AI Unveils Hidden Links in Historical Pi Formulas
A team from the Technion used AI and Conservative Matrix Field (CMF) to analyze 385 unique formulas for π. They discovered that 43% stem from a single foundational formula, 51% belong to broader clusters, and 6% remain unconnected—uniting equations from Archimedes to modern generations.
The method identifies formula equivalence through parallel paths in the CMF grid, where the final point determines the value regardless of route.
Historical Context of Pi Formulas
In the 3rd century BCE, Archimedes applied the method of inscribed and circumscribed polygons (96-gons), bounding π between 3.140845… and 3.142857…. This approximation held sway for 1,600 years.
In the 14th century, Madhava of Sangamagrama introduced an infinite series, though with slow convergence. Euler improved its speed in the 18th century. Ramanujan developed highly efficient formulas in the early 20th century—still relevant today.
Thousands of formulas have accumulated over 2,000 years, plus endless algorithmic generations. Previously, no connections were known.
Research Methodology
Researchers compiled a dataset of 455,050 arXiv preprints over six weeks. GPT-4o and specialized algorithms extracted 385 unique π formulas (10% of the 'Ramanujan Machine' 2019 dataset).
Formulas were unified into infinite series format. CMF (developed by the group in 2023) represented them as paths in a gravitational-like grid:
- Conservatism: Energy difference (or value) depends only on endpoints.
- Equivalence: Parallel paths = transformation from one formula to another.
Algorithms placed formulas within CMF, identified clusters, and formally proved their relationships.
Key Data Processing Steps:
- Parsing arXiv (Tomer Raz’s script).
- Extracting and deduplicating equations (GPT-4o + algorithms).
- Unifying into series.
- Embedding in CMF and clustering.
- Proving equivalence.
Conservative Matrix Field (CMF)
CMF is a universal tool for hierarchical constants. Applied to π, it revealed a base formula as the "common ancestor."
Analogy: Gravitational field on a grid. Paths from A to B yield identical work regardless of trajectory.
Potential: Proofs of irrationality, Riemann Hypothesis. All equations generated via CMF converge to π.
Open Questions:
- Will expanded CMF cover 100% of formulas?
- Do all CMF equations produce π?
Expert Assessments
David Bailey (Lawrence Berkeley): Compared it to automatically assembling the periodic table from scattered elements.
George Andrews (Penn State): "Serious mathematics with discovery potential."
Key Takeaways
- 43% of π formulas are direct derivatives of one core formula in CMF.
- The method unifies Archimedes, Euler, and Ramanujan.
- CMF applies to other constants and hypotheses.
- Dataset: 385 formulas from 455k arXiv preprints.
- Future: Complete classification and new proofs.
— Editorial Team
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