Heptagonal Nested Radicals via Chebyshev T7 and Cubic Field Link — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Nested radical 2cos(2π/7) encodes the heptagon's cubic field via Chebyshev T7, linking a 7×7 determinant to cos(7θ) and enabling periodic nested radical solutions of certain polynomials. | MATH: Chebyshev identity: T7(x) = 64x7 − 112x5 + 56x3 − 7x = cos(7arccos x). For x = 2cos(2π/7), T7(x/2) = cos(2π) = 1 → x satisfies x3 − x2 − 2x + 1 = 0 (minimal polynomial, degree 3, discriminant 49). Nested radical: 2cos(2π/7) = √[3](7 + 21√(−3))/2 + √[3](7 − 21√(−3))/2 (Cardano form). Ramanujan nested radical: √[3]7/2 + 3/2√(−3) + √[3]7/2 − 3/2√(−3). The 7×7 determinant: det(I − 2x A) where A is adjacency of cycle C7 yields T7(x/2) − 1, linking to Chebyshev. | CONNECTION: Heptagon central angle 2π/7 → 2cos(2π/7) ≈ 1.24698. Golden ratio cousin: 2cos(π/7) ≈ 1.80194, \(2\cos(3\pi/7) \approx
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026heptagonal,
title = {{Heptagonal Nested Radicals via Chebyshev T7 and Cubic Field Link — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431335},
url = {https://zenodo.org/records/23431335},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Heptagonal Nested Radicals via Chebyshev T7 and Cubic Field Link — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431335
UR - https://zenodo.org/records/23431335
AB - FINDING: Nested radical 2cos(2π/7) encodes the heptagon's cubic field via Chebyshev T_7, linking a 7×7 determinant to cos(7θ) and enabling periodic nested radical solutions of certain polynomials. | MATH: Chebyshev identity: T_7(x) = 64x^7 − 112x^5 + 56x^3 − 7x = cos(7arccos x). For x = 2cos(2π/7), T_7(x/2) = cos(2π) = 1 → x satisfies x^3 − x^2 − 2x + 1 = 0 (minimal polynomial, degree 3, discriminant 49). Nested radical: 2cos(2π/7) = √[3](7 + 21√(−3))/2 + √[3](7 − 21√(−3))/2 (Cardano form). Ramanujan nested radical: √[3]7/2 + 3/2√(−3) + √[3]7/2 − 3/2√(−3). The 7×7 determinant: det(I − 2x A) where A is adjacency of cycle C_7 yields T_7(x/2) − 1, linking to Chebyshev. | CONNECTION: Heptagon central angle 2π/7 → 2cos(2π/7) ≈ 1.24698. Golden ratio cousin: 2cos(π/7) ≈ 1.80194, \(2\cos(3\pi/7) \approx Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -