dimensionthat

Dimensionthat is a term used in discussions about the scale dependence of dimensionality in complex systems. It describes a proposed measure that captures how the effective dimensionality of a space or dataset changes with the level of observation, bridging notions such as fractal (box-counting) dimension and conventional topological dimension. Unlike fixed notions of dimension, dimensionthat emphasizes context and scale.

Conceptual use: In multiscale analyses, dimensionthat quantifies the rate at which new degrees of freedom appear as resolution improves. In practice, it is described qualitatively as the tendency for a system to cross from lower to higher dimensional behaviour when examined at finer scales.

Relationships: Dimensionthat relates to, but is not identical to, Hausdorff or spectral dimension. Some writers treat it as a derivative or functional of scale-dependent dimension estimates, such as those obtained by box-counting, correlation dimension, or random-walk based measures.

Applications and examples: in physics (quantum gravity models with dimensional running), data science (characterizing intrinsic dimensionality of high-dimensional data), and network theory (scale-dependent embedding).

Status and reception: The term lacks a formal, widely accepted mathematical definition. It appears mainly in speculative or popular-science contexts and some informal online discussions; rigorous treatment remains limited and debated.

Further reading: related topics include fractal dimension, box-counting dimension, Hausdorff dimension, spectral dimension, scale invariance, and dimensional reduction.