P3D Re: Technical question: Missing dimension

[roberts@xxxxxxxxxxxxxxxxx (John W Roberts)]

>Date: Thu, 1 Apr 1999 18:20:41 -0700
>From: Bruce Springsteen <bsspringsteen@xxxxxxxxx>
>Subject: P3D Re:  Technical question: Missing dimension

>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

You left out Case 0 (a 1-dimensional world).

>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).

>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

>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?

In a 3-dimensional world, having two eyes enables stereopsis within a
certain part of the field of view, permitting judgement of relative
distances to parts of the scene. But that's only a tiny portion of the
overall reality of the 3D scene. A hologram effectively employs a huge
number of "eyes" across a portion of a 2-dimensional surface. For true
perception of 3D reality, you would want an eye at every point in the
universe.

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


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