Why do Fibonacci Numbers Appear in Patterns of Growth in Nature?
Bibliographic record
Abstract
While many examples of Fibonacci numbers are found in phenotypic structures of plants and animals, the dynamic processes that generate these structures have not been fully elucidated. This raises the question: What biologic rules and mathematical laws that control the growth and renewal of tissues in multi-cellular organisms give rise to these patterns of Fibonacci numbers? In nature the growth and self-renewal of cell populations leads to generation of hierarchical patterns in tissues that resemble the pattern of population growth in rabbits, which is explained by the classic Fibonacci sequence. Consequently, we conjectured a similar process exists at the cellular scale that explains tissue renewal. Accordingly, we created a model (cell division type) for tissue development based on the biology of cell division that builds upon the cell maturation concept posed in the Spears and Bicknell-Johnson model ("mating”-like design) for asymmetric cell division. In our model cells divide asymmetrically to generate a mature and an immature cell. Model output on the number of cells generated over time fits specific Fibonacci p-number sequences depending on the maturation time. A computer code was created to display model output as branching tree diagrams as a function of time. These plots and tables of model output illustrate that specific patterns and ratios of immature to mature cells emerge over time based on the cell maturation period. Conclusion: Simple mathematical laws involving temporal and spatial rules for cell division begin to explain how Fibonacci numbers appear in patterns of growth in nature.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".