The golden menhir on the mysterious plain

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

EliezerYudkowsky
Posts: 6
Joined: Sun Jan 20, 2013 5:20 am UTC

The golden menhir on the mysterious plain

Postby EliezerYudkowsky » Mon Mar 03, 2014 9:58 pm UTC

I would like to put a Lovecraftian spatial arrangement into a possibly-eventually-forthcoming story (not Methods of Rationality). My ability to visualize curved space isn't what it should be, so I'm turning the following literary question over to you, XKCD.

Suppose you wake up in a mysterious grassy plain with the following property: If you start from any place and walk in almost any direction, you arrive at or near the same point on the mysterious plain, which we'll term the 'Center'. We can suppose the Center is marked by a giant golden menhir (upright stone) which also lets us obscure the exact central point in case it has singularity-like properties. It requires at most 1 kilometer to reach the Center from almost all starting points.

The question is then: What sort of spatial geometries would have the Central Menhir property, and how would things appear as you approached the Center?

In previous discussion (on Facebook), it was suggested that a toroidal geometry in which the "Center" was identified with the inner ring would have the property that almost all straight lines would intercept the Center, though a small positive measure of lines would take arbitrarily long distances to get there. I then observed that if you got near the menhir, stuck out your right hand so that it pointed at the menhir, and walked forward (i.e. initially perpendicular to the line between you and the menhir) you ought to naturally circle the menhir without turning. (Implied: You ought to see your own back when looking in that direction.) But then someone else observed that if you held your right hand toward the menhir and walked in a straight line, you ought to reach your starting point after N meters, the same N *regardless* of where you were on the mysterious plain. This doesn't sound like, waking up in the middle part of the mysterious plain, the plain would initially appear to have locally 'normal' (accustomed) geometry, which is a desideratum. You would be able to see your back from N meters away (the diameter of the Menhir itself) in a certain direction, no matter where you stood.

This space does not otherwise need to obey any physics like the physics we know, unless it helps. I.e. you can ignore General Relativity.

The questions are as follows:

(1) What is a potential geometry with (a) the Central Menhir property that almost any straight path will go to or near the Center after a kilometer or so, that (b) will initially seem normal if you arrive 500 meters from the Center?

(2) Assume that the vertical dimension goes 'straight' up and down. And suppose the source of light on the mysterious plain is a single Sun-like object located 1 kilometer above the Center. What do you see when you look up at the sky? There's probably more than one Sun, but where are they? In a cubic grid? Spreading out in circles? Are there concentric, dimming rings of fire in the sky? How does this change depending on where you're standing?

(3) How do things look in the distance? From which vantage point? How do they change if you run forward?

(4) How do things look in the near vicinity of the Central Menhir?

(5) Does the geometry imply or allow any other special points with their own special geometries, while still preserving the Central Menhir property?

User avatar
Qaanol
The Cheshirest Catamount
Posts: 3035
Joined: Sat May 09, 2009 11:55 pm UTC

Re: The golden menhir on the mysterious plain

Postby Qaanol » Wed Mar 05, 2014 12:05 am UTC

I am not a topologist, but here’s a remark on the torus: what you would “see” in a toroidal geometry is, essentially, a menhir at every integer lattice point of the Cartesian plane, and a copy of yourself in every integer grid square. The only way to break that up, is terrain that impedes your view.

Next, to get a menhir topology, if you want local smoothness and the property that heading away from the menhir brings you back to it, there needs to be some sort of wraparound effect. That could be spherical, toroidal, or something more exotic. For example, you could take the Riemann sphere with straight lines defined as the projection of the straight lines from the plane. Then every straight line is a “circle” on the sphere that passes through the north pole, where the menhir would be.

I feel like you could also do something with the Alexander horned sphere by asserting that the limit point of each branch is the golden menhir, but then you lose any concept of traveling “around” the menhir.
wee free kings

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5959
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: The golden menhir on the mysterious plain

Postby jestingrabbit » Wed Mar 05, 2014 3:03 am UTC

I would probably go for a multiholed doughnut, as a factor of the hyperbolic plane. So, make a regular octagon in the hyperbolic plane, so that its angles are pi/4, sew it up so that opposite sides are identified/sewed together, and you have yourself an object with the menhir property ie almost all geodesics are dense. You can see a geometry that is like this in some of Escher's work, but he doesn't do what you actually need.

Perhaps even more bizarrely, you could use a non compact fundamental polygon, so a regular octohedron, but instead of all the angles being pi/4, you could have them be 0. This way, you can get an arbitrarily large distance from a menhir, and almost all geodesic walks will lead you an arbitrarily large distance from the menhir, but eventually, with probability 1, you will come back towards it.

The lights position thing is hard in these cases, and I can imagine trying to solve it, but... its weird. For instance, as you go further and further from the menhir on the non compact fundamental polygon, you would maybe see the suns strung out, like beads on a necklace wrapping around, getting closer together and further away from you as they skirted the horizon, with a sort of dark patch where the unbounded part of the polygon is, but maybe not. Its weird to think about, intuition is pointless to consult, and the equations are non trivial.

