Zonoitua

Zonoitua is a fictional mathematical construct used in speculative geometry to generalize zonotopes. In this framework, a zonoitua is a compact convex set in Euclidean space that can be described as a parametric Minkowski sum of a continuous family of line segments whose lengths depend on a parameter t in a fixed interval, such as [0,1]. The idea is to interpolate between simple generating elements to produce a broader class of convex bodies.

If the length function is constant with respect to t, the zonoitua specializes to a zonotope. This provides a clear link to the well-established class of zonotopes while highlighting the new feature of continuous variation in the generating elements. In the discrete case, a zonoitua aligns with familiar zonotopal constructions; in the continuous setting, it can exhibit a richer boundary structure and more versatile support functions.

Basic properties attributed to the zonoitua in speculative discussions include central symmetry and a boundary that reflects the integrated contributions of an entire family of directions. In the discrete limit, facets are determined by linear functionals, as with zonotopes; in the continuous limit, the boundary may bend or smooth depending on the variation of the length functions. The resulting bodies remain convex, and their geometry can be analyzed via support functions and Minkowski sums.

Construction typically involves selecting a fixed set of direction vectors and a corresponding continuous length function, forming a limit or union of Minkowski sums over the parameter domain, and then taking a convex hull to ensure a well-defined convex body.

In practical terms, zonoitua are used in speculative contexts to model adaptable spaces, metamaterials, or optimization landscapes, illustrating how interpolation between simple generators can yield flexible convex bodies. See also zonotope, Minkowski sum, convex body. Note: zonoitua is not a standard term in formal mathematics; it appears as a hypothetical concept in discussions and fiction.