I suppose you could get a shape like this if you tried to draw a square by true headings and bearings near the North pole of a sphere. "Turn heading 090, travel 10 miles. Turn heading 180, travel 10 miles." and so forth. Start at a spot close to the pole and this will be your ground track.

Actually no it isn't, because attempting to make a square you'd make four turns in the same direction, this would require turning left, right, right, left.

I guess if we define it as a calculus problem, I can see the point..

I didn’t mean to pun but there it is and I’m leaving it. Any way, there is no infinitely small section that’s perpendicular. Only the tangent at a single (infinitely small) point along a smooth curve, as we approach from either direction. Maybe that’s still called perpendicular.

Two wrongs don't make a right, but three lefts will.

I think that, in order to have this be a projection of a square, the space between the interior right angles of the space from which it was projected would have to be not just curved, but also twisted, like a Möbius strip, such that a person "walking" the square and starting from the rightmost angle leftward would start walking as if they were on your screen (their head coming out away from the screen), but then they would need to have their perspective twist so that they are now walking on the "underside" of the figure (their head now pointing into your phone). This would allow them to perceive the two "external" turns as "internal" turns, as well. Then it just needs to untwist on the way back. We just can't see the twisting, because the lines have no width.

"That's ....like.....just your perspective, man"

You could just use polar coordinates

Philosophers of that era spend a lot of time drunk.

I mean who the fuck dies from laughing to death at a donkey eating figs?