P3D Re: Missing dimension (part 2)

[roberts@xxxxxxxxxxxxxxxxx (John W Roberts)]

>From: Larry Berlin <lberlin@xxxxxxxxx>

>>From: roberts@xxxxxxxxxxxxxxxxx (John W Roberts)
>>>****  Since both eye's observe the same Y information (if set side by side
>>>in the face) it is not the Y dimension that is added. 

>>It is "added" to the case 1 (two-dimensional world) scenario, to make the
>>three-dimensional world scenario (even though the Y dimension is not
>>"actively" involved in human stereo perception).

>****  Wrong. It's there in the 2D scenario just as much as in 3D. It is very
>actively involved in *human stereo perception.* Just try misaligning your
>stereo images before viewing and see how active it is!!! You could say it's
>an *active constant*. 

Wrong. What I was indicating is that while humans may use Y information in
distinguishing objects from one another, the Y information itself is not
used by humans in the primary calculation of depth. If you want to consider
that to be "active involvement", you might as well consider color to be 
"actively involved in human stereo perception". [Note: Y information and color
can serve as non-stereopsis depth cues.]

>Without it being there AND being a constant, you won't
>get stereo that you could enjoy. Without it you can't easily define 2D!

Human stereo vision is inherently keystoned - only those objects in the
vertical plane that bisects the head and passes through the nose are
actually the same vertical size in the left and right fields of view.
So the Y information is generally *not* constant, and yet I enjoy looking
at the world around me in stereo.

>****  Yes it is a definition issue. The truth is we don't know what it would
>look like in a 4D world. Maybe it would manifest as differential coloration
>due to spectoscopic properties, or something else. The point is that to
>model it, we have to specify something as representative. Even in the
>familiar 3D, we don't find ordinary objects connected to physical
>manifestations of the three axes! We can place rulers next to objects, or
>glue them on, but we don't think of a cup without rulers as being any less
>dimensional as one with attached rulers.

The axes are defined by the orientation of the observer.

>If we attempt to model a cup in a 4D world, where do we glue the extra
>rulers? How are we to know it's a 4D cup instead of a 3D one? If 3D is a
>mere projection out of a 4D realm, how do we set up the model for what it's
>like inside 4D (where we can see manifestations of each dimension)?

3D drafting techniques include the use of three 2D projections to represent
a 3D object. This does not give full information on the object, and
ambiguities are possible in certain cases, but it provides a lot more
information on the object than a single 2D projection could give.
Similarly, one would expect that a "typical" 4D object could be represented
with a fair degree of success using four 3-D projections, or possibly
using six 2-D projections. (And yes, we would not *always* be able to
judge the proper 4D shape from these projections. But we're also not always
able to judge the true 3D shape of an object using our stereo vision.)

>>Interesting - perhaps instead of 2 eyes (each N-1 dimensional perception),
>>it takes 2^(N-2) eyes (each N-1 D) to properly perceive in N dimensions.
>>So the 4-D critter might have four 3-D eyes (arranged, for example in
>>a tetrahedron in X,Y,Z) to properly perceive in the W direction.

>Except that the tetrahedron has to exist in an XYZW environment...
>complicating the example a bit since the tetrahedron may not be a
>tetrahedron anymore.

A 2D triangle can still be a triangle in a 3D environment.

>This implies that a 5D creature might require 8 eyes, each of which are
>capable of 4D perception...

>The fact that this model seems to extend towards a steep gradient of growing
>complexity that far outstrips the additional number of dimensions means it
>is perhaps an unlikely scenario.

I don't think we're necessarily qualified to judge whether or not it's
unlikely. We don't have any animals with 4D or 5D vision to consider.
Bamboo is the only 4-dimensional plant I know of, and I don't think it
has "vision" in the way we usually define it.

>You pointed out that even in our 3D world, the ideal 3D view might involve
>multitudes of eyes in all possible locations. Yet our two eyes satisfy the
>minimum requirement of observing in a given instant, a 3D perception.
>Therefore it's unlikely that higher dimensions would require more than some
>minimum number of eyes and not necessarily complexify their internal
>structures with other dimensional perceptions. 

>The problem is to define just what is supposed to be observed and what order
>of sensory mechanism can do the observing.

Indeed - the quesion is whether there's any fundamental characteristic (not
just incremental addition of information) of a 4D space that's "worth seeing",
that requires more than three 3D eyes to see it.

John R


------------------------------