Re: OT: 4D

On Wed, Feb 23, 2005 at 11:34:31PM -0000, Ivan Baines wrote:
> > Hyperspherical planets are curious things. For one, they can have two
> > independent spins *simultaneously*. That is to say, one of the spins
> > can speed up/slow down completely independently of the other.
>
> *Can* do, yes, but surely they wouldn't unless acted upon by an enormous
> force, much like in the 3D world!  Unless there's some truly bizarre
> consequence of an extra dimension on the laws of physics which hasn't
> occurred to me. ;-)
Well, any plausible physics of a 4D world must necessarily be
fundamentally different from ours (even if still analogous in some
ways). One of the major problems being that the atom is unstable in
4D, and would quickly disintegrate. I don't know exactly why, but it
has something to do with the stability of 4D standing waves which are
required for stable electron orbitals around the atomic nucleus.

Then you have the Maxwell equations---or any physics equation
involving the vector cross product, really: in 4D, there *is* no
unique axis of rotation, and there is a cross product but it requires
3 vectors, not 2, so these equations either break down or have to take
on bizarre, unfamiliar forms. This means many familiar phenomena would
no longer be true in 4D, if we generalize 3D physics to it.

Of course, we could always assume (with the usual caveats) that 4D
physics behaves more-or-less analogously to 3D physics. But I'd step
pretty cautiously in that area, since 4D things have a tendency to
exhibit strange unexpected properties.


> Seriously though, I can understand this being possible, but... do you
> have some method for visualising this, or have you just relied on
> calculations and suchlike?
A method for visualising 2 simultaneous spins? Why, yes... :-)

My usual method for understanding 4D is to think in terms of its
projection onto a 3D volume (which is the hypothetical retina of a 4D
eye), and try to analyse it using analogy to projections of 3D objects
onto 2D. In doing such a projection, I assume that I am looking into
the positive W axis (the 4th direction), with the positive Z axis
pointing up, and the X and Y axes defining a horizontal plane (which
would be the planar horizon of a 4D scene) in the projected volume.

Under such conditions, let's say we project the 4D planet onto the our
"retina" such that its center coincides with the origin of the
projected volume. Now, I assume you're familiar with the appearance of
4D rotations from playing with the 4D Rubik's cube; so you'll
understand that if the planet were rotating, for example, in the ZW
plane, it would appear to be turning inside-out along the Z axis of
the projected volume. (E.g., if you were clicking on the top cube face
of the 4D Rubik's cube - you get the turning-inside-out visual effect
along the Z axis, that looks like an hourglass motion.)

Now, note that, in the projected volume, this rotation is along the Z
axis (it's actually in the ZW plane, of course, but in our projection
the W axis has collapsed into a point so the ZW plane leaves only the
Z axis). Note that this is precisely the rotational axis, in 3D of the
XY plane. I.e., it is the part of the object that remains still when
we rotate it in the XY plane. In other words, while our planet is
doing its "hourglass" rotation in the ZW plane, it can simultaneously
rotate in the XY plane (which appears, in projection, to be exactly
the same as a 3D rotation in the XY plane). This rotation does not at
all affect its ZW rotation, because the ZW plane is orthogonal to the
XY plane.

In other words, you could visualize the double-spin of the planet as a
rotating hourglass, where the inside-out motion of the hourglass is
the first spin, and the XY rotation is the second.


[...]
> > http://www.urticator.net/maze/
> >
> > Even if you restrict yourself to making only "horizontal" movement,
> > you can *easily* get completely lost.
>
> Fun.  But difficult.  It reminds me of this:
>
> http://www.superliminal.com/cube/cube.htm
>
> A Rubik's hypercube!  While nobody could seriously be expected to solve
> this, it is doable in small amounts - you can get the game to apply a
> low number of random turns, and try to solve it from there.  I found this
> to be great practice for getting a feel for four-dimensional rotations.
Actually, it *is* quite possible to solve this hypercube. It would
take a lot of time, of course, but a few people have done it, and it
is possible to generalize the so-called "Ultimate Solution" of the 3D
cube to solve this 4D one.

Also note that this particular program is actually showing you the
hypercube inside-out (e.g., remove the front face and look through the
inside at the back face). That could be one of the difficulties, just
as it's rather confusing to do the 3D Rubik's cube the same way.
Ideally, you'd want to view the 4D cube as it would actually appear to
a 4D person, i.e., you can only see 4 faces max at a time, etc.. Then
you'd be able to understand why the cube moves the way it does---you'd
see the corner pieces as (rhombic) dodecahedra composed of 4 colors
each, the edge pieces as hexagonal prisms composed of 3 colors each,
and the face pieces as cuboids composed of 2 colors each---instead of
the current situation where seemingly-unrelated colors always move
about together.

The downside to this more realistic approach, however, is that it
takes true 3D vision to be able to see all of this detail at once.
With our crippled 2D-only retinas, we wouldn't be able to make heads
or tails out of this without some aid, so it may not be significantly
better than the current inside-out approach. Nevertheless, being a
Rubik's cube fan myself, I much prefer to see corner/edge/face pieces
as units rather than colors scattered across different places -- it is
so much more difficult to figure out what is supposed to go where when
it's not immediately obvious which colors actually move as a unit.


