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Strategic Value of a line of symmetry

Original question on worldbuilding.stackexchange by PyRulez

So in my world, there is a line of 180 degree symmetry. What this means if that if we assign it coordinates of (0,0,z), then (x,y,z) and (-x,-y,z) will be the same point. The z=0 plane is the ground.

This world also has a normal season and day night cycle, despite being flat (the sky basically just glows during the day and doesn’t at night).

This line of symmetry provides plenty of entertainment for the royal family, but does situating your country there provide any strategic value?

It should be noted that if your city is a circle, with the line going through its center, then for a given radius, it will be half the circumference of other circles. That means a city centered there would only need to build a wall half as long as otherwise would need to. That being said, it will also be only half the area. All being said, for a given area, you will need the same amount of wall, so this amounts to neither an advantage nor a disadvantage, as far as wall building is concerned.

Answer by Joe Bloggs

Well. There are a few options here, and almost all of them result in:

Avoid this place like the plague

The first option is to treat the world as an infinite, flat, static plane. Your use of cartesian co-ordinates and the assumption of a ‘ground’ level at Z=0 implies this is true. In this case your line of symmetry must effectively be the centre of the world: Everything happens around a ‘pole’ of symmetry that points directly out of the ground. This is not a nice place to be, because any air that approaches this point must be moving in a radially symmetric way, effectively setting up a perfect, permanent cyclone. Whatever quirk of physics is causing this symmetry will also cause some very severe weather conditions.

The second option is similar to the first, but with plate tectonics as well. In this scenario the line of symmetry will become an area of severe, constant vulcanism as any minor plate movement must be immediately and perfectly opposed by it’s counterpart, leading to immense friction and, correspondingly, heat. Couple this with the wind and you get a singularly unpleasant place to be.

The third option is to assume your world is a sphere. In this case the line of symmetry must go through the axis of rotation of the planet, or your little world [will tear itself apart in very, very short order] simply cannot exist. Wind and plate tectonics are still issues. You can switch to a polar co-ordinate system and define Z as a plane passing through the equator if you want to make thinking about this easier.

I’m still working through whether or not this sort of symmetry represents a mathematical singularity or not, but if it does then you have an even more severe problem on your hands: to whit: The laws of physics break down when you reach it.

ADDENDUM:

So. While pondering the awfulness that is the singularity of symmetry I realised that electromagnetism has some unfortunate things to say about this situation.

I present: Amperes right hand grip rule.

This may not seem like an issue at first, however: Your world has radial symmetry. This means any magnetic field it may have must be symmetric around the pole. Now, if we assume a flat world with no meaningful magnetic field we’re OK, until we get to electromagnetic interactions of molecules nearing the singularity. Things like air, or people’s hands, or the ground.

In this situation any magnetic field, no matter how small, *must* form a curl (The magnetic field at x,y must be moving in the opposite direction to the field at -x,-y). This in turn *must* form a current pointing in a direction dictated by the grip rule. That *must* interact with the matter around it. I believe the magnetic flux becomes infinite as you approach the line of symmetry, which means the current does too. I’m genuinely not sure of my own maths/physics here (long day), so please correct me if I’m wrong.

In short, unless I’ve missed my maths: Nothing can reach the line of symmetry, not because it repels itself, but because the electromagnetism responsible for holding molecules together will tear them apart as they approach the mathematical monstrosity that is at the centre of your world. You have a constant line of plasma at the centre of a volcano surrounded by the fiercest winds imaginable.

Seriously. Don’t go here.

Answer by Rolf Kreibaum

You are effectively located in a remote part of the world, because fewer people visit you on accident. A trader from (A) won’t pass your city while they are on their way to a different city (B) because the path from (A) to (B′) (which avoids your city) is often shorter. Here is a crappy picture:

An example diagram of the Situation.

To me this looks like an economic disadvantage, as you need a good reason why people would go out of their way to visit your city.

I suggest that the regional religion encourages pilgrimage to the center of the universe which houses a temple or similar religious institution. Shaking your own hand as a symbol of self discovery (or whatever) seems to be a great ritual. The huge influx of pilgrims should boost your economy.

You might also house a Pope-equivalent and exert influence over your neighbors.