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The world below is flat. However, there are no endless walls of ice or force or magic - Instead, each side of the octagon "wraps" to the other side.

That is, the North boundary wraps to the South boundary at the exact same point. The North-East to South-West, East to West, and South-East to North-West. There is also a featureless "infinitely" (For all reasonable purposes) tall tower at the cornerpoints, and occupies them all simultaneously.

These wraps function exactly as if you printed this on paper, cut it out, and wrapped one edge to its counterpart. This leads to some... weird navigation around the corners. If you were to walk around the tower, you would walk in three full circles (Measured by angle) before reaching your starting point. Otherwise, navigation is fairly straightforward. Going across the north/south wrap doesn't change your east/west position, etc.

Other information that may be relevant or helpful:

  • The planet has two "Suns" - They are always directly across from each other, and rotate around the center of the world.
  • The two "Suns" have a "Sun" face and a "Moon" face, which rotate in synch to provide a day/night cycle. Thus, the entirety of the world is at the same time of day. Assume that the day/night cycle functions otherwise like our terrestrial one.
  • The central landmass is aproxamately 3600 miles wide by 4600 miles tall. Height of the world is thus aproxamately 10,000 miles. This puts the surface area of the world (And oceans) at around 82.8 million miles squared, a little over 42% of the Earth's surface. (If you want to get into the nitty gritty, each pixel on the map is 5mi per side)
  • Gravity is a stable 1g at all points
  • The world is truly "flat", and there are no map distortions like you would find on a spherical world.

Octogonal world

Religion has imparted the knowledge that the world is the shape it is, and tells of the tower in the corner positions. Even so, some wish to prove that the world is this shape, beyond "blindly" accepting religious dogma.

Using European dating and technology, when would humanity be able to prove this world shape?

Things that are beyond the scope of this question:

  • This world affecting technological development. Assume relevant technology and knowledge are exactly what they were historically.
  • Magic or other factors that created the world. Assume there is no magic in the world for the purposes of answering this question.
  • Changes that the world shape would have on weather, wildlife, geography, etc
  • Time needed to prove the world shape. This question is focused solely on technology and knowledge needed to prove it, not how long it would take to accomplish.

A reference for movement across edges: Going North/South across Point 7 takes you to Point 7, North/East from Point 5 takes you to South/West Point 5, etc.

Walking along the coastline, starting at Point 5 heading East, you would go:

  1. 5 to 4, then cross (West to East)
  2. 4 to 1, then cross (SE to NW)
  3. 1 to 2 (Along the "Inner" sea on the top left), then cross back (NW to SE)
  4. 2 to 6, then cross (SW to NE)
  5. 6 to 3, then cross (E to W)
  6. And then you would have a long hike from 3 back to 5 along that very long NW/N coast line.

enter image description here

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    $\begingroup$ Is the civilization in question based on the outer continent or the inner continent? If they have access to the tower it's going to be a lot easier. Without at least seeing the edges for themselves, it's going to be very difficult to prove you aren't on a conventional flat plane. $\endgroup$ – Cadence Oct 6 '18 at 20:51
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    $\begingroup$ What do you mean by "European dating and technology" (sounds suspiciously like online dating) ? I presume you mean to infer some period of European history and it's associated technologies, but what period ? $\endgroup$ – StephenG Oct 6 '18 at 21:05
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    $\begingroup$ Just do you know, your model is incoherent. You claim both that there is no magic, the surface is flat, and that when someone moves north of the northern border they come out the southern border. Those three things cannot be true simultaneously. It's like if I said that people live on a world that both has a sun, and does not have a sun; how long would it taken them to figure out their situation with respect to the sun? Well, no person could be in that situation, so the question is moot. $\endgroup$ – Greg Schmit Oct 6 '18 at 23:44
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    $\begingroup$ @GregSchmit First, there is magic in the world. But I excluded it from valid answers to the question. That's all. I want to know when, technologically/scientifically, people could figure out the shape. Second, you must not have read the question well because there is a tower at the corner points - Its existence from a worldbuilding perspective is specifically to prevent such occurances from happening. $\endgroup$ – Andon Oct 6 '18 at 23:57
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    $\begingroup$ @Andon Whether there is a tower at the "corner" is irrelevant, my example was to show that the model is incoherent. The model is incoherent even with the tower there. In fact, it is even more incoherent because the dimensions of the tower couldn't be defined since the space it occupies is undefined. $\endgroup$ – Greg Schmit Oct 7 '18 at 0:31
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Your not-Europeans would know that the world was flat in antiquity. The Greeks knew the Earth is round ca. 500 BC; this may have originated with or at least around the time of Pythagoras. It's thought to have been based on observations of the stars: the Greeks knew certain stars hung out near the North Pole, but from different places they appear at different heights above the horizon. This is consistent with a spherical world, with the changes in elevation corresponding to changes in latitude.

