Finding the Conjectured Sequence of Largest Small n-Polygons by Numerical Optimization
Bibliographic record
Abstract
LSP(n), the largest small polygon with n vertices, is a polygon with a unit diameter that has a maximal of area A(n). It is known that for all odd values n≥3, LSP(n) is a regular n-polygon; however, this statement is not valid even for values of n. Finding the polygon LSP(n) and A(n) for even values n≥6 has been a long-standing challenge. In this work, we developed high-precision numerical solution estimates of A(n) for even values n≥4, using the Mathematica model development environment and the IPOPT local nonlinear optimization solver engine. First, we present a revised (tightened) LSP model that greatly assists in the efficient numerical solution of the model-class considered. This is followed by results for an illustrative sequence of even values of n, up to n≤1000. Most of the earlier research addressed special cases up to n≤20, while others obtained numerical optimization results for a range of values from 6≤n≤100. The results obtained were used to provide regression model-based estimates of the optimal area sequence {A(n)}, for even values n of interest, thereby essentially solving the LSP model-class numerically, with demonstrably high precision.
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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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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".