Fundamental ratios and logarithmic periodicity in human limb bones
Bibliographic record
Abstract
Fundamental mathematical relationships are widespread in biology yet there is little information on this topic with regard to human limb bone lengths and none related to human limb bone volumes. Forty-six sets of ipsilateral upper and lower limb long bones and third digit short bones were imaged by computed tomography. Maximum bone lengths were measured manually and individual bone volumes calculated from computed tomography images using a stereologic method. Length ratios of femur : tibia and humerus : ulna were remarkably similar (1.21 and 1.22, respectively) and varied little (<7%) between individuals. The volume ratio of femur : tibia was approximately half that of humerus : ulna (1.58 and 3.28, respectively; P < 0.0001). Lower limb bone volume ratios varied much more than upper limb ratios. The relationship between bone length and volume was found to be well described by power laws, with R(2) values ranging from 0.983 to 0.995. The most striking finding was a logarithmic periodicity in bone length moving from distal to proximal up the limb (upper limb λ = 0.72, lower limb λ = 0.93). These novel data suggest that human limb bone lengths and volumes follow fundamental and highly conserved mathematical relationships, which may contribute to our understanding of normal and disordered growth, stature estimation, and biomechanics.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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".