  While planar graphs are flat from a topological viewpoint, we observe that
they are not from a geometric one. We prove that every planar graph can be
embedded into a surface consisting of spheres, glued together in a tree-like
fashion. As a technical ingredient we prove a statement implying an inverse of
the Jordan curve theorem. This statement helps to identify cycles in the planar
graph, corresponding to circles of latitude on the spheres. The tree-like
embedding then allows for an inductive construction of a four-coloring of the
planar graph. Hence, this yields a simple proof of the Four-Color Theorem.