Your astronomers, however, would know that the stars are identical no matter where they are. (They might or might not still move, depending on whether your flat world rotates; it doesn't make a difference here.) They would know that the sea blurs out into an unresolvable haze rather than a distinct horizon, and that distant objects are fully visible when resolved, rather than the uppermost parts being visible first. These are basic observations that could be pieced together with ancient astronomical and other techniques.

Without traveling to the edge or corners, it would be impossible to know about the boundary conditions there. In fact it would be impossible to know you've passed over one of the edges from measurement alone. (It would be clear if you circumnavigated the world, of course, or otherwise ran into known landmarks.) Only by traversing the corners would the full shape of the world become apparent. In this case a number of geometric methods, such as measuring angles or distances involved in walking around the tower, could reveal that shapes there don't work the way they do in "regular" space. Again, this is a measurement that many well-known ancient civilizations (the Greeks and Egyptians spring to mind) could have carried out.

The final puzzle would be in figuring out how the edges are actually mapped together. This might rely on circumnavigation. Essentially there are two "maps" to consider: the order of some set of landmarks as you travel in a circle around the tower, and the directions those landmarks are from the center of the plane. Figuring out the first is trivial, but the second probably requires some people to actually go and find out; I suspect any analytic method will give multiple possible solutions.

TL;DR: Mapping whichever part of the world you're in (the center or the rim) is doable with Bronze Age know-how. Reconciling the two maps will require actually going and finding points of reference, which depends on how easy they find sailing.

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Magellan's fleet finished the first circumnavigation in 1522, after a three years trip. In the following centuries, the European powers send countless expeditions throughout the world, to discover and annex it, thus expanding and precising the map. Maps from the 16th and 17th century already show a round world, even if bits and pieces are still missing since nobody explored them yet.

The oldest globe map to have survived to us dates from the 1490s, although it has many errors and the American continent is, of course, missing.

So, taking into account that technology develop the same way in your world, I'd say they could prove it as soon as their 16th century.

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    $\begingroup$ Re-read the question. The question describes a weird surface, which cannot be physically realized in three-dimensional space. It's most definitely not a sphere. $\endgroup$ – AlexP Oct 6 '18 at 20:38
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    $\begingroup$ My point was that circumnavigation is not enough. The question asks how to prove that the world has the weird shape as described. $\endgroup$ – AlexP Oct 6 '18 at 20:51
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    $\begingroup$ The actual question in bold letters is: 'Using European dating and technology, when would humanity be able to prove this world shape?'. 'When', not 'How'. $\endgroup$ – Sava Oct 6 '18 at 20:54
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    $\begingroup$ Precisely, 'How' is a different question entirely (And one I intend to ask). $\endgroup$ – Andon Oct 6 '18 at 21:44
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    $\begingroup$ I agree with @AlexP , "as soon as their 16th century" doesn't describe a time period, but rather a point of advancement in tech and ingenuity. But these are obviously required for advanced cartography so, the terminology of "when" and "how" is tricky in this question. Perhaps the OP ought to have asked "how" and then decided on the "when". $\endgroup$ – Nahshon paz Oct 7 '18 at 17:55
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You know the geometry of your world when you can walk around a corner. You can infer it when you can tell you have walked across the edges, but you can't be sure of the geometry until you've done the corners.

