#projectivenormalization

3D Structures with Complex Mathematical Forms

• Model:

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: 3D structures with forms, generated using vec3 z = π * p / (exp(p) + exp(-p)) - p / Φ^n with n = 0 to 64, incorporating chp(x) = (exp(x) + exp(-x))/π, chpp(x) = (exp(x/(cosh(x)·π)) + exp(-x/(cosh(x)/π)))/(2π·Φ), shp(x) = (exp(x) - exp(-x))/π, shpp(x) = 1/(exp(x·sinh(x)·π) - exp(-x·sinh(x)·π))/(2π·Φ), ssh(x) = (exp(x·π/.7887) - exp(-x·π/.7887))/(2π), csh(x) = (exp(x·π/.7887) + exp(-x·π/.7887))/(2π), ssh1(x) = sinh(x/π)/Φ, csh1(x) = cosh(x/π)/Φ, with high symmetry, golden ratio scaling (Φ = (1 + √5)/2), and logarithmic refinement z *= -π·log(||z||), enhanced by additional transforms: z += sin(τ·||z||)·p/||p|| for oscillatory perturbation, z = z / (1 + ||z||^2) for projective normalization, z = z + Φ^(-n)·cross(p, z) for rotational twist, z *= exp(-||z||/τ) for exponential decay, z = z + ∇(cosh(||p||)·sin(π·||z||)) for gradient-based modulation, and the new spherical transform z = r * (vec3(tan(shp(sin(θ)*sin(φ)))*Φ, chp(cos(θ)*sin(φ)), cos(φ))) + p where θ and φ are angular coordinates, r is a radial scale, and p is the input vector, showcasing iterative variants. Apply 64.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle over the cotangent bundle, textless !
  • Using base image: No
  • Aspect Ratio: square