Message #7

From: David Vanderschel <>
Subject: Fwd: Re: [MC4D] Re: phew, at last… [4^4 solution]
Date: Fri, 01 Aug 2003 19:31:02 -0000

Date: Mon Apr 8, 2002 8:18 pm

— In, dgreen@s… wrote:
hello christian,

i assume you’re asking me rather than don.
yes, i wanted a completely accurate projection of a 4d cube+mirror
using 4d viewpoint and direction that "sees" both objects such that
mirror mostly includes a 4d "reflected" view of the missing face. The
view you described may not be complete. i think it also wouldn’t be
correct since the current display is of a 4d viewpoint in a direct
from the center of the object through both the closest and furthest
faces. a single mirror somewhat off to that line would not display an
axis-oriented view.

that said, i just realized i may not be making the correct 3d
analogy. i
keep forgetting that our 4d projection is really analogous to taking
lid off a box and looking in so that the bottom of the box is a small
square in the center, and the sides appear foreshortened arrayed
that furthest face. i suppose it would make some sense to think of
putting a mirror inside the box so that you can look back at the
of the lid, but that suggests that the mirror would necessarily
part of the normal view of the puzzle.

it occurs to me that i don’t really know what the "correct" view of
puzzle from the outside would look like. the only thing that seems
obvious is that you could never see more than half of the faces just
like in 3d which, while perhaps more accurate, would be harder to
operate on. though perhaps the addition of a true 4d mirror would
ameliorate that problem, especially if we allowed users to click on
hidden faces "through the mirror".

i’m not sure what you’re getting at with "lines between stickers" but
can say that we never experimented with 4d mirrors but we did
with faithful projections of the invisible face both normally and in
wireframe, but the results were clearly unhelpful.

don, you want to jump in here and tell us what this all means and what
would be possible or helpful?


Christian Lundkvist 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
> 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
> "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
> 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
> been tried before…?With the 3D -> 2D analog, if you rotate
> 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,
> rigid lines will cause the hidden square to rotate in exactly the
> 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 (
> ) 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
> 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
> smaller cube off to the side? Just some musings in the
> night…Cubically yours, Christian
> —– Original Message —–
> From:dgreen@s…
> To:
> 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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