P3D Re: Missing dimension

[Larry Berlin <lberlin@xxxxxxxxx>]

>Date: Thu, 1 Apr 1999 
>From: Bruce Springsteen <bsspringsteen@xxxxxxxxx>
>........................
>
>Gabriel (Nobody's Fool) Jacob puzzles:
>> >If two 2-D pictures equals 3-D, or 2D+2D=3D, how come it's not 4D?
>> >Where did the other dimension disappear to? Another dimension?
>> >Bruce, anyone? ;-)
>
>I'll bite.  The answer is
>xD + xD = (x+1)D
>
>Case 1:  Ever read "Flatland"?  In a 2-dimensional world, a 2D photographer
>views a 2D object with two eyes having 1-dimensional retinas.  One of those
>eyes can only give his brain information about dimension (X) from a single
>vantage point.  But the different (X) information received by two eyes at
>different (X) positions lets his brain mysteriously combine the differences
>and perceive (Z).  Brain now has internal "model" of two dimensions.  He
>scoffs at the "unrealistic" work of conventional 1D photographers.
>1D + 1D = 2D

*****  Problem with this... One dimensional eyes perceive in only one
direction. If you are to add these two and get traditional 2D then one eye
perceives in the X direction and the other eye sees only in the Y dimension.
With both eyes he is able to see in 2D. This might be like the optics in a
scanner. The resulting view is at a constant distance, ie no relative depth
information.

You suggest he might combine a 1D + 1D to observe the Z axis. That would
combine two horizontally arranged vantage points. The resulting 2D
perception is of a plane on it's edge except you may perceive points at
different distances on that plane. This would be like stereo 3D without the
Y dimension because of the depth information.


>
>Case 2:  In our 3-dimensional world the 3D photographer equipped with 2D
>retinas in two eyes adds the (Y) dimension to each eye, but the (Y)
>information in each eye is the same because the two eyes are still only
>displaced in (X).  Net brain gain, 1 dimension.  2D + 2D = 3D

****  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. The specific 2D
elements usually referred to in *this world* are the X and the Y dimensions.
This implies the use of the first of my scenarios above for flatland. Both
views contain X and Y but have differences only between X locations. The
*seeing mind* operates by a constant comparison of differences. The Z
dimension is observed by it's effect on one of the first two dimensions
specified by the axis of sensory duplication and dislocation... (ie:
horizontal displacement of two eyes)

You are right that if you use my second scenario regarding flatland, you are
adding the Y dimension, but this is hardly the typical extension of 2D + 2D
= 3D. We usually think of 2D as having width and height but lacking depth
information. It's true in flat images, but in real life the mind has other
than direct visual factors to work from.


>
>Case 3:  4-dimensional stereographer living in a 4D world (called
>hyperspace) has a 3-dimensional retina in each eye, so each eye can add
>information about one more dimension (W), but each eye has the same (W)
>data because the two eyes are still only displaced in (X), as in cases 1
>and 2.  Net brain gain, 1 more dimension.  3D + 3D = 4D

*****  Here again we run into problems. Do two stereophotographers standing
side by side capture a 4D scene? I don't think so... Each set of two 2D
views yields a 3D scene. Two sets of 3D scenes yields another 3D scene. Even
if each eye could perceive full 3D reality having two of them doesn't create
the fourth dimension. This may work if you specify time as the fourth
dimension, but that exists in the 3D world too, so we have to assume some
dimension other than time.

The hyper cube is a mathematically defined 4D object. It's projection into
3D can be observed in stereo and operates like a shadow. Would stereo
viewing of two 3D shadows allow us to see the 4th dimension? No.

If we assume the problem is the fact that we only have 2D eyes, suppose we
use two persons standing side by side. Each sees 3D. Does their combined
perception see the 4th? Again, no.

So we postulate that the 4th dimension is beyond our direct understanding.
Fine, the creatures in this 4D world with 3D eyes might be different than us
and be able to perceive the 4th dimension. Assuming the 4th is literally
another axis behaving more or less like the familiar X Y and Z axes. What if
it's nothing like that at all? It's effect might be observable on the other
dimensions like we observe Z within changes of X. We have to ask what
observable changes might exist between two 3D views within a 4D realm? If we
use the example of a stereoscopic model of a cycling hypercube, which two
views would represent a side by side displacement of perspective?

http://www.sover.net/~manx/hyprcube.html

An alternate scenario arises. A 4D creature could have three or four eyes
arranged with non-linearity.  With horizontal axis displacement it could
observe 3D. With vertical displacement they could observe another 3D. Is
this combined into 4D? We still have the problem that we are seeing
essentially the same 3D realm. We have two sets of displacement differences,
delta X and delta Y. But is the form of observation legitimately 4D?

Does that make a 10 layer lenticular into a 9D view? Or as many D as there
are possible two layer combinations? (counted over 30)


>
>And so on.
>Now if you start adding EYES along with dimensions, that's a different
>story! 2D + 2D + 2D = ?  Who can answer THAT one.  Gabriel?  George? 
>Larry?
>
>Bruce (Dr. Tesseract) Springsteen
>


*****  Should our assignment of dimensionality be defined by observational
or structural parameters? Does having more eyes mean we see more dimensions,
or do we just see the existing ones better?

Based on the evolution of two eyes in most living creatures, maybe 3D is all
the dimensions there are (that are seeable)? Or other dimensions exist but
at our scale of observation they appear to remain constant, and therefore
aren't seen?

Larry Berlin

Email: lberlin@xxxxxxxxx
http://3dzine.simplenet.com/


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