Iso-Surface Geometry from Tangent-Based Scalar Fields for Generative Design and Fabrication

*Submitted to the Journal of Mathematics and the Arts. Currently under editorial review.

We explore the generation of organic-like sculptural forms from mathematical scalar fields, investigating iso-surfaces defined by u = tan(g) and u = 1/tan(g), where the inner function

g(x,y,z) = z tan(y) + x tan(z) + y tan(x)

is designed to produce rotationally symmetric structures with distributed singular amplification. This cyclic cross-coupled structure was chosen to promote the emergence of interconnected branching geometries reminiscent of natural forms such as coral and fungal mycelium. Unlike conventional sine-cosine constructions, the tangent function introduces periodic singular behavior, enabling iso-surface geometries with localized sharp features that arise intrinsically from the scalar field structure rather than through explicit feature injection.

The gradient decomposes into a geometric variation term from g and an amplification term from the outer trigonometric function; visualization confirms that both contribute throughout the surface with distinct spatial roles. Analysis of gradient magnitude versus distance to the singular set reveals large gradients near singular regions due to amplification, while substantial gradients at larger distances indicate that g independently generates strongly curved structures.

Six models fabricated by 3D printing were selected to explore visually distinct sculptural qualities, including enclosure, branching, and flow; selected models were additionally produced in full color and metal. These results demonstrate that tangent-based scalar fields extend the design space of implicit surface modeling beyond conventional smooth and periodic constructions.