[ARTICLE] Extended Pythagoras Theorem Using Hexagons [Canadian Journal of Mathematics Research / 2021]
Bibliographic record
Abstract
This article presents a new variant of the Pythagorean theorem using hexagons instead of squares. It was first published in 2021 on the Canadian Journal of Mathematics 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.<br>A video is also available (at Figshare) showing the dynamics of the extended Pythagoras theorem using hexagons for varying sides of the 60-degree-angled triangle (the link is also provided below to the benefit of those who may be interested in exploring more). <br><br>****************************************************************************************************ABSTRACTThis article provides the geometric and algebraic proof of the variant equation of the Pythagorean theorem x<sup>2</sup>-xy+y<sup>2</sup>=z<sup>2</sup>. The hypothesis that will be proven is that just as squares govern the original version x<sup>2</sup>+y<sup>2</sup>=z<sup>2</sup>, hexagons are found to govern x<sup>2</sup>-xy+y<sup>2</sup>=z<sup>2</sup>. Both the special case x equal to y, and the general case of x different from y, are examined.<br><br><br>-------------------------------------------------------------------------For more public data, please visit my Figshare profile: https://figshare.com/authors/Luis_Teia/10811244
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.255 | 0.000 |
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 teacher head, 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".