P3D Re: Technical question: Missing dimension

[Larry Berlin <lberlin@xxxxxxxxx>]

>Date: Sat, 3 Apr 1999
>From: roberts@xxxxxxxxxxxxxxxxx (John W Roberts)
>..........(Bruce provided this formula)............
>>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. .....
>
>You left out Case 0 (a 1-dimensional world).

****  A one dimensional world would have to start with eyes of zero
dimensions, or consisting of a single point... Since this world is 1D,
additional eyes must be located on the same dimensional line and can only
observe along this same line. For this situation it is assumed that the
creatures body occupies space on the line connected to the eye or eyes (if two).

A single eye'd creature would know if another point is present or not *in
front of* the observer. (on/off)

A two eye'd creature would add the ability to observe points in either
direction without turning around. This is the ultimate *eye on the back of
one's head* scenario. (on/off + front/back) Thus 0D + 0D = 1D


>
>>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
>
>As you note, binocular stereopsis appears to be fundamentally a 2-dimensional
>phenomenon (the "Y dimension" is not inherently involved).

*****  Incorrect assumptions here...  The Y dimension *is* intimately
involved! It's embedded in the fact that an *identical Y exists for every
observed X* from either vantage point. The Y factor always lies on a line
parallel to the axis of dislocation (of relative observations) which makes
it very much always present. This would be reversed if the eyes were
oriented top to bottom, in which case it would have identical X information
and observe differences along the Y dimension alone. Binocular stereopsis is
a factor of differences along the axis of dislocation, whatever that axis
happens to be. 

Therefore it would be a very strange eye indeed that could only observe 2D
by way of the X axis and the Z axis!!! Imagine a one dimensional retina and
two irises in each eye. In order to get two of such strange eyes to see
effective 3D in which Y is magically added, one eye's retina line would have
to be oriented on the X axis and the other on the Y axis. 

They would each see 2D. One eye sees X + Z, the other sees Y + Z. When
combined, the Z factor remains constant and the X and Y fills out the
width/height of a familiar 3D scene. However with 1D retinas they would have
to each scan the scene constantly otherwise it would only be able to observe
a 3D XY crossed axis. This is similar to the model on which we build
scanners, a 1D receptor scans along an axis perpendicular to it's axis of
perception providing 2D.

Unfortunately, if both eye's see X + Z, having two of them does NOT add the
Y dimension at all! Therefore it would still be a 2D view merely repeated
twice from different vantage points. The different vantage points would not
add another dimension because each separate eye observes the same factors...
In this case it's( X + Z) + (X + Z) = X + Z*2  (assuming the two eyes are
displaced horizontally.)

Feeding this information backwards, the transition from the 2D world to the
3D world is one in which the eye's retina transforms from a 1D retina to a
2D - XY retina. Only then can you add them together to view a 3D world the
way we know it. 

Feeding it forwards, it implies that a 4D world would have eyes with 3D
retinas. Whatever effect the fourth dimension has might manifest as
differences from each vantage point. It raises the questions, which axis to
use for eye displacement and since each eye would have it's own perceptual
axis with built in displacement, which direction to orient each eye, and
would there have to be three eyes? ( I'm writing as I read, so after writing
the above, I see that is precisely where John Roberts goes next...)

>..............
>I'm a little concerned about the case of the 4-dimensinal world. I agree
>that two 3-dimensional eyes would provide information about the
>4-dimensional world not available from just one 3-dimensional eye. But
>might hypothetical residents of a world with 4 spatial dimensions also have
>some fundamental characteristic of perception that requires a minimum
>of *three* 3-dimensional views in order to be properly interpreted?
>
>A 3-dimensional world has 3 fundamental dimensions, and 3 possible
>combinations of pairs of the 3 fundamental dimensions. A 4-dimensional world
>has 4 fundamental dimensions, 6 possible combinations of pairs of the
>4 fundamental dimensions, and 4 possible combinations of triads of the
>4 fundamental dimensions. So as you may have been thinking in the final 
>question, thinking just in terms of 2 eyes in a 4-dimensional universe
>may be a prejudice based on the limits of our 3-dimensional experience.
>
>John R
>

****  I have to agree with John's final conclusion. However it means we'd
have to back up to the 0D and 1D and 2D realms and *not* use two eyes there
by assumption...  If 3D has two eyes, 2D has one eye, and 1D has zero eye's.
What does 0D have? Sight itself seems to only exist beginning with 2D
(proving the primitiveness of the dominant imaging technology in the 20th
Century).

However, nothing in our 3D world prevents us from having more than two eyes.
It merely defines the *minimum requirement* scenario. At the same time,
using less than this minimum results in a failure to observe the full
dimensionality that is present. 

While pondering this scenario, I keep asking myself, what differences would
be observed between any two or more 3D perceptive points (in 4D)? I want to
model it and look at it, only to realize I can only observe it in 3D at any
given moment...

However, it would seem possible to model two 3D versions of some structure,
which themselves are distorted or changed in some way between the two
instances by effect of the fourth dimension. Then observe them sequentially,
thus applying time to that mysterious fourth dimension. This allows the
brain to make comparisons of the two circumstances, (time being already
familiar) which presumably would be simultaneous observations within the 4D
world.

What changes to make? Could one work out all of the situations described by
John for a given solid? I know that the hyper-cube is one such model, but
what factors control which of the many possible views are appropriate to
this multi-eyed synchronous observation from within 4D? Maybe that scenario
hasn't been solved mathematically yet? (a subject for T3D...)

As to the model, it could be presented using anaglyphic 3D combined with
page flipping/interlacing and the use of LCS glasses... Two views would
appear to be simultaneously present, with Red/Blue glasses they would be 3D,
with time sequential operation they would appear to be the same temporal
scene. Arguably the same effect of having two 3D eyes....

Any suggestions as to what modifications are to be imposed with the fourth
dimension? Compression, expansion, distortion of one or more other D's,
inclusion of totally new visual elements not observable in any single 3D
instance (hard to imagine that one!)?

Larry Berlin

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


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