  During the last century, two independent theories using the concept of
dimensional reduction have been developed independently. The first, known as
F\"oppl-von K\`arm\`an theory, uses Riemannian geometry and continuum mechanics
to study the shaping of thin elastic structures which could become as complex
as crumpled paper. The second one, known as Kaluza-Klein theory, uses
Minkowskian geometry and general relativity to unify fundamental interactions
and gravity under the same formalism. Here we draw a parallel between these two
theories in an attempt to use concepts from elasticity theory of plates to
recover the Einstein-Maxwell equations. We argue that Kaluza-Klein theory
belongs to the same conceptual group of theories as three-dimensional
elasticity, which upon dimensional reduction leads to the F\"oppl-von
K\`arm\`an theory of two-dimensional elastic plates. We exploit this analogy to
develop an alternative Kaluza-Klein formalism in the framework of elasticity
theory in which the gravitational and electromagnetic fields are respectively
associated with stretching-like and bending-like deformations. We show that our
approach of dimensional reduction allows us to retrieve the Lagrangian
densities of both gravitational, electromagnetic and Dirac spinors fields as
well as the Lagrangian densities of mass and charge sources.
