Cross-Ratio Dynamics and the Anharmonic S3 Orbit in Universal Hyperbolic Geometry
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2026
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| author | Coady, John D. |
| author_facet | Coady, John D. |
| contents | <p class="ng-star-inserted"><span class="ng-star-inserted">This paper develops a unified projective and Lie-theoretic framework for the kinematics of parameterized null configurations on the absolute conic in Universal Hyperbolic Geometry (UHG). Building upon the metrical and algebraic structures established in the previous papers of this series, the work maps the projective plane of UHG to the trace-zero Lie algebra sl(2,F), proving that the Lorentzian quadratic form is equivalent to the matrix determinant and that the J-function (hyperbolic cross product) is isomorphic to the matrix commutator bracket. Under this representation, the anti-symmetry of the Lie bracket collapses to a projectively commutative relation, directly mirroring the commutativity of the projective join.</span></p> <p class="ng-star-inserted"><span class="ng-star-inserted">The paper establishes three central results:</span></p> <ol class="ng-star-inserted"> <li class="ng-star-inserted"> <p class="ng-star-inserted"><span class="ng-star-inserted">The concurrency of altitudes in a triangle (the Orthocenter Theorem) is shown to be the projective geometric expression of the algebraic Jacobi identity in sl(2,F), projectively formalizing and extending an observation originally attributed to V. I. Arnold and N. V. Ivanov.</span></p> </li> <li class="ng-star-inserted"> <p class="ng-star-inserted"><span class="ng-star-inserted">The sixfold metrical reciprocal quadrance orbit is proved to be naturally isomorphic as an S3-set to the classical anharmonic cross-ratio orbit on the projective line, showing that metrical reciprocity in UHG is a direct projection of the projective symmetry group of the underlying null quadrangle.</span></p> </li> <li class="ng-star-inserted"> <p class="ng-star-inserted"><span class="ng-star-inserted">Up to projective scaling, the scalar Cayley transform is proved to be the unique non-singular Mobius transformation that conjugates the reciprocal metrical involution to a linear sign-reflection, making it the optimal computational coordinate on the moduli space of null quadrangles.</span></p> </li> </ol> <p class="ng-star-inserted"><span class="ng-star-inserted">Additionally, the work computes the setwise stabilizer groups of the moving quadrangle under the adjoint action of PGL(2,F) (showing they are generically the Klein four-group V4, and exceptionally the dihedral group D8 and alternating group A4) and classifies the number-theoretic orbit collapse over finite prime fields F_p.</span></p> <p class="ng-star-inserted"><span class="ng-star-inserted">Within the finitist program initiated by N. J. Wildberger, this paper demonstrates how metric structures, orthogonal projective frames, and parameterized null configurations can be modeled exactly over the rationals or finite prime fields without reliance on real-analysis limits, smooth manifolds, or transcendental coordinates.</span></p> <p class="ng-star-inserted"><span class="ng-star-inserted">This manuscript forms the fifth paper in an ongoing research series investigating reciprocal quadrance structures, null quadrangles, and algebraic dynamical systems in Universal Hyperbolic Geometry.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20385615 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Cross-Ratio Dynamics and the Anharmonic S3 Orbit in Universal Hyperbolic Geometry Coady, John D. Universal Hyperbolic Geometry sl(2,F) Lie algebra Jacobi identity Orthocenter theorem Anharmonic orbits Cross-ratio dynamics Cayley transform Stabilizer groups Orbit collapse Finite prime fields Finitist mathematics Projective geometry <p class="ng-star-inserted"><span class="ng-star-inserted">This paper develops a unified projective and Lie-theoretic framework for the kinematics of parameterized null configurations on the absolute conic in Universal Hyperbolic Geometry (UHG). Building upon the metrical and algebraic structures established in the previous papers of this series, the work maps the projective plane of UHG to the trace-zero Lie algebra sl(2,F), proving that the Lorentzian quadratic form is equivalent to the matrix determinant and that the J-function (hyperbolic cross product) is isomorphic to the matrix commutator bracket. Under this representation, the anti-symmetry of the Lie bracket collapses to a projectively commutative relation, directly mirroring the commutativity of the projective join.</span></p> <p class="ng-star-inserted"><span class="ng-star-inserted">The paper establishes three central results:</span></p> <ol class="ng-star-inserted"> <li class="ng-star-inserted"> <p class="ng-star-inserted"><span class="ng-star-inserted">The concurrency of altitudes in a triangle (the Orthocenter Theorem) is shown to be the projective geometric expression of the algebraic Jacobi identity in sl(2,F), projectively formalizing and extending an observation originally attributed to V. I. Arnold and N. V. Ivanov.</span></p> </li> <li class="ng-star-inserted"> <p class="ng-star-inserted"><span class="ng-star-inserted">The sixfold metrical reciprocal quadrance orbit is proved to be naturally isomorphic as an S3-set to the classical anharmonic cross-ratio orbit on the projective line, showing that metrical reciprocity in UHG is a direct projection of the projective symmetry group of the underlying null quadrangle.</span></p> </li> <li class="ng-star-inserted"> <p class="ng-star-inserted"><span class="ng-star-inserted">Up to projective scaling, the scalar Cayley transform is proved to be the unique non-singular Mobius transformation that conjugates the reciprocal metrical involution to a linear sign-reflection, making it the optimal computational coordinate on the moduli space of null quadrangles.</span></p> </li> </ol> <p class="ng-star-inserted"><span class="ng-star-inserted">Additionally, the work computes the setwise stabilizer groups of the moving quadrangle under the adjoint action of PGL(2,F) (showing they are generically the Klein four-group V4, and exceptionally the dihedral group D8 and alternating group A4) and classifies the number-theoretic orbit collapse over finite prime fields F_p.</span></p> <p class="ng-star-inserted"><span class="ng-star-inserted">Within the finitist program initiated by N. J. Wildberger, this paper demonstrates how metric structures, orthogonal projective frames, and parameterized null configurations can be modeled exactly over the rationals or finite prime fields without reliance on real-analysis limits, smooth manifolds, or transcendental coordinates.</span></p> <p class="ng-star-inserted"><span class="ng-star-inserted">This manuscript forms the fifth paper in an ongoing research series investigating reciprocal quadrance structures, null quadrangles, and algebraic dynamical systems in Universal Hyperbolic Geometry.</span></p> |
| title | Cross-Ratio Dynamics and the Anharmonic S3 Orbit in Universal Hyperbolic Geometry |
| topic | Universal Hyperbolic Geometry sl(2,F) Lie algebra Jacobi identity Orthocenter theorem Anharmonic orbits Cross-ratio dynamics Cayley transform Stabilizer groups Orbit collapse Finite prime fields Finitist mathematics Projective geometry |
| url | https://doi.org/10.5281/zenodo.20385615 |