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Crooked space

Space with left, right, up, down, front and back built in
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Real space is isotropic. Left, right and other directions mentioned in the summary are arbitrary. We needn't decide left-hand X coordinates are lower than right-hand ones.

If parallel lines always converge on the same side and diverge on the opposite side of a line, space has direction. A coordinate grid is then not a series of contiguous cubes but of crooked hexahedra with irregular quadrilateral faces, and that space has a left, right, up, down, front and back.

Although it has absolute coordinates and an origin, there can be five other spaces divergent in different combinations of dimensions, so in these spaces parallel lines might diverge to the left rather than the right or down rather than up and so forth. Nevertheless, there is only no variation in parallel lines if they straddle the origin, so space is usually isotropic in one way or another. Also, space stays edgeless.

This means that it would be possible to define proper directions in space.

So to summarise, this is a non-Euclidean geometry which has directions built in almost everywhere.

nineteenthly, Dec 22 2011


       Baked by gravity, I think.
RayfordSteele, Dec 22 2011

       Gravity pulls in different directions depending on where you are and curves space. This kind of space is not curved.
nineteenthly, Dec 22 2011

       I see you've posted this in [science:mathematics]. Are you suggesting a mathematical treatment of such space, or actually creating a universe which has such space in it?
lurch, Dec 22 2011

       In the same sense as a crooked style, a crooked house and the like? A square has one obtuse corner opposite an acute one, for example.
nineteenthly, Dec 22 2011

       I'm suggesting that this is a particular kind of geometry, though not one which describes any kind of real space.
nineteenthly, Dec 22 2011

       Draw a space with 3 orthogonal time dimensions.
lurch, Dec 22 2011

       Maybe time is two dimensions arranged in this way.
nineteenthly, Dec 22 2011

       This is either brilliant or not, and I'm going with the former.   

       As a hypothetical contruct it's nice. As a possible description of the universe, it's also nice but I don't see how space would arise with this asymmetry.   

       If space *were* like this... hang on, let me see if I understand. The idea is that space converges as you go north, say, is that it? If so, then if space *were* like this, how would one tell?
MaxwellBuchanan, Dec 22 2011

       It is not impossible that real-existing space has some gentle curvature built into it. This would be a logical consequence of the Big Bang, and the notion that space is expanding away from that origin-point.   

       Our observed 3D space could be the "surface" of a 4D hypersphere; it is the hypersphere which is expanding, and so our space gets larger as that happens (like the surface area of a balloon gets larger as it is inflated).   

       Anyway, the larger the hypersphere, the "flatter" its 3D surface would appear to be. Remember all those centuries in which people thought the of Earth was purely 2D/flat --a consequence of the Earth being quite large, from the average human-on-surface perspective.   

       So far we have not observed any overall 4D curvature to the generally 3D space of our Observable Universe (I'm ignoring local gravitational effects here, such as was measured by Gravity Probe B). But IF overall space has some slight 4D curvature, we would have one built-in direction-line, at least: toward/away-from the origin-point of the Big Bang.
Vernon, Dec 22 2011

       Working through the consequences of this would either lead to something trivial and uninteresting, or else to some rich and interesting branch of math.   

       One approach might be to rewrite Euclid's postulates in a way that distinguished right from left, and then work systematically through the Elements, re-doing everything with the new postulates. Similar to Lobachevsky's strategy*   

       The first 4 postulates don't seem to lend themselves to this in any obvious or natural way, but the 5th, at least in Wolfram MathWorld's version, explicitly refers to "sides" -- and so might be rewritten as a pair of postulates, one for right side and the other for left side.   

       *No, not "Plagarise plagarise plagarise." The other one.
mouseposture, Dec 22 2011

       //Similar to Lobachevsky's strategy//   

       Ah yes. Full frontal Lobachevskomy.
MaxwellBuchanan, Dec 22 2011

       is this like directions from point A to point B based on polyhedra, rather than a vector/distance ?   

       //if parallel lines always converge// aaaaaand you lost me.
FlyingToaster, Dec 22 2011

       I've a notion that this is really about non-metrizable spaces[1]. Which may not have "geometry," exactly[2]. Topology, maybe.   

