[ARTICLE] Extended Pythagoras Theorem Using Hexagons [JMR / Luis Teia 2021]
Bibliographic record
Abstract
This article presents a new Pythagorean theorem using hexagons instead of squares. It was first published in 2021 on the Canadian Journal of Mathematical Research under a Creative Commons Attribution 4.0 International License (CC-BY 4.0) - see reference at bottom for original source - and, it is made available again here to the benefit of all that might be interested in such a topic. ****************************************************************************************************ABSTRACTThis article provides the geometric and algebraic proof of the variant equation of the Pythagorean theorem x2-xy+y2=z2. The hypothesis that will be proven is that just as squares govern the original version x2+y2=z2, hexagons are found to govern x2-xy+y2=z2. Both the special case x equal to y, and the general case of x different from y, are examined.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.020 | 0.004 |
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".