Hyper-Triskelion (4-D Parametric Family)

This work introduces the Stone Hyper-Triskelion, a 4-dimensional parametric geometric family defined over with inherent three-fold rotational symmetry, time-inverted radial evolution, and mixed-frequency vertical oscillation.Traditional triskelion structures exist only as 2-D or 3-D rotational symbols. In contrast, the Hyper-Triskelion extends the concept into four dimensions through a unique combination of: a time-inversion delta , a phase-perturbed angular field, a radius modulated by future-state influence, and a new 4-D half-frequency oscillator pair . The family is defined explicitly as: Let δ_t = t_future − t (t goes from 1 down to 0) x(θ, φ, t) = (r + δ_t cosθ) · cos(θ + φ + sin(δ_t)) y(θ, φ, t) = (r + δ_t cosθ) · sin(θ + φ + sin(δ_t)) z(θ, φ, t) = δ_t sinθ + cos(θ/2) w(θ, φ, t) = δ_t sin(θ/2) For the 3 arms: φ = 0, 2π/3, 4π/3 Use: r = 1 t_future = 2 delta_t = t_future - t x(theta, phi, t) = (r + delta_t * cos(theta)) * cos(theta + phi + sin(delta_t)) y(theta, phi, t) = (r + delta_t * cos(theta)) * sin(theta + phi + sin(delta_t)) z(theta, phi, t) = delta_t * sin(theta) + cos(theta/2) w(theta, phi, t) = delta_t * sin(theta/2) phi = 0, 2*pi/3, 4*pi/3 r = 1 t_future = 2 This structure does not correspond to any known triskelion, knot, attractor, torus embedding, or 4-D classical surface. It constitutes a new mathematical object with potential applications in symbolic geometry, 4-D dynamical systems, computational morphology, and AI-generated geometric priors. Figures and Python simulation code are included for reproducibility. Authored by Shane Edward Stone, Simulations conducted by AI collaborator Kaelen. Creative Commons Attribution 4.0 International (CC-BY-4.0)