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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.

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  • $\begingroup$ While perfectly circular cities might not need more wall per area, there's always natural formations that would be doubly useful. $\endgroup$ – Andon Jan 16 '18 at 1:13
  • $\begingroup$ If you reach into the line of symmetry, would your wrist and elbow come to occupy the same space? $\endgroup$ – Willk Jan 16 '18 at 1:32
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    $\begingroup$ @Will no, you arm bumps into itself $\endgroup$ – PyRulez Jan 16 '18 at 2:54
  • $\begingroup$ @Will This means you can shake your own hand! $\endgroup$ – PyRulez Jan 16 '18 at 6:05
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    $\begingroup$ This world has a rotational symmetry, not a mirror symmetry. As I understand it, you can walk in a circle around the center while watching your mirror image next to you. (Hence the entertainment for the royal family.) $\endgroup$ – Rolf Sievers Jan 16 '18 at 14:52
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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.

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  • $\begingroup$ "The third option is to assume your world is a sphere." Note that the rest of the world also needs to have symmetrical geometry, otherwise on the symmetry on the sphere will break down. $\endgroup$ – PyRulez Jan 16 '18 at 16:43
  • $\begingroup$ @PyRulez: if your sphere is spinning on an axis centred at x=0, y=0 then it can have 180 degree symmetry with no issues, aside from at or near the poles. $\endgroup$ – Joe Bloggs Jan 16 '18 at 16:59
  • $\begingroup$ I believe that there is some sort of singularity at 0. It is likely still continuous and maybe even manifold but the curvature might not be defined. $\endgroup$ – Rolf Sievers Jan 16 '18 at 16:59
  • $\begingroup$ @RolfSievers: I’m pretty certain the pole would be a column of ionised plasma, with the singularity itself being fundamentally unreachable. Turns out physics is pretty unforgiving if you force symmetry on it. $\endgroup$ – Joe Bloggs Jan 16 '18 at 18:20
  • $\begingroup$ I'm not sure that you would have such crazy weather. While you would definitely have air moving in a radially symmetric way, the two air columns cannot interact with each other, it's more like having a huge brick wall than anything. The main effect of this (I think) would be to make rainfall much more common when the wind is blowing towards the mirror, including some ice/snow from the moisture blown up high. The opposite effect would be a sudden cold-snap and lots of fog. $\endgroup$ – bendl Jan 16 '18 at 19:10
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If you were to stand with your back to the symmetry line, you would never have to worry about being flanked. You would, in effect, have your own back. A king whose throne is backed up to this point would have quite some value indeed.

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    $\begingroup$ But this would be no different than if the world ended in a giant wall or something. It is also a major disadvantage to have no way to retreat. While this could be useful in very specific circumstances within a story, but having your own back will not matter that much in reality. $\endgroup$ – Raditz_35 Jan 16 '18 at 10:32
  • $\begingroup$ This is exactly like the Battle of Helm's Deep. Your whole population is here, and they will never be able to escape a besieging force. Even if they leave ahead of time, they have to walk TOWARDS the enemy to leave the city. $\endgroup$ – bendl Jan 16 '18 at 19:12
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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.

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  • $\begingroup$ The question talks about a line of symmetry you seem to focus on a center of symmetry $\endgroup$ – L.Dutch - Reinstate Monica Jan 16 '18 at 15:43
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    $\begingroup$ The line is vertical and my answer uses this line of rotational symmetry. I believe, that you are thinking about a plane of (mirror) symmetry. $\endgroup$ – Rolf Sievers Jan 16 '18 at 15:56
  • $\begingroup$ I didn't get it. Thanks for explaining $\endgroup$ – L.Dutch - Reinstate Monica Jan 16 '18 at 16:27
  • $\begingroup$ Surely the two paths are symmetric. If you are walking the short path an anti-you is walking the opposing short path and the economies of A and and anti-A are the same? $\endgroup$ – Joe Bloggs Jan 16 '18 at 18:29
  • $\begingroup$ Yes, the economics of A and A' are the same. It is just one city. In this space the "strait line between two points" isn't unique. There are two strait lines from A to B and the shorter line avoids your city. $\endgroup$ – Rolf Sievers Jan 16 '18 at 19:26
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Internal economics would be slightly more efficient. The reason why is that you have two paths any internal point (and you don't need to leave the city to get to it). You can take the shorter path.

This also helps with dealing with emergencies (such as foreign invasions).

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