A Theory of the Connectivity Dimensionality Field in Edge- Vertex Graphs and Discrete-Continuous Dual Spaces

Over the years, a number of measures have been defined for the purpose of determining the number of independent dimensions contained in a space. The most common dimensionality measures are the topological dimensionality and various kinds of fractal dimensionalities. While each of these dimensionality measures is useful in its own right, none of them accurately quantifies the effective number of independent directions passing through locations contained in a local region of a space. This article introduces a new dimensionality measure, called the connectivity dimensionality field, which is the true measure for the effective number of independent directions passing through locations in a space. In contrast to the fractal dimensionality, the connectivity dimensionality field is a topological property because its value at each material location is invariant to deformations of the space preserving connectivity. The connectivity dimensionality field is a fundamental concept that applies to many different kinds of discrete spaces, continuous spaces, and discrete-continuous dual spaces. A discrete space is a space in which positions cannot be varied differentially, and a continuous space is a space in which positions can be varied differentially. A discrete-continuous dual space has complementary discrete and continuous representations, and a process called discretecontinuous dual matching relates the discrete and continuous representations to each other. This article formally defines two basic types of discrete-continuous duality: (a) asymptotic and (b) strict. A rigorous method is provided for computing the connectivity dimensionality field in edgevertex graphs, continuous spaces, and discrete-continuous dual spaces. Many examples are given to illustrate the key concepts. Single points, unbranched lines, and periodic lattices are examples of discrete-continuous dual spaces in which the connectivity dimensionality field is a constant nonnegative integer. In other types of discrete-continuous dual spaces, the connectivity dimensionality field contains inherent uncertainty. For the first time, a comprehensive theory is derived that predicts the inherent uncertainty associated with the connectivity dimensionality field in discrete-continuous dual spaces. The study of discrete-continuous dual spaces with variable connectivity dimensionality fields transcends variable-based mathematics.