Regarding your other questions, the points near the point at infinity are the obvious for number 5, but... I'd have to think a lot more to answer your other questions.

This is all geodesic orbits on closed surfaces of constant negative curvature, if you want to look something up.

Eta: here is a pretty good explanation of hyperbolic geometry, and if you crane your neck you can sort of see the spaces I'm talking about.

http://www.josleys.com/article_show.php?id=83

Eta2: oh, and the brightness of light would attenuate differently. Again, non trivial, but the idea would be that vertically, the change in brightness would look like it does in Euclidean space, because you want that direction to be "straight", but the brightness would drop as something like C/e^r where C is a constant and r is the radius. Otherwise, if you just used the old C/r^2 formula, total light at source would be dwarfed by total light at a distance.

Eta3: and that's before you start summing up the light from all sources to calculate some sort of ambient brightness. There might be weird bright spots and dark spots, or the sums might diverge, meaning you'd need a better model. I think my back of the eyelid calculation tells me that the light incident at a point would diverge, but I'd need to actually do the calculation.

Eta4: okay, in a toroidial space like the one suggested by Qaanol, you definitely get infinite buildup of light. The total light incident at a point would be something like

CodeCogsEqn.gif
CodeCogsEqn.gif (941 Bytes) Viewed 4904 times


where h is the height of the light, and the unit of distance is "minimum distance to walk back to the menhir in a straight line". I've got an argument that says that is unbounded, but I expect wolfram alpha or similar can spit out an answer. If its giving you trouble, replace h with 1, and the answer will be unbounded iff the answer you get for the simpler sum is unbounded. (this assumes the light has always existed, if it comes into being at a fixed time, then the light incident at a point will grow with ln(t)).

As for what happens in the hyperbolic case, much harder to work out, which is a good sign.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

User avatar
snowyowl
Posts: 464
Joined: Tue Jun 23, 2009 7:36 pm UTC

Re: The golden menhir on the mysterious plain

Postby snowyowl » Wed Mar 05, 2014 12:33 pm UTC

Can someone with more knowledge tell me what you break if you lose the assumption that geodesics are the shortest paths between two points?

Because I was thinking of the Riemann Sphere. A straight line between two points is the circle that passes through them and the point at infinity. Effectively, every straight line passes through a common point - you could put the Menhir there. At the same time, it's a sphere, so it's compact and (with the right metric) no circle can have a greater radius than the sphere itself. And it maps conformally to the real plane, so locally everything is fine and everyday and even flat.

If you do that map, the mysterious plain becomes a roughly-circular bubble in an infinite expanse of Menhir-substance, which is how it appears from every point on the inside. But in that visualisation, you get the odd property that objects become larger the closer they are to the Menhir. Any attempt at measuring the Menhir's size gives a sensible result, indeed it would appear to be surprisingly small. But there's nowhere actually in the sphere that you could stand to see how tiny it really is - it will always appear to surround you in all directions. So even if it does work, it might not be narratively suitable.

Alternatively, the Menhir need not be actually at the singularity, just nearby. If there's a 3D version of this metric, the singularity could be directly above or directly under it. Then it would appear enlarged, but not actually turned inside-out. I have no idea whether that would work - gravity would probably need to be houseruled to point in certain directions. And the singularity would still appear to surround everything. Well, that's not necessarily bad - embed it in something opaque and sky-blue, and the place might actually look like it was on Earth at first glance.

But in 3D, you don't walk along a general geodesic, you have to fly or tunnel for that. So now we need a layout for the 2D ground that more-or-less matches the last one. I hope we can find one - I've got a sketch on my desk of an off-centre sphere of dirt in a larger sphere of "sky".
The preceding comment is an automated response.

lalop
Posts: 210
Joined: Mon May 23, 2011 5:29 pm UTC

Re: The golden menhir on the mysterious plain

Postby lalop » Wed Mar 05, 2014 2:26 pm UTC

I'm not a geometer, but it seems to me you just take a closed disk (for the surface) x R (for height), identify all the points on the boundary of the closed disk, and call that the Center. Alternatively, identify all antipodal points on the boundary. The Central Menhir would then be a cylinder obscuring [part of] the boundary.

More rigorously:
Spoiler:
Let r be the radius of the disk, and define clD = {(x,y) : x^2 + y^2 ≤ r^2}. The boundary, aka the center, is then bdD = {(x,y) : x^2 + y^2 = r^2}. Define the equivalence relation ~ to be: (x_1,y_1,z_1) ~ (x_2,y_2,z_2) iff (x_1,y_1),(x_2,y_2) ∈ bdD. Then take the model of the world to be clD x R / ~.

The previous "identified boundary" is easiest, but leaves undefined the notion of "if I go straight toward the boundary, from which direction do I come out?" If this question is important, one can instead use, say, an antipodal equivalence relation on bdD: (x_1,y_1,z_1) ~ (x_2,y_2,z_2) iff (x_1,y_1) = ±(x_2,y_2) and z_1 = z_2.


To define the central menhir, let 0 << m < r. The central menhir would then be the torus between radii m and r: {(x,y) : m^2 ≤ x^2+y^2 ≤ r^2} x H (where H is the set of heights the menhir covers).
Edit: Even more rigorously: see post below.



