Message #110

From: David Vanderschel <>
Subject: Presenting 3D Cube looking from inside out
Date: Sun, 27 Mar 2005 12:20:45 -0600

I am wondering if anyone can provide the reference for
a way of presenting 3D Rubik’s in 2D by regarding it
as an empty box and displaying the view looking inside
with one face removed. (I recall that the method has
been mentioned on this list previously, but without
much elaboration.)

I know that I have seen a thorough presentation using
that method of showing the state of the Cube. I have
done some fairly exhaustive Web-searching on the
subject, and I have not come up with anything. So I
am inclined to suspect that I probably saw it in a
magazine. My best bet is that it was in one of
Douglas Hofstadter’s Metamagical Themas columns in
Scientific American. If that is the case, I probably
have the old issue still lying about if someone could
just tell me which one. (Yes, I have them going back
to the ’70s.) Of course, something on the Web would be
even better.

More detail on the presentation method: Think of the
Cube as an empty box with 9 transparent stickers stuck
on each of 6 transparent faces to the Cube. Looking
down from the top, remove the Up face and set it to
the side of the box with the same orientation. Now,
for the 2D viewing projection, the vertical sides of
the box (and the stickers on them) project as
trapezoids; but, looking from the inside out this way,
it is easy to see which stickers are adjacent to any
given one. It is not even difficult to imagine how
the stickers on the set-aside ‘lid’ relate to those in
the top rows of the vertical faces. This solves the
problem that a normal projection of the 3D Cube will
leave at least three faces unobservable, so additional
mechanisms for ‘looking around’ are needed with these
normal projections.

What I was really hoping for was a software simulation
of Rubik’s Cube based on this looking-from-the-inside
model. It would be far simpler to implement than
MagicCube4D. A big advantage of having such a
simulation is that this way of looking at the 3D Cube
is much more nearly analogous to the way MagicCube4D
presents the 4D Cube. I think folks might catch on to
MagicCube4D’s way more quickly if they could first
play with a 3D Cube presented in the ‘lid off’ manner.

David V.