Message #407

From: Alexander Goldberg <>
Subject: On Rendering the Elusive 8th "Face" of the 4D Puzzle
Date: Fri, 17 Aug 2007 20:29:50 -0000

For those of you who have also been curious where those 8th "face"
facets disappear to when they get cycled outward(?), this post is open
to your thoughts. I hope some discussion of this subject will be
aroused here.

I’ve copied some discussions I had with Melinda on the subject below,
and am going to upload a couple of renderings I made to depict how the
8th face’s facets could be rendered to complete the image of the

There is currently no way to draw the 8th face. We tried that before but
the correct projection puts it inside the bounds of the rest of the
puzzle rather than outside which just makes a mess. We don’t want to
draw an unrealistic projection so it’s best to just not draw it.

I’ve attached a couple of renderings of a quick 3d model I made
showing what the 8th face would look like. Is this how you saw it?
I’m not sure I understand the realistic vs. unrealistic projection.

I like your images and agree that it could make for a workable puzzle.
What I mean by "realistic" is that there needs to be a natural
projection from 4D into 3D of a 4D puzzle that includes face and sticker
shrinks in 4D. In other words we do not allow ourselves to perform
non-linear transformations on the 3D projections. Given that
restriction, the outer face’s stickers would not end up outside the
others but rather would be turned inside-out and would intersect other
parts of the visible puzzle. It would just be a mess. One could do
something like what you depict but since it can’t be done within the
constraints we’ve given ourselves, we aren’t likely to want to try to
implement it.
p.s. You may want to join the MC4D mailing list and bring this
discussion to the rest of the community who I’m sure would enjoy it,
especially your pictures.

I don’t really understand what you’ve described. Well, I should say I
understand it, but I’m having trouble imagining how to visualize it.
I’m under the assumption that you visualize the paths the outer facets
(pink) take when they get cycled inward, and similarly the paths the
another face’s facets take when moving outward. The solid state
rendering doesn’t portray that at all. Unfortunately the program I
used is too simple and doesn’t support animation. The shapes of the
facets would transform as their face would be cycled too (as they do
when cycled from the "top" (center) face to any other position, only
slightly more warped when cycled outward.

Notice also that right before a face becomes the invisible face it gets
large and almost completely flat. It is actually turning inside out. At
that point it begins to get smaller and move back into the space
occupied by the projections of the other faces.

The images I have provided should portray the 8th face (inside-out)
surrounding the inner seven faces. The only facet that remains
invisible or ‘unrenderable’ is the single-color center for the 8th
face, which is the only facet I do not see a point in attempting to
render in 3D. The other 26 facets of the 8th face are pink in my
renderings; the overall color scheme in this model is similar to a 3D
Rubik’s Cube, with two new colors (gray and pink) for the added faces
in 4D.

Alex Goldberg

(I posted this once sending from Gmail, but am unsure if it arrived or
will arrive properly so I am posting this again from Yahoo!Groups to
be sure. The two image files 4Dcubewith8thface(1).jpg and
4Dcubewith8thface(2).jpg are already in the ‘Files’ folder.)