What is the minimum radius of a circular corridor for the walls to appear straight?

Question

Where am I? It is so dark around here... Hopefully I have a torch. It looks like I am in a very long corridor.

Time passes

What? No this couldn't be! I walked straight in front of me during the whole time and now I am back to the start point!

Oh yes, it is possible. It is just a circular corridor like this one :

Thus, what is the minimum radius of a circular corridor for the walls to appear straight ?

The inside of the corridor is in the dark, the character (human) sees around her in a radius of 20 meters thanks to her fire torch. She has to think that the walls are straight because of both her vision and her touch if she touches the wall. Thus, she thinks it is a very very long corridor, but in fact, she is walking in a circle! The corridor width $R - r$ is about 2 meters.

Answer 1

TL;DR: Really, really big (exact numbers below)

Seeing the wall "crossing" is hardly the issue. Seeing the outer wall bend even slightly toward the inner wall - looking as if the hallway gets narrower further on - and the point at which that visually happens moving ahead at the rate at which you are moving already breaks the illusion of a straight surface.

That is not to say it cannot be done, but it is made much harder by the fact that you have TWO surfaces that need to appear to have the flat trait and appear parallel to each other. Apart from the above, this is what is known as a horizon problem.

Imagine the logical alternative: instead of building your round corridor on a flat plane, you build it along the axis of the planet. Depending on the height of your corridor, even this will be noticeable after even a short while. It will likely have a much lower ceiling than for example a tunnel meant for cars, and the horizon problem is actually a structural concern for the longest of those already.

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But, you provide a stipulation that may make this a little easier: the light limit. Science Focus tells us that the smallest discrepancy in visual data the human eye can perceive is two points that are 0.01 degrees apart in the field of vision. Therefore, 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.

Have fun walking!

EDIT For completeness sake: In humans, visual resolution at point-blank range (comparatively the only one that matters) is up to a factor 1000 times higher than tactile resolution. Therefore, any solution to this problem that will fool the eye will fool the hand (which, if in doubt, a coarse wall texture will resolve this without this extra data). Auditory spatial resolution is even lower, so echo will not be an issue.

Answer 2

SRM points out that even in a circle long enough that you won’t reach your starting point again if you're walking for an hour, and where you can’t see very far, one can still tell that the walls are curved.

So, hide that better. Make the path wavy, and the walls are even rougher. Any given visible wall may be convex or concave to varying degree. The bias is unnoticed. It may even use tricks to hide the systematic bias of curving in one direction by using differently sized bends so it “feels” more like they are balanced. The two walls may get closer or farther from each other, and the difference in the two bends is opposite from the appearance of which direction they are bending.

The size is given by the other constraints: how far does the hero walk in an hour?

The walls are not a simple uniform smooth path, so you can’t see the curve by looking at a small patch.

We assume the hero doesn’t have a compass.

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Being more flexible on the length (please make your question text accurate!), I think 1 or 2 hours is doable. The shorter the circuit the larger the curvature bias to overcome; so it's a matter of how wavy and how aggressive you need to be in introducing other techniques. For example, if you want the walls to remain the same distance apart, you might use a longer path. My gut feeling is that 12 miles, a 4-hour walk, would be easy to hide the curvature, with halls that are not more zigzag than a natural pathway, and straight enough that walking is easy.

For a definitive answer, you’d have to test people. The psychological aspects can’t be simply computed. A good description of the hall and the experience of walking through it would make these figures quite believable. And a reader won't be able to repudiate it, anyway, without testing such a model. And the written form is not exact, so you have to take the author's word that the gentle meandering and rough walls hid the slight bias of turning left more often than right, without having detailed measurements for those curves.

Answer 3

If you can afford to make it not straight, make it zig-zag. You can make the path shorter than a kilometer then!

Give the corridor a right-angle turn every $s$ meters. Make turns strictly alternate - left-right-left-right so the person could be sure without counting that the turns cancel each other out.

But the constructor (you) can cheat - make the left turns $d$ degress more than right-angle and right turns - $d$ degrees more.

Make all the corners rounded so the person couldn't measure the angle in corners.

The math

If you cheat $d$ degrees on each corner, you need $\frac{360}{2d} = \frac{180}{d}$ pairs of turns. If the distance between turns is $s$ meters, your tunnel would be $\frac{180n}{d}$ meters long. The path would be approximately $\sqrt{2}$ meters longer than the circle that it approximates, so you'd have a zig-zag path approximating circle circumference of $\frac{180n}{d\sqrt{2}}$ that corresponds to circle of radius $\frac{90n}{\pi d\sqrt{2}}$.

