Physical theories address different numbers of degrees of freedom depending
on the scale under consideration. In this work generalized mathematical
structures (nonlinear $\mathcal{B}_{\kappa}$-embeddings) are constructed that
encompass objects with different dimensionality as the continuous scale
parameter $\kappa \in \mathbb{R}$ is varied. Based on this method, a new
approach to compactification in unified physical theories (e.g. supergravity in
10 or 11-dimensional spacetimes) is pointed out. We also show how
$\mathcal{B}_{\kappa}$-embeddings can be used to connect all cellular automata
(CAs) to coupled map lattices (CMLs) and nonlinear partial differential
equations, deriving a class of nonlinear diffusion equations. Finally, by means
of nonlinear embeddings we introduce CA connections, a class of CMLs that
connect any two arbitrary CAs in the limits $\kappa \to 0$ and $\kappa \to
\infty$ of the embedding. Applications to biophysics and fundamental physics
are discussed.