This solution is reminiscent of the joke:

An engineer, a physicist, and a mathematician are shown a pasture with a herd of sheep, and told to put them inside the smallest possible amount of fence. The engineer is first. He herds the sheep into a circle and then puts the fence around them, declaring "a circle will use the least fence for a given area, so this is the best solution." The physicist is next. She creates a circular fence of infinite radius around the sheep, and then draws the fence tight around the herd, declaring, "This will give the smallest circular fence around the herd." The mathematician is last. After giving the problem a little thought, he puts a small fence around himself and then declares,"I define myself to be on the outside."

in this case, the Center is defined to be the boundary of the model... yeah. I'm assuming that the world somehow corresponds to, or is perceived as, is constructed to be an "inverse" of this model, with the menhir a small rock in the center rather than a surrounding chasm.


The following thought experiments come from assuming considering that distance follows the model rather than what you'd normally expect.

In previous discussion (on Facebook), it was suggested that a toroidal geometry in which the "Center" was identified with the inner ring would have the property that almost all straight lines would intercept the Center, though a small positive measure of lines would take arbitrarily long distances to get there. I then observed that if you got near the menhir, stuck out your right hand so that it pointed at the menhir, and walked forward (i.e. initially perpendicular to the line between you and the menhir) you ought to naturally circle the menhir without turning. (Implied: You ought to see your own back when looking in that direction.) But then someone else observed that if you held your right hand toward the menhir and walked in a straight line, you ought to reach your starting point after N meters, the same N *regardless* of where you were on the mysterious plain. This doesn't sound like, waking up in the middle part of the mysterious plain, the plain would initially appear to have locally 'normal' (accustomed) geometry, which is a desideratum. You would be able to see your back from N meters away (the diameter of the Menhir itself) in a certain direction, no matter where you stood.


If the menhir covers all height, all points being identified would be hidden, and you'd get no identification effects such as seeing yourself (though still weird distance effects; see 3,4) and could just identify all boundary points.

If the menhir doesn't cover all height, there'd be identity issues. If you (or light) hit the boundary at some point, you'd need to specify where and how you'd come out; to this end, the antipodal identification (for example) is more useful than identifying the entire boundary. If you looked straight (i.e. radially) at the Center, you could see your own back (if you could see far enough). At an angle, you would eventually see the menhir from another angle, and looking past that the menhir from another angle, and so on.

(2) A menhir that does not cover all height would, of course, imply "infinite suns" at the Center. However, depending on its height, the menhir could obscure most of them (you'd only see the ones close enough to "angle" over the menhir"). You'd get the sun and progressively smaller copies of the sun down to the top of the menhir. If there were no menhir, this would continue on to height zero for infinite suns. I haven't done the math yet, though I strongly suspect that would imply infinite light.

(3)-(4) Angular movement (i.e. how far it takes you to circle x degrees around the menhir) would require more distance the closer you get to the Center, since in the model you're walking around the edge of the disk. Radial movement would remain "normal".

Another way of looking at this: as you approached the Center, you would get thinner in the angular direction (i.e. perpendicuar to the radius). After all, you "cover" only a small angular portion of the boundary by the time you reach it. You would retain the same height, however. In the radial direction, I think it can go either way (you can get thinner or not) depending on how exactly the model corresponds to the world.

If grass, rocks, etc are native to the model, then they would similarly shrink (i.e. look "small" with respect to the menhir from far off) and the local environment would be the size you expect. If, on the other hand, they're native to the world (i.e. look "normal" with respect to the menhir from far off) then they would grow relatively to you as you get thinner, the closer you get to the Center. I'm guessing the latter is more desirable from a story standpoint.

This also implies the Center has to be a vertical cylinder rather than a vertical line: if it wasn't, you'd have to shrink to a line yourself, and I'm pretty sure the math doesn't work at those scales.

(5) Shapes other than circles are also possible. For example, take the square with opposite sides identified (aka the torus). If you defined the whole boundary of the torus (rather than just one boundary like last time) to be the Center, and in the world made it correspond to as a "central rectangle" rather than a "central cylinder", most of the ideas should carry over. However, flattening would not be as simple as "in the angular direction". If you approached the middle of one edge of the central rectangle, you would become thinner in the direction parallel to that edge, and if you moved away from the middle, you would become thinner in the perpendicular direction as well (since, close by, it only takes a short perpendicular distance to get yourself close to the other boundary).
Last edited by lalop on Wed Mar 05, 2014 9:17 pm UTC, edited 6 times in total.

lalop
Posts: 210
Joined: Mon May 23, 2011 5:29 pm UTC

Re: The golden menhir on the mysterious plain

Postby lalop » Wed Mar 05, 2014 8:38 pm UTC

Here's a more rigorous refinement to my suggestion above (with actual space rather than just "somehow corresponds to"!) Can someone with knowledge in geometry please check that what I'm saying is sane?


Let cD be a closed disc of radius r_d, bD be the boundary of that disc. Let ~ be the relation on bD identifying all antipodal points. We start with a "model space", M = cD/~. The points of bD are called the Center of M.

