Re: OT: 4D

On Wed, Feb 23, 2005 at 08:32:49PM -0000, Ivan Baines wrote:
> Interesting.  I assume we'd be talking about 4D beings who
> live on hyperspherical "planets"...?  Then, there'd be
> gravity.
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. That
should make for some interesting phenomena in the night sky. :-) This
means that there would be no north/south poles. It also means that
there would be a linear region (a "great circle") on the surface of
the planet where one of the two rotations has no effect (the closest
equivalent to a north/south pole), and vice versa. These two "great
circles" are perpendicular to each other and encircle the planet, yet
they never intersect!  (The only intersection of these stationary
planes is a single point at the center of the planet. There is no
intersection on the surface.)

From the viewpoint of an observer floating in 4D space beside the
planet, if he traces the path of a fixed landmark on the surface as it
rotates with the planet, it would trace out a spiralling path (the
exact shape of which depends on the relative rates of the two
independent spins). If one spin were significantly faster than the
other, the path would appear either as a tightly-wound spiral or
disjoint "racing" across the diameter of the spherical projection of
the planet, which (the diameter) slowly rotates through 360 degrees.
If the two rates were similar, the path is ribbon-like.


> In much the same way as we stand on a 2D floor which is
> actually the surface of a sphere, these 4D guys would live
> "standing on" a 3D "floor" which is actually the "surface"
> of a hypersphere!
Yes. Having 3 degrees of freedom is really fun. And really confusing,
too, if you're not used to it. For example, try this 4D maze game
sometime:

	http://www.urticator.net/maze/

Even if you restrict yourself to making only "horizontal" movement,
you can *easily* get completely lost.


> So you'd have words for up and down, as well as for forwards
> and backwards.  But instead of just left and right, there'd
> be four words conveying this kind of "sidestepping" motion.
> You could be standing up, looking at a given target, and
> "sidestep" freely in two dimensions while never getting any
> further forwards or backwards along the direction you're
> facing.
Indeed.


> Not only this, but weird rotation stuff is possible too.
> For example, on a 3D world, assuming your vertical orientation
> is fixed (you're standing up and not allowed to lean over), and
> the direction you're looking in is fixed, then you can't really
> do much other than sidestep left and right... but in a 4D
> world, not only can you sidestep left, right, ana and kata (to
> use the terms you used in your post), you can do other things
> too, such as rotate freely in a certain plane of rotation while
> remaining bolt upright and staring straight at your target.
> Weird!
Also, there would be two ways to turn your head so that you face
backwards, each of which gives you a different "sweep" of your
surroundings. From the perspective of an observer, assuming you had 2
eyes like we do in 3D, 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.

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


> I think a lot of this kind of stuff would be possible, and it
> would hopefully need all kinds of interesting words to cover
> it! :-)
[...]

Definitely. :-)


T

--
Knowledge is that area of ignorance that we arrange and classify. -- Ambrose
Bierce

Reply