       [1] If so, then it's an existing branch of mathematics.
[2] Corrections welcome.
mouseposture, Dec 23 2011

       // //if parallel lines always converge// aaaaaand you lost me.//   

       [FlyingToaster] Look up Euclid's fifth postulate.
spidermother, Dec 23 2011

       Buuuuu-uut... turns out, Euclid wasn't quite on the mark, so to speak. Re-read the first part of General Relativity, maybe?
Alterother, Dec 23 2011

       Exactly; my point was that if you start reading about the parallel postulate you'll almost immediately stumble upon talk of its exceptions. And maybe be less lost.
spidermother, Dec 23 2011

       One thing that occurs to me is that the type of artistic perspective involving three orthogonal vanishing points represents a projection of your space onto a plane.   

       Just to be picky, //crooked// seems to contradict //not curved//; I know what you mean, though.
spidermother, Dec 23 2011

       It's hypothetical, [MB], not a description of the physical world. Yes, you do get it: this is space with compass directions. It might even be, in a sense, space where spheres could be said to have east and west poles. You could probably tell because the cosmological constant would increase most along a particular line, and this would seem to manifest itself as the apparent weakening of fundamental forces in one direction and their strengthening in the opposite. It might also manifest itself through relativistic effects and i think the Michelson- Morley experiment would have different results. The flaw in thinking of it this way is that physics as it's currently understood depends on teeny rolled-up extra dimensions, so i'm pretending Newtonian physics is sort of true in this space.   

       I really don't think it does contradict. This is not curved space, it's bent space. Maybe this is the space Disaster Area's accountant discovered at the University of Maximegalon. There are true straight lines in this space which are not influenced by mass.   

       [Vernon], yes, in reality space appears to be expressible as curved, though since i see space as a relationship whose necessary conditions include direction and distance, it isn't so much that space is curved as that the parallel postulate has exceptions and that these exceptions ultimately amount to there being a maximum distance between two objects at a given time and that once this distance appears to be exceeded the relative directions suddenly reverse. Therefore the hypersphere may not exist. On the other hand, predicted observations would not rule it out. If it doesn't exist, there is no centre and space is not anisotropic in that sense. I would also say that gravitational lensing is quite good evidence for this kind of geometry applying to what can be observed.   

       [FT], the polyhedra just illustrate what happens in such a space. It means that the corners of an equilateral quadrilateral cannot all be right angles, and that a quadrilateral whose corners are all right angles cannot be equilateral, then everything else follows from that, so there are in a sense no squares and therefore no cubes or true Platonic solids. However, perhaps interestingly, there are two other polygons which are like squares. One has right angled corners, the other has sides of equal length.   

       In fact, and i know i'm going on and on here, every Euclidean Platonic polyhedron has a set of variants in this space, but no exact equivalents. A Euclidean cube corresponds to two polyhedra, one with equal angles, the other with equal-area faces and equal-length edges. Each of these variants has versions which are asymmetrical along different axes, so there are left- and right-handed cubes, right-way-up and upside-down cubes, back- to-front and front-to-back cubes and every combination of these. Each of these cubes, and the same applies to other polyhedra, has a space in which it's more at home, and a number of foreign volumes.   

       This is where east, west, front and back poles come in with spheres. Spheres can be slightly flatter on one side than the other, so in a way they have three possible poles and different orientations. These can also be oriented differently according to rotation on an axis, so they have intrinsic poles and axes of rotation. These poles also exist with other shapes, so there are actually infinite sets of wonky shapes along with ideal, "perfectly" wonky shapes.
nineteenthly, Dec 23 2011

       Is it helpful to point out that if you consider spacetime as an entity, it does indeed have a direction - albeit only along the time axis - so there is a precedent here. Extending that, I quite like the polar notion of time having 'poles' - i.e. a Future Pole at which point everything else is in the past, and a Past Pole, where everything else is in the future. That does somehow make it easier to think about the big bang and associated crunch.   

       Yes, it's weird to consider being at a point on the globe where there's no North anymore, but it's a lot easier to think of than a point in time where there's no future. Considering the end of the Universe as being a 'mere' pole softens the blow a little, I think.   

