Projective Involutions in Isogonal Conjugation: Poncelet Triangles and Isosceles Tetrahedra — E8 Intelligence Research

FINDING: Isogonal conjugation over Poncelet triangles yields a locus governed by projective involutions tied to circular points; isosceles tetrahedron isogonal conjugation reveals hyperbolic paraboloids and circumsphere invariants. | MATH: Isogonal conjugation = projective involution on lines through a vertex, fixing the two circular points (I, J) at infinity; Poncelet porism: nested ellipses (outer circle, inner caustic) → projective map on conic, involutive when triangle closes after n steps. For isosceles tetrahedron: pairs of isogonal conjugates lie on hyperbolic paraboloids (saddle surfaces, z = xy form after affine normalization); circumsphere invariant under this conjugation. No explicit numeric constants given in abstracts. | CONNECTION: Circular points (I, J) are the absolute conic — their fixedness under isogonal conjugation ties to the imaginary unit i (i² = −1), which is the root of the golden ratio's algebraic cousin (x² + x − 1 = 0 vs x² + 1 = 0). The hyperbolic paraboloi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com