P3D Re: Missing dimension (part 2)

[roberts@xxxxxxxxxxxxxxxxx (John W Roberts)]

>Date: Sat, 3 Apr 1999 20:47:10 -0700
>From: Larry Berlin <lberlin@xxxxxxxxx>
>Subject: P3D Re:  Missing dimension

>>Date: Thu, 1 Apr 1999 
>>From: Bruce Springsteen <bsspringsteen@xxxxxxxxx>

...
>>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. 

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).

>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.

I think this discussion was specifically on direct visual factors, and
on stereo perception, so prior knowledge on the part of the observer of the
dropoff in intensity of smell in a 2-dimensional universe (linear) or the
apparent versus actual size of a 4-dimensional wogglebug would not be
relevant to the issues being discussed here.

>>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... 

>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.

The human observers have 3-dimensional mental models, so anything they see
they interpret as being 3-dimensional. A 4-D creature with two 3-D eyes
might have a 4-D mental model, which can be built up from the two 3-D images.

>So we postulate that the 4th dimension is beyond our direct understanding.

That's not necessarily an incurable condition. Medical diagnosticians can
sometimes view time-sequential 2-D slices of a 3-D view to build up a mental
3-D model. It may (or may not) be possible for a human to learn to build
up a 4-dimensional model, and to build up that model by viewing
time-sequential 3-D slices of that 4-D view. That would be machine-aided
perception rather than direct human perception, but it would at least
involve understanding and perceiving a 4th dimension in a fashion.

>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. 

This appears to be a definition issue.
A "dimension" could be anything - color, temperature, taste, electric charge,
etc. But if we're talking about spatial perception, I think it's sort of
assumed that the dimensions described will all be spatial dimensions.

In the technical field of scientific visualization, many more than three
dimensions are often depicted in a single image. A newspaper weather map
is a classic example. But the extra dimensions shown are not intended to
represent *spatial* dimensions.

>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?

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.

John R


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