       What you're describing in spatial terms already exists, when you look at the geometry of spacetime over time - all parallel lines do indeed converge at time's Southern Pole (whether they diverge - or do something else - at the Northern Pole is a topic for debate) - and so it is currently possible to draw a line across spacetime and say, confidently - this way is one direction, and that way, is another.   

       Once you've determined one direction in time, you should be able to infer directions from other possible translations. For example right-hand rotations are distinguishable from left-hand rotations (though it's not necessarily possible to say which is which, until you know which way is up) I'm not sure whether (or how) you can get from there, to being able to state categorically, which way is up (and similarly, be able to lay down markers pointing in each of the other cardinal directions) Perhaps it's this edgeless property itself that prevents us from doing this.   

       Going back to the Earth and the geometry that we've arbitrarily imposed upon it, based on its spin. If the Earth didn't spin, we'd likely use some other concept other than latitude and longitude to carve out its representation. Without the cardinal directions, you'd need at least two reference points from which you could describe a third point as being some measure of angle and distance away from.   

       Indeed any unit of measure on a 2 dimensional surface must be built up from at least 3 things. A distance, an angle, and an orientation against which the angle is referenced.   

       Without all 3 of those things, you can't get a proper set of measurements.   

       So you're saying that if somehow you were to build a geometry in such a way that directions were built-in - but how do you tell the difference between going North, and going East, or going Up? If each of those directions points towards areas where parrallel lines converge (and South, West and Down all pointed to divergent spaces) you still have to arbitrarily lay down your coordinate representation somehow. And while these points of convergence and divergence are off in the far- and essentially infinite distance, it wont be necessarily possible to lay down a set of axes so that they are in-line with those distant points. If Up and North and East all look similarly convergent, would there not be a super- convergent point somewhere in the Up-North-East where everything converged?   

       And if in the opposite Down-South-West corner, everything diverged, would those divergent lines all meet in the Up-North-East corner? I'm not sure what shape that makes space, but I quite like it's long-term symmetry. Indeed, it kind of turns all of space into a single one-dimensional line (if you look at it from far enough away) who's end is also its beginning - which takes us back to the situation we see every day where time is a kind of linear dimension with a very obvious direction - time - it might be nice to think of it as being Ouroboros- like though, and that is what this idea ultimately suggests.
zen_tom, Dec 23 2011

       Regardless of your explanations, [19thly], your directions are still purely arbitrary to an observer inside the space you describe, as there is no relative referential frame by which you navigate in a cohesive manner with another observer also inside the space. i.e. In space, there is no-one to tell you you're "upside down."   

       You have to be an external observer to this space to be able to see any direction/s, no?
infidel, Dec 23 2011

       No, you would be able to detect it over a big enough distance, because for example, if you had a material "cube" which was more pointed in the direction of maximum divergence, rotating it a hundred and eighty degrees through an axis perpendicular to that direction would lead to the matter composing it undergoing some kind of stress with one vertex becoming compressed and the opposite becoming attenuated. A sufficiently large divergence would mean that ordinary matter would be destroyed by such a rotation. Regarding the large-scale structure of the universe in such a space, matter can only coalesce near the origin. This would be a universe where the Big Rip was spatial rather than temporal. Stars would evolve differently in different parts of the universe, for example, because of differences in the relative strength of the strong force and gravity. There may be something in there which would imply that this kind of geometry is incompatible with physical reality somehow.   

       A universe with this kind of geometry has limits of distance beyond which atoms cannot exist.
nineteenthly, Dec 23 2011

       Ahh ok, I quite like the 'stress-through-rotation' concept - but again, that's kind of similar notionally to this time dimension. It's easier to break an egg than it is to unbreak it - i.e. rotating an egg-shaped object through spacetime is easier in one direction than the other.   

       If Gravity is a result of the shape of spacetime, and objects experience resultant forces - then if spacetime (or just space) were arranged in this way, analogously, wouldn't all of matter be gently squeezed over time towards the convergent (or divergent - I can't decide which) pole?
zen_tom, Dec 23 2011

       Hmm, maybe, depends on various things. It does sort of feel like it would be unstable in some way, though i can't put my finger on why. Having said that, the electromagnetic force is stronger than gravity and can repel, and whereas it might collapse, it might also take a very long time to do so. Also, if it's small enough, the quantity of matter might mean there's no Olbers Paradox.   