We construct Lovecraftian World W homeomorphic to M as follows. Let r_c be the desired radius of W's Center. Let A be the closed annulus with inner radius r_c and outer radius r_c + r_d, bA be the boundary of this annulus, and I be the relation on bA that identifies:

  1. All points of the outer boundary (the points at the outer radius) to the same point
  2. Each pair of antipodal points of the inner boundary (the points at the inner radius)

Define W = A/I.

Define a homeomorphism f : M → W in polar coordinates as follows: f(r,theta) = (r_c+r_d-r,theta). In particular, the point (0,theta) [i.e. the center point of M, not to be confused with the Center of M] is mapped to the outer boundary of the annulus, and the points (r_d,theta) of the Center are mapped to the inner boundary of the annulus (considered the Center of W).

Let M have the Euclidean metric. We define a corresponding metric on W by pulling the points back through the homeomorphism: d_W(x,y) = d_M(f^{-1}(x),f^{-1}(y)). Since the geodesics on M are the straight lines, all of which pass through M's Center, we have the geodesics in W as images of the straight lines (f(L) where L is any straight line in M), which also pass through W's Center by construction.

In W, as you get nearer and nearer to the Center, you undergo length contraction along the angular direction but not the radial direction. Different choices of f will allow for contraction or expansion along the radial direction, however.

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5959
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: The golden menhir on the mysterious plain

Postby jestingrabbit » Thu Mar 06, 2014 1:36 am UTC

I don't think the Riemann sphere works here, nor do I think lalops circle with the circumference identified with itself makes sense. To explain why, I'm going back to the OP to extract the specification in mathematical language.

EliezerYudkowsky wrote:Suppose you wake up in a mysterious grassy plain
the plain would initially appear to have locally 'normal' (accustomed) geometry


From this we have that every point that we can wake up on has to be part of a 2-manifold ie a surface. This basically means that if you cut out part of the mysterious plane and fitted it to a Euclidean plane, you might have to stretch the outer edge a little, but not much ie its locally Euclidean.

If you start from any place and walk in almost any direction


There's an underlying assumption here that you can pick a direction and walk in it. That means that the manifold isn't just a manifold, it needs to be a Riemannian manifold, and the straight lines that you walk along would be what are known as geodesics.

Now, some manifolds are nicer than others, and in particular, I think constant curvature would tend to make you believe you were on a plane, rather than some stranger, lumpier place.

If you start from any place and walk in almost any direction, you arrive at or near the same point on the mysterious plain, which we'll term the 'Center'.


This, I think, is best identified as the requirement that almost every geodesic is dense. That means that, with probability 1, if you pick a direction and walk in it, you'll get within epsilon distance, of every point on the surface. It doesn't guarantee exactly what you want, the time that it will take to get back to the centre will vary, but... something's gotta give.

So, to summarise, if I were you, I would want:

1) a Riemannian manifold,

2) with constant curvature,

3) with a preponderance of dense geodesics.

There are lots of spaces that satisfy these three conditions, two of which have already been mentioned.

The first is the torus, realised as R^2/Z^2 ie the square (or rectangle) with opposite sides identified, the so-called flat torus. The only geodesics that aren't dense here correspond to lines with rational gradients (if we are talking about the square).

The second is compact surfaces with constant negative curvature, which can be realised as taking a polygon in the hyperbolic plane, and identifying edges in a nice way, like the way that the edges of a square are identified to obtain the flat torus.

In both of these cases, the centre is just a place, its not going to be some sort of mind twisting weirdness by itself. They both have the property that for almost all geodesics you will spend (Area of Centre)/(Total Area) in the area defined as the centre if you walk in a straight line, with probability 1.

In previous discussion (on Facebook), it was suggested that a toroidal geometry in which the "Center" was identified with the inner ring would have the property that almost all straight lines would intercept the Center, though a small positive measure of lines would take arbitrarily long distances to get there. I then observed that if you got near the menhir, stuck out your right hand so that it pointed at the menhir, and walked forward (i.e. initially perpendicular to the line between you and the menhir) you ought to naturally circle the menhir without turning. (Implied: You ought to see your own back when looking in that direction.) But then someone else observed that if you held your right hand toward the menhir and walked in a straight line, you ought to reach your starting point after N meters, the same N *regardless* of where you were on the mysterious plain. This doesn't sound like, waking up in the middle part of the mysterious plain, the plain would initially appear to have locally 'normal' (accustomed) geometry, which is a desideratum. You would be able to see your back from N meters away (the diameter of the Menhir itself) in a certain direction, no matter where you stood.


I've been trying to wrap my head around this. Its not clear if you're talking about the flat torus or not, and... it just seems weird. The centre should just be a point, imo. But regardless, the torus can work, it just needs to be flat, and the centre a single point.

This space does not otherwise need to obey any physics like the physics we know, unless it helps. I.e. you can ignore General Relativity.


I think this is a huge mistake.