The numbers

Drainage right-angle bends are commonly 87.5 degrees. I have worked with those and it's hard to notice they are not right-angled. Example:

So you could try to cheat $d=2.5$ degrees on each turn. If you make a turn every $s=10$ meters, your path will be $\frac{180n}{d} = 720$ meters long which would approximate circle of circumference about $509$ meters and radius of about $81$ meters.

Yes, this path is not straight at all. But with this solution you could make the person return to go full circle within 5 minutes not many hours/days. Or you can increase the distance/decrease the angle cheating as you like to make the trip longer.

Answer 4

One other option, is a simple optical illusion. Paint vertical stripes on the walls (or use alternating colors of bricks). Make the stripes on the inner wall slightly wider than the stripes on the outer wall. The eyes and brain will interpret this as going straight. It is a variation of this idea: but imagine not being able to see the end of the room and nothing on the floor so that being on the . Unless it was a very tight curve, it would be very difficult to know the curve was there. You would most likely be convinced that it is perfectly straight. The only other requirement would be that small r be greater than 20 meters (2r=40m) so that the convergence of the two walls is out of sight. Of course as a few others have mentioned bigger would help hide shadow/reflection issues. Another option, as far as that goes, which would allow your maze to be as basically as small or large as you would like would be to be to paint the hallway with vanta black: http://www.maxim.com/news/scientists-vantablack-20-blackest-black-2017-4 (seriously cool stuff). Then even the horizon issue wouldn't be a problem, visually you could have big R be 2 meters(little r=0) and it would be fine (though you would have to worry about fitting the internal maze into 0 space, and I think taking two steps and running into a wall would be telling, so you might go bigger than that.

Answer 5

It is easy to fool people into mistaking a single slightly curved wall for a straight one, but this becomes much harder in corridors - if one wall visibly "crosses" over the other in the distance, or even becomes noticeably closer to it, the illusion will be shattered and it will become obvious that the hallway is curved. Therefore, the minimum radius of the maze will increase as the width of the hallways decrease, making the size of the maze extremely large either way.

You have a few options to make the maze continue to appear straight: give the maze a lot of right-angle turns with no long "straight" corridors. The person may count their right and left turns, making it seem as though they are going in a single direction until they arrive back where they started. If each "square" of the maze consists of a small square (actually trapezoidal) "room" with either two or three possible paths, but no single path more than three "rooms" in length, you can get away with a much more reasonably sized maze.

Alternatively, make the maze foggy, so it is impossible to see more than a few meters ahead even with a torch.

EDIT: Another option that could help - make the walls jagged, as if built roughly out of large stones. If the walls are rough around the edges, the off-centerness of the hallways could be chalked up to the roughshod nature of the maze rather than a deliberate choice of design, especially if the person in the maze is conditioned to expect grid-based mazes.

Answer 6

Your question immediately made me think of stories of people who walk in circles when they are lost in the desert. According to the first article in those search results

lacking external reference points, [humans] curve around in loops as tight as 66 feet (20 meters) in diameter, all the while believing they are walking in straight lines.

Therefore, all you need to do is take away your character's torch and make sure the walls of your corridors are sufficiently uneven so as to throw off the overall circular feel if one were to run their hand along it.

Something like this:

With visual references however, it would be much easier for her to tell that she is inside a circular corridor - as other answers have mentioned, one wall eventually crosses in front of the other when looking forward. Even if the light does not shine far enough to see them meet, it would need to be an enormous circle to eliminate the perception of the two walls at least getting closer to each other. Again, texturing the walls would help a bit, but probably not enough in a 20m radius.

Answer 7

You want the curvature of the walls to be unnoticeable.

This is equivalent to say that the deviation from the straight line after 20 meters has to be small, let's say 1 cm.

1 cm deviation at 20 meters correspond to an angle of 0.02 degrees.

Angular resolution of human eye: about 1 arcminute, approximately 0.02° or 0.0003 radians,1 which corresponds to 0.3 m at a 1 km distance. (source)

This means that this equation has to be verified (with $d=20 m$)

$R-\sqrt(R^2-d^2) < 0.01$

A radius R of at least 20 km will satisfy the condition.

Answer 8

I once got similarly disoriented despite being able to see the angles ahead of me as I walked the streets.

As a child, I was used to the regular street grid with 90 degree intersections in Germantown, Philadelphia. Our summer house was on Windsor Avenue, Cape May, New Jersey, in the western part of the city.

You should be able to find Windsor Avenue in a map of Cape May.

As you can see, Windsor Avenue goes off at about a forty five degree angle from Beach Avenue. But I thought of it as 90 degrees because I could only imagine right angle intersections. One day I left the house in the third block back from the beach and turned away from the beach for a walk. Windsor Avenue made a little turn to the left in that block that made the angle to Beach Drive even more different from the right angle I imagined.

