Message #6

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Fwd: Re: [MC4D] Re: phew, at last… [4^4 solution]
Date: Fri, 01 Aug 2003 19:29:19 -0000

Date: Mon Apr 8, 2002 6:44 pm

— In MC4D@yahoogroups.com, "Christian Lundkvist"
<christian.lundkvist@t…> wrote:

Hi!

I’m not sure of what you want here. Did you want the "visible" faces
of the 4D-cube to look like they do now, and just have the "hidden"
face "mirrored"? Or did you want to make a completely accurate
projection of the cube+mirror down into 3D? An accurate projection
would probably look very strange; imagine being a 2D person studying
the drawing in your message. It would be very difficult figuring out
what it was…
If you project the rubik’s cube down to 2D "MC4D-style" you would
have the picture on the MC4D-icon. If you "mirrored" the "hidden"
face it would probably be best to make it look like the center face,
that is: a square (and place it to the side). The analog with the
MC4D would of course be a cube that looked like the center cube,
placed off to the side. As I understand it, this has been tried
before…?
With the 3D -> 2D analog, if you rotate (whithout twisting) the cube,
the "mirrored" face would rotate in the same manner. How would
the "mirrored" "face" in the MC4D rotate when the whole 4D-cube is
rotated? I’m not sure here… My guess is that it would rotate
exactly like the center cube of the MC4D. In the 3D->2D analogy, you
can think of it like you have suspended your projection on a plane in
3D. Below this plane, you have another plane with the
projected "hidden" face on it. You can draw parallel rigid lines
between the stickers of the middle square and the hidden square
(perpendicular to the planes). When you rotate the center square, the
rigid lines will cause the hidden square to rotate in exactly the
same manner as the center square.
With the MC4D, you can suspend your projection on a hyperplane in 4D
and draw the rigid lines connecting the stickers of the center and
hidden cubes. The lines are perpendicular to the hyperplanes.
Now, if we assume that the coordinates of one of the cube-stickers in
the projection is ( x,y,z,w ) it must be the case that the coordinate
of the analogous cube-sticker on the "mirrored" "face" is (
x,y,z,w’ ) because the vector between these points must be
perpendicular to the hyperplanes. Therefore, it would rotate exactly
like the center cube. (Guess my guess was correct…. :-) )

Was I right in assuming that you had experimented with this kind
of "mirroring" before, that is: placing a smaller cube off to the
side?

Just some musings in the night…

Cubically yours,

Christian

—– Original Message —–
From: dgreen@s…
To: MC4D@yahoogroups.com
Sent: Monday, April 08, 2002 5:46 AM
Subject: Re: [MC4D] Re: phew, at last… [4^4 solution]

Don Hatch wrote:
> my
> dream would be to model the precise 4d analog of a 3d mirror
and place it in
> the 4d scene such that the invisible face becomes visible from
a different
> direction. i guess that the 4d "reflection" would appear as a
3d object
> embedded within a 3d mirror appearing like a solid block of
glass. i have no
> idea how to do this though. perhaps you’ll figure it out!
Can you draw a lower-dimensional analogue (Rubik’s cube)?

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