You're talking about a story where there are two characters: are a person walking around a strange terrain; and the terrain through which the person is walking. Good authors talk about the characters surprising them. For that to occur, they need an internal truth, a set of rules that govern their perceived reality, actions and feelings. For a real person character, this means an emotional honesty, their actions and emotions arising naturally out of their interactions with circumstances they find themselves in.

If you're not trying to make it obey things like conservation of energy/mass, and if light can travel along whatever paths you want... you lose the element of reality being truly surprising. I mean, the fact that a standard light source in TxR (the flat torus cartesian product the reals (for the vertical direction)) ends up limitlessly increasing the illumination at every point is, I think, fascinating, and the sort of thing that you could use to make a person go mad, if the light had some other property.

Similarly, in the hyperbolic case, with an uncurved vertical dimension, you end up with things seeming taller than they are. That is, if one were equipped with a surveyors tools, and you measured the height of the menhir using a theodolite and normal trig, you would end up different heights when you measured from different places, with greater distance leading to greater height. This is because of the path of photons going along shortest paths.

menhir.png
A poor diagram to illustrate what is going on. The red line is the path of a photon from the top of the menhir, bending downwards, making the surveyor percieve the height to be greater than it is.
menhir.png (10.34 KiB) Viewed 4697 times


This could lead to the surveyor thinking that they are different menhirs, having different heights, when really it is one menhir viewed from different vantages.

I'll provide details if you're interested, but these sorts of surprising elements can only arise if you give the mysterious plane an "inner truth".

__

anyway, my two cents on the sort of things that you could reasonably initially identify as a plane, that have some of the strangness required. Neither the Riemannian Sphere, nor lalops suggestion make sense imo. To have the requisite geodesics, they would end up with the central menhir being a central concave wall, I believe.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

User avatar
Xanthir
My HERO!!!
Posts: 5202
Joined: Tue Feb 20, 2007 12:49 am UTC
Location: The Googleplex
Contact:

Re: The golden menhir on the mysterious plain

Postby Xanthir » Thu Mar 06, 2014 1:53 am UTC

jestingrabbit wrote:anyway, my two cents on the sort of things that you could reasonably initially identify as a plane, that have some of the strangness required. Neither the Riemannian Sphere, nor lalops suggestion make sense imo. To have the requisite geodesics, they would end up with the central menhir being a central concave wall, I believe.

Being a central wall is just as good as being a central cylinder, no?
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5959
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: The golden menhir on the mysterious plain

Postby jestingrabbit » Thu Mar 06, 2014 2:10 am UTC

Xanthir wrote:
jestingrabbit wrote:anyway, my two cents on the sort of things that you could reasonably initially identify as a plane, that have some of the strangness required. Neither the Riemannian Sphere, nor lalops suggestion make sense imo. To have the requisite geodesics, they would end up with the central menhir being a central concave wall, I believe.

Being a central wall is just as good as being a central cylinder, no?


One leaves you feeling hemmed in, the other you might believe you can escape. I'd prefer to offer false hope if I was an eldritch horror I expect. And the spec was for a menhir.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

lalop
Posts: 210
Joined: Mon May 23, 2011 5:29 pm UTC

Re: The golden menhir on the mysterious plain

Postby lalop » Thu Mar 06, 2014 5:11 am UTC

Come to think of it, if light travels along the geodesics of M, then even though you'd get all the previously analyzed length contraction and whatnot in W, you'd still see the world as if it were M.

But if (this probably breaks physics), light travels along the "straight line" geodesics of W under the Euclidean metric, but otherwise the distance metric is d_W, I think you would actually see the Center as the small cylinder that it is, see other peoples' length contraction from afar, etc. In other words, sight-wise, W is an annulus with identified points, but distance-wise, W is M "inside out".

User avatar
snowyowl
Posts: 464
Joined: Tue Jun 23, 2009 7:36 pm UTC

Re: The golden menhir on the mysterious plain

Postby snowyowl » Thu Mar 06, 2014 10:41 am UTC

jestingrabbit wrote:
EliezerYudkowsky wrote:Suppose you wake up in a mysterious grassy plain
the plain would initially appear to have locally 'normal' (accustomed) geometry


From this we have that every point that we can wake up on has to be part of a 2-manifold ie a surface. This basically means that if you cut out part of the mysterious plane and fitted it to a Euclidean plane, you might have to stretch the outer edge a little, but not much ie its locally Euclidean.
I think all the suggestions so far meet this requirement, except that some of them are curved. But since the space is kilometers across, the curvature could be locally small enough that it's not noticeable.

jestingrabbit wrote:There's an underlying assumption here that you can pick a direction and walk in it. That means that the manifold isn't just a manifold, it needs to be a Riemannian manifold, and the straight lines that you walk along would be what are known as geodesics.

The manifold needs to have an affine connection, but this is a slightly weaker condition than being Riemannian. I'm nitpicking, but there are non-Riemannian manifolds which still have geodesics, and I think they're worth investigating.

jestingrabbit wrote:
If you start from any place and walk in almost any direction, you arrive at or near the same point on the mysterious plain, which we'll term the 'Center'.

