodo.23539438
UR  - https://zenodo.org/records/23539438
AB  - The foundational architecture, geometric intuitions, and motivating philosophy of this framework originated independently with the author prior to and separate from any AI involvement. Subsequent mathematical derivation, connection to established physics literature, computational verification, and error-checking were conducted through extended technical work with AI systems (Anthropic’s Claude and Google’s Gemini). The author directed this process, evaluated and selected among proposed derivations, and takes full responsibility for the accuracy and originality of the final work.
ER  -