Harmonic Lattice Hidden in Conic-Driven Isogonal Involutions — E8 Intelligence Research

FINDING: Isogonal conjugation and projective involutions converge on a fixed-point structure tied to circular points at infinity, revealing a hidden harmonic lattice in conic-driven transformations. | MATH: Isogonal conjugation is a projective involution on the pencil of lines through a point; its fixed lines correspond to the isotropic directions (circular points at infinity, \(I, J\)). For a conic, the involution on a line induced by projection from a point has fixed points where the line meets the conic's polar — the cross-ratio of the four fixed/paired points satisfies \((a,b;c,d) = -1\) (harmonic). In the tetrahedral case (arXiv:2601.22042), isogonal conjugation in an isosceles tetrahedron yields three hyperbolic paraboloids; the bimedian symmetry implies the conjugate points lie on a ruled quadric with Gaussian curvature \(K = -1\), and the circumsphere condition forces a constant ratio of edge lengths: for isosceles tetrahedron with edges \(a, a, b, b, c, c\), the circumradius \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com