Can regular maps, no matter their complexity, be topologically transformed into circular maps or rectangular maps? Can regular maps, no matter their complexity, be topologically transformed into circular maps or rectangular maps?

For rectangular maps, for example, I intend maps that are made from overlapping rectangular faces as shown here: Rectangular and circular maps.
And here is the proof.

I don't know if this is a new result or not (I did not find it on internet) and I don't even know if this may be useful to help finding a "pencil and paper", elegant and human checkable proof for the four color problem.

What is known about this? Has it been studied before?
Your "proof" is too vague to convince me, but here's a description of how these maps can be formed. Given a 3-regular 3-connected planar graph, its dual is a maximal planar graph. Every maximal planar graph has a canonical ordering (see references in my earlier comment): an ordering on the vertices of the graph such that the following properties hold:


The first three vertices in the ordering are adjacent
Each prefix of three or more vertices in the ordering defines a subset of the vertices whose boundary is a simple cycle
Each vertex after the first three has as its neighbors a contiguous path of the boundary cycle of the previous vertices


This should allow you to define your rectangular drawing top down. Find a canonical ordering of the dual graph, and make side-by-side rectangles for the first two dual vertices in the canonical ordering (corresponding to two adjacent faces in your graph). Then, add rectangles one by one in the given ordering, below these first two rectangles. For each successive rectangle r, corresponding to a dual vertex v, choose the top vertices on the left and right sides of r to be points on the rectangles corresponding to the endpoints of the contiguous path of neighbors of v from the canonical ordering.

The invariant that makes this work, at each step, is that the boundary cycle of the canonical ordering is exactly the same as the sequence of rectangles that you would see if you stood underneath your drawing and looked upwards.

The circular map should be exactly the same, just starting at the outside and working inwards.

As for the other questions at the end of your post: is this interesting? I don't know. There's some related work in the graph drawing community on representing maps by regions with rectilinear boundaries; the phrase to search for is "rectilinear cartogram". You'd have to show some concrete advantage of your method over previous rectilinear cartogram construction methods such as the one in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.6582&amp;rep=rep1&amp;type=pdf — but suggesting ways to achieve this is probably beyond the scope of this exchange.