[...]
> > one turn would have your eyes moving off to the
> > side one after another (as in the analogous 3D view), while the other
> > would have them both move off at the same time.
>
> Could you explain this further?  Is it maybe something to do with
> whether the plane of rotation is parellel or perpendicular to the
> line drawn between your eyes?
Well, the point is that there are 2 independent planes of rotation by
which you can turn your head so that you face the opposite direction.

Using our 3D projection volume again, let's say we have an
approximately spheroidal lump in the center representing your 4D head.
Let's say this head has 2 eyes, for familiarity's sake. Suppose the
head was looking at you. In this case, then, the two eyes would, in
the projection, be located *inside* the 3D volume corresponding to the
projected head. Let's say they are located, in the projection, along
the X axis, one on the +X side, and one on the -X side, with a roughly
conical lump between them representing the projected 4D nose.

Now, one ought to understand, in visualizing such projections, that
the *center* of the projected volume represents the focal point of
your eyes--what your eyes are looking at. The volume surrounding the
center are actually peripheral. So the nose at the center of the
projected head is actually the closest point to you, in 4D. The back
of the head is not visible, because it is behind the face in the +W
axis.

Now, suppose this head turns to look behind it (at the +W direction).
There are two ways this rotation can happen. The first way is to
rotate along the XW plane, which corresponds with "turning inside-out"
along the X axis in the projection. There are, of course, two ways to
turn on this plane, so let's say the head turns in the direction of
the +X axis in the projection. As this happens, the eyes will move
along the X axis towards the positive direction, one behind the other,
until the first one reaches the boundary of the 3D projection of the
head. That is when the first eye has reached the 90-degree angle from
our viewpoint. After that point, it vanishes behind the head,
followed by the second eye, and the head continues to turn until the
back of the head now takes its place at the center of the projected
volume of the head. So if the head has hair at the back, you'd see a
bunch of hair filling the volume of the head in the projection at this
point.

The other way for the head to turn is along the YW plane, or, in terms
of the projection, along the Y-axis. Let's say the head turns such
that in the projection we see the "inside-out" turning in the +Y
direction. Since both eyes begin at Y=0, as the head turns they will
move together along the direction of increasing Y, until they both
reach the boundary of the projected volume of the head at the same
time. This is the 90-degree angle from our viewpoint. After that
point, they both vanish behind the head as the back of the head turns
into view, until the bunch of hair has centered on the origin of the
projection (i.e., we are now looking at the back of the head).

Note that the head could also turn along, say, the XY plane, but as it
does so, notice that the eyes and nose still remain around the origin
of the projected volume---i.e., the face is still facing us, even
though it's turning! Think of it as another way of "tilting your head
to the side"  while still keeping your head upright, made possible by
the extra dimension in 4D. :-) (This is how the rotation-while-
maintaining-eye-contact thing you described would look in projection.)


[...]
> > Also, your eyelids would have 2-dimensional edges, and frown "lines"
> > on your brow would be planes, which give you whole new dimensions of
> > facial expressions. Not to mention 3D lips, which allow for whole new
> > ways of pouting... :-P
>
> The potential for body language with the extra degree of freedom is
> awesome in itself.  It's since occurred to me that the rotation-while-
> maintaining-eye-contact thing I mentioned would make an awesome 4D analog
> of, say, shrugging, or rolling one's eyes...
Not that there's any shortage of eye-rolling possibilities in 4D. I
mean, with the extra degree of freedom with which you can roll your
eyes, you have a LOT of options open, indeed. :-) As usual, in the 3D
projection, your eyes would appear to be truncated spheres, and your
eyeball would actually be in the *center* of the spherical white. That
means you have 2 full dimensions in which to look away, and 3 whole
dimensions for eye-rolling gestures. Add to that the possibility of
conical eyelids for the wide-eyed look, and inverted conical eyelids
for anger, with all their nuances thereof, and you've literally got
endless expressive possibilities. :-) (Or endless ways of looking
insane, esp. with all that eyeball rolling. :-P)


> Mind you, just imagining the possible body-plans for a 4D lifeform
> breaks my brain!
[...]

This is actually a very fascinating, though difficult, topic. So far,
I've tried to stick with essentially a 3D anatomy suitably generalized
to 4D. However, that probably wouldn't quite capture the aesthetics a
natively 4D creature would have. Designing a truly 4D creature that is
crafted specifically for 4D (as opposed to generalized from 3D) is a
truly mind-bending task, but extremely interesting. For example, in
Garrett Jones' 4D discussion forum, we came up with the conclusion
that although bipedal creatures such as humanoids are certainly
possible, it would probably be much more common to find quadrupeds or
hexapeds, especially the latter, since the additional dimensions imply
that 6 legs are a lot more stable than 2. There are a LOT more ways to
fall over in 4D. :-)

We've also decided that 4D creatures probably need more fingers per
hand, in order to be able to grasp 4D objects adequately. Or, at the
very least, 2 orthogonal groups of fingers opposite an opposable
thumb. Basically, a 4D hand needs a way to adequately grasp a 3D
volume in order to get a sufficient hold on objects.

Also, like you said, I've pondered about the need for more joints per
limb, just so you can bend your limbs adequately to reach all 4
dimensions. Although you can probably get by with only a shoulder and
an elbow per arm, I suspect it would be much more convenient to have a
shoulder and two elbows, just so you have 3 degrees of freedom for
positioning your arm.


T

--
The only difference between male factor and malefactor is just a little
emptiness inside.

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