Re: OT: 4D

On Tue, Feb 22, 2005 at 09:59:02PM -0500, Paul Bennett wrote:
> So, I want to produce a believable 4D world and people, and of
> course, a language for them.
Oh boy. So I am not alone. :-P


> I can just about imagine 4D objects as a collection of 3D objects
> (in much the same way as an architect or engineer can notate a 3D
> object as a set of 2D drawings), but I really can't think in 4D yet.
That's not a very useful way to think of 4D. IMNSHO, a much better way
to understand 4D is to think in terms of 3D projections. For example,
it is not really helpful to think of a 4D hypercube as a stack of 3D
cubes --- it doesn't, for example, immediately reveal that the 4D
hypercube is bounded by 8 cubical cells, or the fact that maximum
number of cells visible at any one time is 4 (unless the hypercube is
transparent).


> I know several listmembers are into 4D, so I figured I'd ask a few
> questions, if anyone cares to provide answers.
I'll try my best. :-)


> First, obviously, where can I find a good online primer to 4D?
Try our very own listmember, Garrett Jones', website:

	http://tetraspace.alkaline.org/

<shameless plug> I have also written up some (incomplete) notes about
how things might appear to someone who can see 4D:

	http://eusebeia.dyndns.org/~hsteoh/4d/vis.html

</shameless plug>


> Second -- actually, no, my second question is too hard to formulate
> at the moment, but it has to do with sound waves emenating as
> spheres in 3D, hitting the recipient virtually as 2D surfaces, and
> being percieved as 1D vibrations.
I suspect this is related to aural perception in 4D?

Keep in mind that even though our world is 3D, and sound waves are
actually 3D waves, our ears are capable only of perceiving them as 1D
waves. Our eardrums are (to some approximation) like a point (zero-D)
sensor in the 3D volume of space where 3D sound waves travel. What we
pick up are the readings of this "sensor" as the 3D waves pass over
it.

My point is that, in order to avoid incomprehensibly difficult
phonologies in your prospective conlang, you might want to make the
assumption that even though sound waves in your con-world would
actually be travelling through 4D, they are perceived as more-or-less
1D vibrations.

It might be useful, in any case, to realize that because there is one
extra dimension for the wave to propagate it, waves will dissipate
(diminish) much faster than they do in 3D --- there is just so much
more surface volume per unit space in 4D than there is surface area
per unit space in 3D. So you might want to make appropriate
adjustments to your conlang so that it is more resistant to distortion
or noise over distances, which would be much more prominent in 4D than
in our own world.


> Third -- Each "pixel" of a 4D person's eyes is going to be a cube
> instead of a square, right? Or is it going to still effectively
> behave like a square, being as light rays are still zero-width
> lines?
Lines are still lines, of course. A 4D person's eyes would have a
retina that covers a 3D volume, just as a 3D person's eyes have
retinas that cover a 2D volume. But no, it will not behave like a
square. A 2D figure in 4D is as substantial as an infinitely thin
line. Remember that in 4D, 3D volumes are "flat".

My thesis is that 4D isn't as hard to imagine as it may appear at
first, simply because we are quite capable of imagining 3D, even
though our eyes really only see 2D images. We infer the 3rd dimension
based on indirect information, such as light and shading, from the 2D
images our eyes see. In the same way, one can imagine 3D projections
of 4D objects, and, with some practice, infer the 4th dimension from
it.

One of the things you'll have to get used to, when it comes to 4D, is
to think in terms of 3D volumes rather than 1D lines or 2D surfaces.
In projection, 4D objects will look like assemblages of 3D volumes.
You have to "see" the internal structure of how these 3D volumes
connect to each other. It is pretty useless to still adopt the
2D-centric viewpoint we are used to in the 3D world -- for example,
one of the projections of the 4D hypercube is a rhombic dodecahedron
(a 12-sided figure with identical parallelograms for each side).
However, if you only look at the outer surface of this figure, it
gives you as much information about the hypercube as a hexagon gives
about the 3D cube. (One projection of a 3D cube into 2D is a hexagon.)
I.e., next to nothing. You have to examine the internal structure of
the projection to realize that the rhombic dodecahedron is actually
composed of 4 parallelopipeds, which are, in fact, "flattened" cubes
that correspond with 4 of the cells that bound the hypercube. The 3D
analogy is when you look at the internal lines of the hexagonal
projection of the cube, and realize that it is made of 3
parallelograms joined together, each of which corresponds with a
square face of the cube.

In other words, surfaces in 4D are as lines are in 3D. To truly
understand a projection of 4D into 3D, you have to understand that
you're looking at volumes, just as when we see a 2D image of a 3D
scene, we deduce 3D objects from the 2D regions that compose the
image.


> Fourth -- I have seen the names "ana" and "kata" given to the two
> extra directions, but I need words like forwards:front::ana:??? and
> so on.
That's where your conlanging tendencies could come in useful. :-)

Personally, I dislike "ana" and "kata" as sounding too arcane. I
prefer "inward" and "outward" as being more descriptive of what a 4D
person might see in her retina when she moves forward/backward in the
4th direction. But then again, that makes about as much sense as
calling our 3D 'forward' and 'backward' as 'inward' and 'outward' just
because that describes what happens to the 2D image in our retina when
we move forward/ backward in the 3rd dimension. So it's really up to
you what terms you use. Since you're making up a conlang anyway, you
might as well coin your own directional terms from scratch. In your
lexicon, I suggest you use mathematical notation rather than imprecise
words (like "ana" and "kata", which not everyone understands). For
example, you could say that if the observer is facing the +W direction
(looking into the positive W axis), then <up> is +Z, <down> is -Z,
<left> is -X, <right> is +X, and <??> is +Y, and <??> is -Y. <Forward>
would be +W, of course, and <backward> would be -W.


T

--
Just because you survived after you did it, doesn't mean it wasn't stupid!