  Using geometric algebra and calculus to express the laws of electromagnetism
we are able to present magnitudes and relations in a gradual way, escalating
the number of dimensions. In the one-dimensional case, charge and current
densities, the electric field E and the scalar and vector potentials get a
geometric interpretation in spacetime diagrams. The geometric vector derivative
applied to these magnitudes yields simple expressions leading to concepts like
displacement current, continuity and gauge or retarded time, with a clear
geometric meaning. As the geometric vector derivative is invertible, we
introduce simple Green's functions and, with this, it is possible to obtain
retarded Li\'enard-Wiechert potentials propagating naturally at the speed of
light. In two dimensions, these magnitudes become more complex, and a magnetic
field B appears as a pseudoscalar which was absent in the one-dimensional
world. The laws of induction reflect the relations between E and B, and it is
possible to arrive to the concepts of capacitor, electric circuit and Poynting
vector, explaining the flow of energy. The solutions to the wave equations in
this two-dimensional scenario uncover now the propagation of physical effects
at the speed of light. This anticipates the same results in the real
three-dimensional world, but endowed in this case with a nature which is
totally absent in one or three dimensions. Electromagnetic waves propagating
entirely at the speed of light can thus be viewed as a consequence of living in
a world with an odd number of spatial dimensions. Finally, in the real
three-dimensional world the same set of simple multivector differential
expressions encode the fundamental laws and concepts of electromagnetism.
