I still don't buy it. Once you cut a person vertically in two half with a guillotine, you get two topological disks of skin. Each disk can be mapped into the 2D surface of the brain. It doesn't matter if the parts of the skin are in the same plane or rotated 90°.
Obviously some parts of the skin are stretched, so a 2D map will cause a lot of deformation. Also some parts of the skin are more sensitive than other and will need more brain surface. But this is what is happening, there are a few maps in the brain https://en.wikipedia.org/wiki/Cortical_homunculus and they are quite deformed, and they even have a few cuts here and there.
Once you decide to cut the map in two parts, each part can be projected in both orientations without geometric problems.
Maybe related, maybe not, but I noticed this quote is incorrect:
> Odd things happen when we [project 3D space onto a 2D surface]. On a 2D map, an airplane taking the most direct path between two cities appears to travel in an arc, and satellites orbiting the globe appear to oscillate in a sinusoidal path
If you use the Gnomonic projection then all great circles become straight lines.[0] The only "catch" is that you need to cut the Earth into two hemispheres.
Once you cut the person in two halves, it's easy to map the half to a plane.
Imagine a sphere. You cut it in two halves, and you get something like this https://en.wikipedia.org/wiki/Nicolosi_globular_projection and use a scissor to cut it in the middle. Now imagine the initial sphere is made of rubber and it contracts to a body glove thigh around the person.
There are no discontinuity inside each half. Just a huge discontinuity between the two halves, but each one is processed by a different side of the brain anyway.
The projections in each half can be made in both orientations, the one that is like the skin and the mirror one. The non-mirrored is difficult to wire, but the mirrored one is easy to wire.
About the discontinuity, there is a huge discontinuity between the two sides of your body that are processed by the two sides of your brain, and you don't notice it.
Moreover, in each eye, there is a discontinuity because each half of each of your eyes is proceed in a different side of your brain https://www.quora.com/Which-side-of-the-brain-does-the-optic... With that hardware I expect to see a black vertical line in the middle of my eyes, but the transition is quite smooth.
Obviously some parts of the skin are stretched, so a 2D map will cause a lot of deformation. Also some parts of the skin are more sensitive than other and will need more brain surface. But this is what is happening, there are a few maps in the brain https://en.wikipedia.org/wiki/Cortical_homunculus and they are quite deformed, and they even have a few cuts here and there.
Once you decide to cut the map in two parts, each part can be projected in both orientations without geometric problems.