This, I think, is best identified as the requirement that almost every geodesic is dense. That means that, with probability 1, if you pick a direction and walk in it, you'll get within epsilon distance, of every point on the surface. It doesn't guarantee exactly what you want, the time that it will take to get back to the centre will vary, but... something's gotta give.

I don't think this interpretation really works. The description implies that the Menhir is interesting because it has certain mysterious properties. If every point on the surface shares those properties, then it's not quite as interesting. The Menhir is just a rock now.

jestingrabbit wrote:To have the requisite geodesics, they would end up with the central menhir being a central concave wall, I believe.
This is a very good point, and I agree with everything else that you said.

----

lalop wrote:Come to think of it, if light travels along the geodesics of M, then even though you'd get all the previously analyzed length contraction and whatnot in W, you'd still see the world as if it were M.

But if (this probably breaks physics), light travels along the "straight line" geodesics of W under the Euclidean metric, but otherwise the distance metric is d_W, I think you would actually see the Center as the small cylinder that it is, see other peoples' length contraction from afar, etc. In other words, sight-wise, W is an annulus with identified points, but distance-wise, W is M "inside out".


If this works, it's pretty cool.
The preceding comment is an automated response.

cphite
Posts: 1135
Joined: Wed Mar 30, 2011 5:27 pm UTC

Re: The golden menhir on the mysterious plain

Postby cphite » Thu Mar 06, 2014 8:46 pm UTC

EliezerYudkowsky wrote:I would like to put a Lovecraftian spatial arrangement into a possibly-eventually-forthcoming story (not Methods of Rationality). My ability to visualize curved space isn't what it should be, so I'm turning the following literary question over to you, XKCD.

Suppose you wake up in a mysterious grassy plain with the following property: If you start from any place and walk in almost any direction, you arrive at or near the same point on the mysterious plain, which we'll term the 'Center'. We can suppose the Center is marked by a giant golden menhir (upright stone) which also lets us obscure the exact central point in case it has singularity-like properties. It requires at most 1 kilometer to reach the Center from almost all starting points.

The question is then: What sort of spatial geometries would have the Central Menhir property, and how would things appear as you approached the Center?


None. You end up back at the giant golden menhir because you do, and your puny human mind would shatter if you even began to understand why. Best not to think about it, and pay no attention to what sounds like a flute playing madly in the distance...

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5959
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: The golden menhir on the mysterious plain

Postby jestingrabbit » Fri Mar 07, 2014 4:48 am UTC

snowyowl wrote:
jestingrabbit wrote:There's an underlying assumption here that you can pick a direction and walk in it. That means that the manifold isn't just a manifold, it needs to be a Riemannian manifold, and the straight lines that you walk along would be what are known as geodesics.

The manifold needs to have an affine connection, but this is a slightly weaker condition than being Riemannian. I'm nitpicking, but there are non-Riemannian manifolds which still have geodesics, and I think they're worth investigating.


No, no, nitpicking is good. Parallel transport is enough for a sense of straightness, but you do need an underlying metric imo: the unit "paces" would naturally arise. I don't believe you can get that without a Riemannian manifold, but this mathematical territory is full of pathological stuff that I really hate, so feel free to prove me wrong.

snowyowl wrote:
jestingrabbit wrote:
If you start from any place and walk in almost any direction, you arrive at or near the same point on the mysterious plain, which we'll term the 'Center'.

This, I think, is best identified as the requirement that almost every geodesic is dense. That means that, with probability 1, if you pick a direction and walk in it, you'll get within epsilon distance, of every point on the surface. It doesn't guarantee exactly what you want, the time that it will take to get back to the centre will vary, but... something's gotta give.

I don't think this interpretation really works. The description implies that the Menhir is interesting because it has certain mysterious properties. If every point on the surface shares those properties, then it's not quite as interesting. The Menhir is just a rock now.


Its a rock in a strange place. In the spec, the plane is mysterious and the menhir is golden. The menhir could be mysterious, but the spec doesn't require it. The spec seems more about the mehir property than the menhir itself being weird.

lalop wrote:Come to think of it, if light travels along the geodesics of M, then even though you'd get all the previously analyzed length contraction and whatnot in W, you'd still see the world as if it were M.

But if (this probably breaks physics), light travels along the "straight line" geodesics of W under the Euclidean metric, but otherwise the distance metric is d_W, I think you would actually see the Center as the small cylinder that it is, see other peoples' length contraction from afar, etc. In other words, sight-wise, W is an annulus with identified points, but distance-wise, W is M "inside out".


I think this breaks the "Lovecraftian" assumption. There would be no growing sense of weirdness/horror. In a world like this, reaching down and plucking a flower would be a feat of tremendous concentration and disorientation, like writing on your face in the mirror but worse. I mean, if you break the ability to pick something in the distance and walk towards it, I don't think you get to call where you are a plane, and that is what decoupling line of sight geodesics and walking geodesics does.

But, some of what you were talking about suggested some other possibilities to me, as did the "strange menhir" requirement.