If you were to walk around the tower [at the corners], you would walk in three full circles (Measured by angle) before reaching your starting point.

Unless I'm misreading your text, this is wrong (or at least contradictory).

See the diagram below. You walk from 1 to 2, then 2 to 3, then 3 to 4 and finally 4 to 1. That's four interior angles of the Octagon ( $135^\text{o}$ each ).

So you walk about a path $4\times135^\text{o}=540^\text{o}$ or one and a half times a normal circle (or the interior).

Your idea seems to be that the corners all overlap (which would lead to your "three full circles" idea). But you also want east-west and north-south wrap around. If you try to do this then the corners cannot all be together.

Now because four corners are involved in each tower circumnavigation, you only need to circumnavigate at two corners (one of the East-West and and one of the North-South set).

So when you people can manage the two trips and share their information they can work out the geometry of your world.

How soon they can do this depend on factors outside the question :

  • How well do people (governments) share information ?
  • How detailed is their map making ?
  • How large is the world and what means of transport do they have ?
  • Are their obstacles in the way (rivers, seas, mountain,) or weather problems ?
  • How reliable and fast is the spread of information ?

Note that navigational information would be considered potentially secret by governments.

This is all very detailed and there's no point in discussing it. But, e.g. the Vikings could have performed this navigation weather and distances allowing. If the corners are landlocked it can be done by anyone local. But how soon anyone can piece all the information together depends on the politics of your word as well as the technology.

enter image description here

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    $\begingroup$ I believe you misread what OP wrote about how the edges are linked: if North is Linked to South, then East is linked to West, North-East is linked to South-West and North-West is linked to South-East. Thus your points #1 and 3 on the map are wrong. $\endgroup$ – Sava Oct 6 '18 at 21:57
  • $\begingroup$ @Sava is correct. I've updated the question with a diagram and such to make it easier. $\endgroup$ – Andon Oct 6 '18 at 22:14
  • $\begingroup$ As a side note, I feel very sorry for whatever poor soul who would try to draw a map not centered on the central landmass... $\endgroup$ – Sava Oct 6 '18 at 22:20
  • $\begingroup$ @Sava that's because the model that the OP presented is incoherent. $\endgroup$ – Greg Schmit Oct 6 '18 at 23:45
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Even so, some wish to prove that the world is this shape, beyond "blindly" accepting religious dogma.

Look at the planet's shadow on it's moon. The shape of the shadow tells you the shape of the planet.

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    $\begingroup$ The planet cannot create on shadow on it's 'moons' given the set up as they orbit above the plane of the world. $\endgroup$ – Sava Oct 7 '18 at 17:56
  • $\begingroup$ "as they orbit above the plane of the world." I'd need to see a picture, but I'm pretty sure that's not how orbits work. $\endgroup$ – RonJohn Oct 7 '18 at 18:32
  • $\begingroup$ There's magic in this world, OP said it. $\endgroup$ – Sava Oct 7 '18 at 18:40
  • $\begingroup$ Please enlighten me as to why OP's work is 'poorly-thought' in your opinion? If you say that it is because the world OP describes does not conform to known laws of physics, I will begin to question why you are here on an SE dedicated to the construction of imaginary worlds and settings. $\endgroup$ – Sava Oct 7 '18 at 19:05
  • $\begingroup$ @Sava I commented on your instant use of magic to solve every poorly-thought scenario. "an SE dedicated to the construction of imaginary worlds and settings." Constructing a world means rational rules. Heck, even magic has rules: coppermind.net/wiki/Sanderson%27s_Laws_of_Magic Quoting, "If characters (especially viewpoint characters) solve a problem by use of magic, the reader should be made to understand how that magic works. Otherwise, the magic can constitute a deus ex machina." $\endgroup$ – RonJohn Oct 7 '18 at 19:14

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