P3D Re: Missing dimension

[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>

>>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. 

Why should that be? Bruce appears to be modeling what human-type vision would
be like in scenarios of different numbers of spatial dimensions. A human-
type eye with a 1-dimensional retina would map a large wedge (e.g. 120-180
degrees) of the two-dimensional plane of the universe, onto a line segment
(or curve segment).

>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.

That's nothing like the human-like model of vision that Bruce appears to be
describing. You seem to be thinking of a model like the compound eyes of an
insect, except that instead of the directions of view of the component eyes
extending radially outward from a central point (real insect), you're
thinking of the directions of view of all the component eyes being parallel
(so the only difference in views is the lateral displacement), and the
eye images some tiny fraction of its universe at a 1:1 scale. (And for two
eyes, the direction of view of the two eyes always being 90 degrees apart.)
I do not think that is the eye model Bruce had in mind.

>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.

I think that's what Bruce had in mind. The discussion thus far is analogous
to one person stating that apples grow on trees, and another person replying
that that's generally not true, that most trees (oak, maple, etc.) do not
grow apples, and that only in the specific and special case that it's an
apple tree is there any chance of apples growing on it. The second statement
is true and adds information, but there wasn't necessarily any problem with
the first statement.

John R


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