--

The first idea is a simple Euclidean disc with a menhir in the centre, and opposite points on the circle identified. Now, none of the geodesics are dense, but, no matter what straight path you choose, there will be some distance D such that you pass within D of the menhir regularly, but overall its a pretty boring idea. One interesting thing, the menhir would appear to be reflected every time you come back to it, because when you cross the circle, left and right are exchanged. So there's something in it that's fun, but overall, kinda boring.

The surface is topologically a projective plane, as is the next example.

Much more interesting is a Euclidean ellipse with two menhirs, one at each of the foci, and opposite points on the circumference identified. Walking around this now resembles elliptical billiards.

http://cage.ugent.be/~hs/billiards/billiards.html

However, when you arrive at the boundary, instead of bouncing off it, your position is given as though you were rotated 180 degrees around the centre of the ellipse, and your new direction is as though you had bounced off the boundary. That explanation feels kinda vague, but hopefully its clear enough Essentially, the billiards dynamics is very instructive for the dynamics that we're interested in.

In particular, if you ended up going to a menhir, and then running directly away from it, the dynamics is such that your path will eventually start to very closely match the major axis and get closer and closer to it as you cross the boundary more and more times ie by trying to run directly away from a menhir, you are guaranteed to revisit it endlessly, which I thought was kinda neat. You will also be bouncing between the two menhirs, with there being a distance D such that every D along your path you encounter a menhir, taking turns between the two, over and over again.

And you could have corkscrew shaped flowers with corkscrew proboscis things eating from them, and the orientation thing makes it seem that there are different species for the different orientations, but you could have a moment of realisation thing for that too.

--

Another idea, satisfying the "mysterious menhir" suggestion, is a punctured torus with the menhir at the point at infinity. So, to be clear, a diagram.

punctured_torus.png


So, this is a top down diagram of what is known as the punctured torus. The large disc is the poincare disc model of the hyperbolic plane, so the 4 large quarter circles drawn in the disc are geodesics, and the idea is that opposite sides of the "square" in the middle are identified, so that you have a torus, but the point at infinity isn't part of the torus, so its punctured, and is called the punctured torus. You could put the menhir anywhere and still have the menhir property, *but* you could also put the menhir "at infinity" basically, you could make the red patches in the diagram "menhir", so its faces are concave, and it has 4 faces, and as you walked around the menhir, you would see them in the same order over and over as you tried to walk around it, and it would have finite circumference.

However, it would give a person some sort of direct experience of a horocycle.

https://en.wikipedia.org/wiki/Horocycle

The blue circle in the diagram is a horocycle, and the corners of the menhir would touch it infinitely many times as you walked along it. Horocycles are one of those things that really tell you you're in non-euclidean geometry, and they're interesting sorts of things. You would also be able to see yourself or at least the back of your head, quite near if you stepped back form the menhir a bit.

It would be hard to put a light above the menhir, it would sort of need to be at infinity, but... whatever.

--

As an added bonus, at the bottom of this page, you can see some really great pics of the octogons I mentioned in my first post.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

lalop
Posts: 210
Joined: Mon May 23, 2011 5:29 pm UTC

Re: The golden menhir on the mysterious plain

Postby lalop » Fri Mar 07, 2014 4:08 pm UTC

I just realized that the decoupled geodesics would lead to some pretty weird effects for objects near the center [not to be confused with Center] of M, aka the identified outer boundary of W. For instance, an object at the center in M would appear as a giant wall surrounding everyone in W. So there would be two weird points, rather than one: the center and Center. This isn't necessarily a dealbreaker - I think it's pretty darn interesting - but I'll try to find an alternative that avoids the center while preserving the Center and other properties.

jestingrabbit wrote:In a world like this, reaching down and plucking a flower would be a feat of tremendous concentration and disorientation, like writing on your face in the mirror but worse.


From what I can tell, far away from the center and Center, everything should be close to Euclidean.

If you are at M coords (r,0), and a 2m-meter width rectangle is s meters closer to the Center, then in M coords that rectangle would be (sqrt((r+s)^2+t^2),arctan t/(r+s)), where -m ≤ t ≤ m. In W, that would become f(sqrt((r+s)^2+t^2),arctan t/(r+s))=((r_c+r_d)- sqrt((r+s)^2+t^2),arctan t/(r+s)), which in cartesian coordinates is (((r_c+r_d)- sqrt((r+s)^2+t^2))cos arctan t/(r+s), ((r_c+r_d)- sqrt((r+s)^2+t^2))sin arctan t/(r+s)) = ((r_c+r_d)(r+s)/sqrt((r+s)^2+t^2) - r-s, (r_c+r_d)t/sqrt((r+s)^2+t^2) - t). The visual endpoints of this rectangle would be at:

t = m: ((r_c+r_d)(r+s)/sqrt((r+s)^2+m^2) - r-s, (r_c+r_d)m/sqrt((r+s)^2+m^2) - m)

t= -m: ((r_c+r_d)(r+s)/sqrt((r+s)^2+(-m)^2) - r-s, (r_c+r_d)(-m)/sqrt((r+s)^2+(-m)^2) - (-m))=((r_c+r_d)(r+s)/sqrt((r+s)^2+m^2) - r-s, -(r_c+r_d)m/sqrt((r+s)^2+m^2) + m)

and the middle of the rectangle at:

t=0: (r_c+r_d - s - r, 0)

Let's plug in some numbers. Say r_d = 500m (so you end up at the Center within 1km), r_c = 1m, r = 250m (you start equidistant between center and Center), m = 1m, s=1m (rectangle of width 2m, 1m closer to the Center). Then, your position is: (251m,0m) and the rectangle's endpoints are at:

t = m: (501*251/sqrt(251^2+1^2) - 250-1, 501/sqrt(251^2+1^2) - 1)=(249.996m,0.996m)

t= -m: (249.996m,-0.996m)

and the middle at:

t=0: (250m,0m)

Visual width contraction is by about 8mm out of 2m; reasonably promising.

