Squaring the circle is easy, you just need to know what you want to do. My personal favorite method is to use elliptic functions defined on rectangular tori to map rectangles to disks, as shown below for a square. These maps don’t preserve area (which is what the Greeks had wanted), but they preserve angles.
I had some leftover architecture images from Columbus and wanted to see how they look when made circular. Here, for instance, is the AT&T building
and this is a circular version:
There are three degrees of freedom one can play with (the dimension of the automorphism group of the hyperbolic plane), which means that one can squeeze parts of the image towards the boundary cirle. Here are two other versions of the same image.
Another favorite of mine is the atrium of the Cummins office building with its wonderfully intricate play with straight lines and black and white.
Now we only have to find architects and builders who create buildings that have these curves in reality.