       Entropy seems to be connected to time having a direction, so as i say, there might be some kind of use for this notion in working out why time passes, though i doubt it. I also think that given the "stress" thing, even disregarding the velocity of light, there would come a point where atomic matter would be destroyed if it moved above a certain speed in some directions, either because it would get pinched and become neutronium or a black hole, or because it would get ripped apart, just because the parallel movement of its individual particles wouldn't be able to stand the strain.   

       Getting back to the anisotropy, we are actually aware that time passes, so a being in this universe could be aware of different directions in an analogous way.
nineteenthly, Dec 23 2011

       I rather prefer your concept of oriented space, nineteenthly, built out by stacked geometries that disobey Euclidian rules or in which direction can be inferred by order. Very much in keeping with hypergeometric constructions. You might agree, but I sense that such disproportions could be represented mathematically as geometries defining the shape of probability clouds. Such a construction would be elegant, but would seem feasible.
reensure, Dec 29 2011

       Would it be asymmetrical in the right way? Care to say more?
nineteenthly, Dec 29 2011

       About asymmetry, that is exactly what I was thinking. A good example would be a sphere, a fair example would be would be a cube (as you've described above). My concern is knowing what rules apply when correcting for orientation within a community of like objects.   

       The effect of rules within such a community (let's say hypercubes) was writ large by Vernon over a space/time curve, but is transposable to small groups of unit cubes and by extention to a single cube. The larger the number of cubes included in one's community of cubes , the lesser one's community assymetry will be reflected in the assymetry of a single cube. My logic breaks down at this point, because of divergent reasoning (lateral thinking?). Again, I'm not sure about the rules -- is a sole unit of a community allowed more or less overall assymetry than the community set? Is a state of disproportion only stable under certain conditions? For example, one might conclude that disproportionate sets may exist but must adhere to some complementary set of opposite assymetry.   

       This is the kind of mental spin cycle I enter when reasoning through basic "truths" such as the requirement that protons and neutrons co-exist in stable matter.
reensure, Dec 30 2011

       A nice quote on correlation, from Physics.com:   

       "My complete answer to the late 19th century question 'what is electrodynamics trying to tell us?' would simply be this: Fields in empty space have physical reality; the medium that supports them does not.   

       Having thus removed the mystery from electrodynamics, let me immediately do the same for quantum mechanics: Correlations have physical reality; that which they correlate, does not.   

       N. David Mermin, 'What is Quantum Mechanics Trying to Tell Us?' "
reensure, Jan 01 2012

       A wonderful idea that works 'almost everywhere'? left and right as well as up and down are arbitrary relationships found in any space of three or more parallel dimensions (as you describe). this relationship is defined by the vectors of four lines. Your system proposes to allow relationships to exist in only two dimensions on planes that intersect the origin. You cannot prevent three lines intersecting at the origin being able to arbitrarily describe the relationship with a fourth line unless you only have two dimensions parallel.I believe that your idea contains clear contradictions.   

       All relationships between an origin and a line are relative and arbitrary. The intersection between the L/R dimensions that your are proposing is just another line with arbitrary relationships to the origin.
WcW, Jan 01 2012

       // All relationships between an origin and a line are relative and arbitrary //   

       Relationships are relative? Really?
Alterother, Jan 01 2012

       the polarity of up/down or left/right is arbitrary. It does not have a geometric or mathematical basis.
WcW, Jan 02 2012

       That's exactly my point, [WcW]. This is a completely hypothetical example of a space which, while it hasn't exactly got left and right, up and down and so forth in the sense that left could not be right for example, is asymmetrical in all axes. In fact, it might have a real up and down because i think matter will tend to fall into one corner in each volume and tend to disintegrate as it moves away from the "origin".
nineteenthly, Jan 02 2012

       But then you stipulated that relative to the origin the dimensions are symmetrical. Can you explain how we avoid the polarity problem if we have symmetrical dimensions? That they might curve and form complicated spaces doesn't matter so long as we can navigate them in a reversible fashion.
WcW, Jan 02 2012


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