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.

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?

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?

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.

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.

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.

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.

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.

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.

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?

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.

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?

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.

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.

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.

"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?' "

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.

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".

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.