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How large (in terms of height and radius) would an outdoor circular wall (like a city wall on a colossal scale) need to be before a casual observer would perceive the wall as "straight"? I found "What is the minimum radius of a circular corridor for the walls to appear straight?" and based on this answer:

...the tunnel must bend less than 0.01 degrees over the proposed 20 meters visual range. Therefore, the tunnel would need to be (360/0.01)*20 meters in length, or 720 kilometers. This will put the inner radius at about 115 kilometers.

some examples of distance to the horizon courtesy of Wikipedia, and some hasty back-of-the-envelope math I came up with a minimum diameter of ~53,000km which seems... high, given that the Earth itself only has a diameter of ~13,000km. Is my math just completely off, is this formula incorrect for this problem, or is a wall that big just not possible?

Caleb Brinkman
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  • While the questions are similar (I actually referenced the possible dupe in my question) they are different in that I am not concerned with a corridor (implying an enclosed space) but with an outdoor wall; I will clarify in my question. – Caleb Brinkman Jun 26 '19 at 00:01
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    Yeah, in the original question there was an explicit 20-metre visual range, in this one it's as far as the horizon. – F1Krazy Jun 26 '19 at 00:06
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    Our [help/on-topic] explains that questions must be specific and answerable, must include context, must include restrictions/requirements, and should include research. This is technically a straight physics question, so we usually want to know why your asking (context) so we feel kinda warm and fuzzy that we're not simply helping you with homework. We're lenient with new users, but please keep it in mind for future questions. Thanks! – JBH Jun 26 '19 at 04:57
  • @JBH apologies, and understood; honestly didn't have much context other than the question I linked which got me thinking about mine – Caleb Brinkman Jun 26 '19 at 05:04
  • Oh, that's cool. Frankly, telling us that a previous question inspired this question is sufficient context. It just needs to be said to keep all the Ts crossed and Is dotted. – JBH Jun 26 '19 at 05:10

3 Answers3

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Relevant facts and formulas:

In order for a circular wall to appear straight, the wall needs to deviate from straight by no more than one minute of arc over the course of the visible section. Since we're dealing with small sections of absurdly huge structures, we can use the small-angle approximation and treat the arc of the circle as a straight line. This gives a formula for the radius of the circle as $r \approx 3400 d$

The absolute limiting factor on how far you can see is the atmospheric opacity, not the horizon distance. Under exceptional seeing conditions, you can see high-contrast objects about 300 km away. Plugging that into the formula gives a circular wall with a radius of around a million km. You're going to need a medium-large star to build that wall on (maybe build it at night?).

But let's say we don't care about a full circle, we just want an arc that disappears over the horizon without looking horizontally curved. Assume a two-meter-tall observer trying to spot the curvature of a two-meter-high wall: the top of the wall disappears below the horizon at a distance of 10 km (5 km observer-to-horizon + 5 km horizon-to-wall). You'll need a radius of curvature greater than 34000 km.

Mark
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  • Interesting; I was definitely way off. Would the height of the wall in the 2m high observer scenario (which is more in line with what I was thinking) affect the minimum radius? Intuitively I imagine that making the wall higher would make the curvature more difficult to detect but I'm not basing that on anything solid – Caleb Brinkman Jun 26 '19 at 03:02
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    Making the wall higher will actually make the curvature easier to detect. For example, if the wall is four meters high with a radius of curvature of 34000 km, you can sight down the length of the wall for 15 km before it vanishes below the horizon, and the visible deviation is above the critical 1 minute of arc. – Mark Jun 26 '19 at 03:07
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    Ok, I understand; thanks for the clarification, and for "maybe build it at night" :) – Caleb Brinkman Jun 26 '19 at 03:53
  • How could it require a greater radius than the Earth, considering the Earth appears straight (flat)? – komodosp Jun 26 '19 at 10:36
  • This may be a candidate for a separate question, but what if I were unconcerned with the curvature of the planet, only whether detecting the wall curves in to form a circle? I.e. An observer could see the wall disappear over the horizon and determine that it is curved with the planet, but would the observer also be able to tell that the wall is circular? Or is that actually the same question and I'm just not understanding clearly? – Caleb Brinkman Jun 26 '19 at 15:07
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    @colmde: The Earth appers flat only when the observer's view is obstructed by objects situated relatively nearby. In good conditions, e.g., at sea, the curvature of the Earth is clearly visible. Mariners have known that the Earth is spherical since just about always, and they certainly did in the 5th century BCE. (At sea, the horizon is clearly closer than the distance where an object would be rendered invisible by atmospheric opacity; a sailor will immediately notice that as objects became more and more distant they sink below the horizon.) – AlexP Jun 26 '19 at 16:15
  • @CalebBrinkman: You are near the wall, and look parallel to the wall. If the wall is straight it will appear to go the horizon and descend below it. But if the wall is curved it clearly appears to end before going below the horizon. – AlexP Jun 26 '19 at 16:18
  • @AlexP I completely understand; thanks for the clarification! – Caleb Brinkman Jun 26 '19 at 16:21
  • Final clarification (I promise): would there be a difference for an observer "inside" the curve of the wall as opposed to "outside"? – Caleb Brinkman Jun 26 '19 at 16:43
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    @CalebBrinkman, someone "outside" will have an easier time. Outside, you just look for the wall ending in a vertical line rather than dropping below the horizon; inside, you need to visualize what a straight wall would look like and notice that the wall deviates from that. – Mark Jun 26 '19 at 20:17
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The simple answer is that your wall should curve downwards rather than around, and thereby form a Great Circle with the same diameter as the planet/moon it's built on but still appearing perfectly straight.

If you build it on the Moon, its diameter would be smaller than if you build it on Earth.

You can actually stretch this technique slightly to have your wall form a Small Circle, such as one of the lines of latitude. In this case the wall would technically curve slightly to one side as seen from the surface, but it might not be noticeable.

Chromatix
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First suggestion:

My answer in this question What is the minimum radius of a circular corridor for the walls to appear straight?1 discussed various architectural methods to trick someone into thinking that a circular corridor was actually straight. Most of those methods that might work indoors wouldn't work so well for an outdoor wall, but possibly something similar might work to some degree out side.

What if the wall is not perfectly circular?

Perhaps the wall has semicircular towers no higher than the wall itself attached to it at regular intervals. So when the viewer of the wall looks to right or left the sections of straight wall between towers will look shorter and shorter with increasing distance until the viewer will only see the front edges of the semicircular towers one behind the other in the distance. And the farther away each tower is from the one nearer to the viewer, the less the front of it will seem to project beyond the front of the tower near to the viewer. Finally the farthest off towers will be impossible to tell one from the other.

Or perhaps the wall will have independent bastions in front of it at regular intervals, connected to the main wall by bridges or walls. So as a traveler gets closer and closer to the wall and is closer to being able to sight along the length of the wall to see if it is straight or curved, the traveler will have to look through any openings that might be in the bridges or wall connecting the bastions to the main wall.

If there are no openings in walls connecting the bastions to the main wall, each space between two bastions will be a separate bay of the wall structure. Once a traveler is closer than the outer edge of the bastions he will be able to see only a short stretch of the main wall.

Possibly a traveler reaching the city or country enclosed by the wall will travel across a plain and the wall will slowly appear as a dark line on the horizon and appear taller and taller as the traveler approaches.

The wall might have a moat many miles wide in front so the traveler will have to approach along a causeway or bridge for miles, with the nearest other causeway or bridge miles away over the horizon to the right or the left. Thus the traveler will be unable to walk around the outside of the wall and notice from the sun in the day and the stars at night that the wall is curved.

Second suggestion:

If your story is a fantasy you can set it on a flat world that could be as large as needed and your wall can be as many thousands or millions of miles or kilometers in diameter as it needs to be to be circular and appear straight. And in a science fiction setting you might be able to create an analog of a flat Earth in some gigantic artificial construct in space built by a highly advanced society. Something like some of the constructions discussed in Larry Niven's "Bigger Than Worlds": http://www.isfdb.org/cgi-bin/title.cgi?133302

M. A. Golding
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