Windsor Ave. ended after the next block so I turned left at a 90 degree angle on South Broadway. I thought that I had been going straight away from Beach Ave., so I thought the 90 degree turn should put me in a course parallel to Beach Ave. There as another slight turn in South Broadway which I ignored, thinking I was still walking parallel to Beach Ave.

And then I saw the Boardwalk and Beach Avenue straight ahead of me, and I wondered how I could have turned 180 degrees when I had only really noticed the 90 degree turn!

So in board daylight someone who sees every turn he makes can still find himself thinking his course is 90 degrees from what it actually is, if he has inaccurate preconceptions misleading him.

So imagine how easy it would be for someone in the dark with only a small light they carry to be accidentally or deliberately misled about their surroundings and direction.

1. I would suggest having pillars or columns or piers in the wall every ten or twenty feet. If they each jut out a foot or two and the person can only see a few tens of feet ahead, the person will only be able to see a few of the pillars or columns or piers ahead. The nearest pillars or columns or piers on each side will cast long shadows over the walls beyond them.

   The pillars or columns or piers can alternate in how much they stick out in a pattern. They might alternate like this:

   One foot, two feet, one foot, two feet, one foot, two feet, one foot, two feet.

   or:

   One foot, two feet, one foot, three feet, one foot, two feet, one foot, three feet.

   Or:

   One foot, two feet, one foot, three feet, one foot, two feet, one foot, four feet, One foot, two feet, one foot, three feet, one foot, two feet, one foot, four feet.

2. The walls can be made of artificial bricks or cinder blocks or artificial stone that bulges out a lot on the surface facing out. Thus feeling it will feel a surface going in and out, in and out, in and out, over and over again. Nobody will be able to feel how straight or curved the wall as a whole is over long distances. And the visual effect will be of a bumpy wall, not one those curvature is easy to measure.

3. There could be air conditioning that cools the corridor down to uncomfortable temperatures, and heating to warm it back up. And the heating could be hot air coming from grates spaced along the sides of the corridor, air hot enough to make waves in the air and make everything beyond look wavy. Thus the victim will see the more distant parts of the corridor waving in the heat waves and will be unable to judge how straight or curved the corridor is.

4. The walls can be built or painted with vertical panels of different colors. The floor and ceiling can have colored bands crossing from side to side the same width as the wall panels. Thus there will not be lines pointing along the direction of the corridor to follow into the distance to see if they are curved or straight.

5. Or everything can be painted black and the victim only sees very dim reflections from everything, thus not being able to see curves or straight lines ahead very well.

6. Or the corridor could be made of segments like little rooms. Each little room could be an oval about 2 meters wide by four meters long, with openings about one meter wide in the two short ends. Since the walls in each segment are curved, it will be impossible to see if the corridor as a whole is straight or curved. And each little segment will be tilted a fraction of a degree off the ones behind and ahead of it. That will be done by slightly changing the thickness of right and left walls on the side of each opening between segments.

   Each opening will be closed by a curtain that the victim will have to open. The curtains will be transparent plastic, with a lot of vertical folds that will distort the images of the segments ahead, so the victim will not expect everything to line up anyway.

   And there can be support columns in the center of each segment so the view directly through opening after opening after opening will be blocked, making it harder to see how straight or curved the corridor is.

I think that a combination of several of those suggestions should be enough to fool the victim.

Answer 9

Edit: The question seems to have changed slightly while I wrote this, let me rethink.

I'm no expert.. but I don't think it would be possible without the maze being incredibly vast.

Let's assume your character starts in the very centre of the maze in your example picture. If that chamber was any less than 40m wide (diameter of the circle) then her torch would illuminate it and it should be obvious it's circular. Even twice that size it should still be obvious when close to the walls that they are circular, the opening should make this even more obvious.

Going out to the next ring of the maze you would need the circumference to be large enough that it appears that each 40m section is straight so the character can not notice the curvature. It's hard to know what size this would make it but at a rough estimate I'd say you'd need the 40m sections to be about 1/50 to 1/60 of the total circumference (looking at where the wall segments start to look like a straight line) to even begin to make the curvature less noticeable. So for the very first ring of your maze you'd be looking at between 2km and 2.4km in circumference and a radius of approx. 380m.

If you extrapolate that outwards by the eight rings your example maze has your total radius should be 3km, making the outer wall nearly 19km long.

And to be honest I'm still not sure you wouldn't notice the curves at that size, plus the maze itself (as in the number of corridors and turns) isn't really that big so to make a more complicated maze you'd need it even bigger than that.

So why does the maze need to be curved, and why does the character need to not notice? If it's just to explain how they continually walk back on themselves... surely that's the entire point of a maze anyway?

