# 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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— End forwarded message —