[FIGURE] Relativity of Pi as a function of the N-sided unit regular polygon circumscribing a circle [published in JMR / 2022]
Bibliographic record
Abstract
The picture is part of a recently (2022) published article in the Canadian Journal of Mathematics Research (please see the link below for the full paper). Consider the following perspective: Take a unit square and inscribe in it a circle. It's perimeter will be π(=3.1415926536) with a diameter D=1. Now take a unit regular triangle, and inscribe in it a circle. It's perimeter changes to 1.8137993642. Do the same for a unit regular pentagon, an hexagon and a heptagon, the perimeter of these inscribed circles changes to 4.3240313299, 5.4413998027 and 6.5235843645. Each unit polygon and inscribed circle represent a point in the new perspective of π as a function, where the ratios of the perimeter of a circle (the arc length) and the perimeter of the regular polygon that circumscribes it (the linear length) equates to the ratio of the corresponding above π and the number of sides of the polygon. In this perspective, the linear reference that defines the proportion that is π varies with the side of the chosen circumscribing polygon (where D=2R is the particular case of the side of the polygon square). The article shows that both the perimeter and area of a circle can be equated via the appropriate π to the side length of the circumscribing polygon selected.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.059 | 0.008 |
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".