Vertex Spirals in Fullerenes and Their Implications for Nomenclature of Fullerene Derivatives
Bibliographic record
Abstract
Abstract The IUPAC nomenclature of fullerene derivatives is based on the vertex spiral, but not all fullerenes possess one. There are 1495 isomers with ≤20 vertices (66 isolated‐pentagon isomers with ≤150 vertices) that are vertex‐aspiral, but only four for which all vertex spiral starts succeed. Vertex‐aspiral trivalent polyhedra are common (40 190 with ≤24 vertices); infinite series include truncates of all trivalents larger than the tetrahedron. An alternative and readily automated breadth‐first‐search numbering scheme is proposed to deal with all trivalent polyhedra including fullerenes and incidentally to give robust, efficient determination of the automorphism/point group. For fullerenes, the new scheme has two clear advantages over the vertex spiral nomenclature: it is exception‐free, removing the need for complex prescriptions to deal with exceptional cases, and it is mathematically simpler in that the numbering can be found in linear rather than quadratic time (as all canonical numberings for a fullerene begin on a pentagon).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".