Answer 10

Even with a radius 6371 km (radius of the earth), your corridor will appear to bend downward, as does the surface of the sea. Since on our planet you cannot make the radius any larger, the problem has no (terrestrial) solution.

With a torch one would only notice if the torch is held VERY close to the wall, or, in case of a full earth radius, to the floor. Since that's impractical, having a radius of say about 1 km would probably be enough.

To verify, visit the Large Hadron Collider...

Answer 11

Sadly, there isn't one answer for this.

The most popular answer here is based on the angular resolution of the eye. It, in effect, asked "how much curvature can there be before there is a detectable difference in the photons that hit the eye." However, there's an incredible amount of signal processing which goes on after that. Differences like that will simply go unnoticed.

As an example, consider the first two pictures here:

Would you believe the first two pictures have exactly the same geometry? Our eyes can fool us.

Our ability to see the corridor actually depends greatly on the nature of the walls and floor. We can detect curvature in the corridor faster if the texture helps us. For example, consider looking out at the ocean. The horizon looks flat. You can't see the curvature of the earth. However, if you put a boat on the horizon, you can see that the bottom part of the boat vanishes due to the curvature. That lets you see that the Earth is round.

So if the floor has texture to help you see the curvature, such as parallel lines running down the length of the corrdior, you're going to see the curvature much sooner than if it were a featureless corridor. Rounding the corners could make it even harder to tell.

Answer 12

Active corridor.

The corridor is a ring 1400 meters in diameter, at a reasonable pace you make one round every hour. Every twenty meters there's a narrow sconce in the wall, with an idol and a small votive light in front of it. The hero cannot see the whole corridor, but he can spot several lights in front of him in the darkness, and they describe two straight lines, converging in the middle.

He looks back, and he sees the same.

So he is sure the corridor is straight.

When he walks forward, the two large LCD screens (or magic constructs) that lurk just outside the torch's range move as well, displaying what an endless, straight corridor would look like were you to look at it from the hero's point of view.

This requires some Kinect-level magic to reliably tell where the hero's eyes are, but has the advantage of not needing any recognizable cues to thwart the perception, which could make the hero suspicious ("Why all this clutter? I almost can't tell whether the corridor is straight! Errr... hold that thought...").

A similar trick

using a grav engine and a corridor bending on the vertical plane

was pulled on a guy called Hulon in Theodore Sturgeon's What Dead Men Tell, and a wholly different principle was used on a larger scale in James P. Hogan's novel Endgame Enigma.

Flexible corridor.

The corridor appears to be solid, and unmoving. It actually is neither. It is a racetrack-shaped running mill, three hundred meters long, built to tolerances small enough that near the hero it appears to be solid. One hundred and fifty meters behind the hero, the slices making up the corridor unlock, bend a full 180 degrees like baggege conveyor belts in airports, and are rolled back in the opposite direction. If the corridor is sufficiently soundproofed and the movement is smooth enough, the hero will notice nothing from the inside, and he'll be in the middle of a thousand yard corridor that is perfectly straight.

Magic (or technological) light bending

By supplying vertical laminar flows of air heated and cooled at different temperatures through grilles in the floor and ceiling - they can be deactivated by the hero's pressure on the floor, to provide a more comfortable environment while he walks - it is possible to bend the light so that the corridor appears to be straight, even with better lighting than a fire torch.

This is the same effect that makes the sky reflect on a road on a hot day.

Of course the corridor will appear to shimmer, but it's unlikely that the hero is conversant enough with physics (or vertical mirages) to cotton up to what's happening.

Answer 13

Could you work with a path that is a polyline, with many left and right turns, not attempting to conceal the fact that the one traversing it is not moving straight ahead, but not betraying the fact that she is moving in a closed path, to the starting point? Unless one has a good "internal compass", after a few turns, they should lose track of their overall bearing. I think the simplest case which would achieve this is a 5-point star: you have 5 acute turns and 5 obtuse.

Answer 14

You need to specify something: was it the intent of the tunnel builders to make it confusing so someone thinks the tunnel is straight, or did they make a simple tunnel that just happens to be so big you only think it's straight?

If it's the latter, you run into the horizon problem mentioned by others: it's hard to pull off. If it's the former, that's easy. For example, and some of this was mentioned in other answers, don't make it a simple straight path. Have the tunnel jog around a bit. Have sections that are really straight; say you have a series of three identical rooms with doors in a line so you can see them, but you had to go sideways a bit entering the sequence and existing. If you look back, you see a perfectly straight path, but then you get into a section with tight turns and narrow passages where you can't see very far and you're gradually being turned in one direction, and then, look, another series of straight rooms.