Let's make the rectangle smaller, say flower-sized at around m=0.001m. Intuition says the results should be even more Euclidean, and I think that's borne out:

t = m: (501*251/sqrt(251^2+0.001^2) - 250-1, 501*0.001/sqrt(251^2+0.001^2) - 0.001)=(249.999999996m,0.000996m)
t = -m: (249.999999996m,-0.000996m)
t = 0: (250m,0m)

Let's go back to the original 2m rectangle, but this time position it s=10m away:

t = m: (501*260/sqrt(260^2+1^2) - 250-10, 501*1/sqrt(260^2+1^2) - 1) = (240.9963,0.9269m)
t = -m: (501*260/sqrt(260^2+1^2) - 250-10, 501*1/sqrt(260^2+1^2) - 1) = (240.9963,-0.9269m)
t = 0: (241m,0m)

Visual width contraction is 14.62cm out of 2m; at that distance, barely noticable.

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5959
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: The golden menhir on the mysterious plain

Postby jestingrabbit » Fri Mar 07, 2014 6:25 pm UTC

lalop wrote:Visual width contraction is 14.62cm out of 2m; at that distance, barely noticable.


If I threw you a ball, I think you would notice it.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

lalop
Posts: 210
Joined: Mon May 23, 2011 5:29 pm UTC

Re: The golden menhir on the mysterious plain

Postby lalop » Fri Mar 07, 2014 7:37 pm UTC

jestingrabbit wrote:
lalop wrote:Visual width contraction is 14.62cm out of 2m; at that distance, barely noticable.


If I threw you a ball, I think you would notice it.


I don't think I would. A ball of radius m=5cm, s=10m away, would appear to have radius 4.63cm, not a difference I'd notice off the bat.

To try a different tact, the visual angle of something of width 2m, at distance s closer to the Center, would be: V = 2 arctan((apparent width)/(2s)). The apparent distance a of the object therefore comes from solving:

V = 2 arctan((actual width)/(2a)) = 2 arctan(m/a)
2 arctan((apparent width)/(2s)) = 2 arctan(m/a)
(apparent width)/(2s) = m/a

[Maybe there was an easier way of reasoning about it just by using proportions, but... whatever.]

a = 2ms/(apparent width) = 2ms / (2( (r_c+r_d)m/sqrt((r+s)^2+m^2) - m) /(2s)) = s/((r_c+r_d)/sqrt((r+s)^2+m^2) - 1)

In the case of this ball, a = 0.05*10/((1+500)0.05/sqrt((250+10)^2+0.05^2) - 0.05) = 10.79m

By the time the ball reached s = 5m, its apparent distance would be a = 5/((1+500)/sqrt((250+5)^2+0.05^2) - 1) = 5.18m.

By the time the ball reached s = 2m, its apparent distance would be a = 2/((1+500)/sqrt((250+2)^2+0.05^2) - 1) = 2.02m.

By the time the ball reached s = 1m, its apparent distance would be a = 1/((1+500)/sqrt((250+1)^2+0.05^2) - 1) = 1.004m.

The trick ball maybe, strangely, slows down a tiny bit more than you'd bargained for, but by the time it's a few meters away, it's practically as you'd expect. Someone should write a game for oculus rift and see how these numbers affect accuracy. Is it so bad that you would actually notice, if you weren't practicing ball day by day? Or would you simply consider that you were out of practice, maybe even get used to it, and try to ignore that naggy feeling that something has changed?

All this is a bit contrived, though; why the heck are you playing ball in the first place? You're trapped in some unknown landscape where some of the angles seem wonky; the first time you'd notice any of this is when trying to throw rocks at some unknown.. creature..

Caesar
Posts: 60
Joined: Fri Oct 24, 2008 12:34 am UTC
Location: Germany

Re: The golden menhir on the mysterious plain

Postby Caesar » Fri Nov 21, 2014 5:17 pm UTC

Now this thread is kinda old and I know next to nothing about manifolds... but I noticed that you treated the character in the plane basically as a point. But when our character, who is in fact a volume, passes a curved "boundary" I think it would do weird things to him.
Take the euclidean disc for example. Every time you pass the border it would scatter you like a lens and you would grow larger. Also, if you walk straight into the boundary, your shoulders would touch first and your center later. That means that when you emerge your center would get squashed and your sides stretched.


Return to “Mathematics”

Who is online

Users browsing this forum: No registered users and 10 guests