Rational Harmonic Arithmetic: The First Universal Rational Framework for Constants, Sequences, and Geometry across All Bases
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| Natura: | Recurso digital |
| Lingua: | inglese |
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Zenodo
2025
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| _version_ | 1866902295289004032 |
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| author | Hassanine, Sam |
| author_facet | Hassanine, Sam |
| contents | <p>This document introduces the canonical version of Rational Harmonic Arithmetic (RHA), a fully rational and self-contained arithmetic framework built on base 7. All classical constants—pi, phi, e, square root of 2, etc.—are redefined as rational expressions of the form k/7, derived from a minimal generator phi_7 = 8/7. Each base b > 1 defines its own constants phi(b), pi(b), and e(b), all projected rationally from the base 7 system. This allows consistent geometric, numerical, and dynamic constructions in any base.</p> <p>This framework includes:</p> <ul> <li> <p>Exact rational definitions of pi, phi, e, delta, and pi squared</p> </li> <li> <p>Harmonic sequences and exponential dynamics</p> </li> <li> <p>Rational geometry (circles, spirals, rectangles)</p> </li> <li> <p>A projection principle for inter-base consistency</p> </li> <li> <p>A formal score of harmonic coherence, maximized at base 7</p> </li> </ul> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15379622 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Rational Harmonic Arithmetic: The First Universal Rational Framework for Constants, Sequences, and Geometry across All Bases Hassanine, Sam base 7 rational arithmetic golden ratio Euler's constant pi harmonic recurrence alternative number systems symbolic computation <p>This document introduces the canonical version of Rational Harmonic Arithmetic (RHA), a fully rational and self-contained arithmetic framework built on base 7. All classical constants—pi, phi, e, square root of 2, etc.—are redefined as rational expressions of the form k/7, derived from a minimal generator phi_7 = 8/7. Each base b > 1 defines its own constants phi(b), pi(b), and e(b), all projected rationally from the base 7 system. This allows consistent geometric, numerical, and dynamic constructions in any base.</p> <p>This framework includes:</p> <ul> <li> <p>Exact rational definitions of pi, phi, e, delta, and pi squared</p> </li> <li> <p>Harmonic sequences and exponential dynamics</p> </li> <li> <p>Rational geometry (circles, spirals, rectangles)</p> </li> <li> <p>A projection principle for inter-base consistency</p> </li> <li> <p>A formal score of harmonic coherence, maximized at base 7</p> </li> </ul> |
| title | Rational Harmonic Arithmetic: The First Universal Rational Framework for Constants, Sequences, and Geometry across All Bases |
| topic | base 7 rational arithmetic golden ratio Euler's constant pi harmonic recurrence alternative number systems symbolic computation |
| url | https://doi.org/10.5281/zenodo.15379622 |