Message #412

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: [MC4D] On Rendering the Elusive 8th "Face" of the 4D Puzzle
Date: Tue, 21 Aug 2007 01:03:05 -0500

On Monday, August 20, "Roice Nelson" <roice@gravitation3d.com> wrote:
>First is that while I recognize this should be
>considered a matter of personal preference (and my
>preference is therefore no better than another’s),

Note that what I advocated was an _option_ for
displaying the Out face shifted to the side. I was
not trying to impose my preference on everyone. But I
think there are others who would like the option.

>I find the "box with lid off" presentation
>undesirable because it introduces an artificial
>manipulation of the projection.

Face shrink and sticker shrink strike me as being
rather artificial as well. The motivation for
introducing these artifical tweaks was to improve
one’s ability to see what’s where. It also strikes me
that refusing to render the Out face at all is
extremely artificial. The puzzle is _already_ being
displayed with the "lid off". There are some of us
who would like to see what the lid looks like, but
without obscuring our view of the other faces. This
desire is certainly compatible with the original
desire to be able to see what’s where and which
original desire led to introduction of the shrinks.

>To me, there is not much difference between moving
>the lid off to the side vs. doing a display somewhat
>like Alex presented …

I agree. See my ‘revelation’ near the end.

>To give a last bit of further weight to my position,
>I’d like to point out the programming model becomes
>less simple/elegant if one wants to introduce either
>of these behaviors (I’m not saying it becomes too
>difficult, just less elegant because of the special
>casing).

I think the logic about "Which side of this face are
we seeing now?" must already be in MC4D because the
stickers come into view and go out of view at the
appropriate times during animation when the puzzle is
turned so that which face is Out changes. All that
remains is to introduce a pair of virtual centers
relative to which the stickers are plotted depending
on which side is being viewed. This is not a major
change. It does not even strike me as inelegant.

>Finally, while I agree with David’s explanations of
>what is actually happening with the outer face
>relative to the eyepoint, I don’t find the
>description of it turning "inside out" offensive.

I believe that the word I used was "meaningless". If
the eyepoint can be placed close enough to the puzzle
that stickers can ‘hit’ it, then maybe "inside out"
might apply while the eyepoint lies inside a sticker
(and then only with respect to that one sticker).
However, I don’t think you can make this happen. (I
certainly made a point of preventing the analogous
occurrence in MC3D.) In any case, stickers hitting
the eye is not what I am talking about. I am only
talking about the hyperplane in which the stickers are
embedded hitting the eye. The mere passing of the
eyepoint from one side of the hyperplane in which a
face is embedded to the other does not create a
situation for which the phrase "inside out" strikes me
as meaningful. A 3D analogy would be a 2D scene
painted on a flat transparency. You can turn that
transparent ‘paper’ in 3 space. When your eye lies in
the plane of the transparency, the ‘picture’ collapses
into a line. There is nothing "inside out" about the
collapsed situation. As the turning proceeds, you are
then viewing a mirror image of the scene from the
other side of the transparency. The word "flip" seems
to have better connotations for what I perceive as
happening than "turn inside out". What is true is
that the eye is moving from the inside side to the
outside side (or vice versa) of the hyperplane of the
face; but, though using similar words, that fact is a
bit different from an "inside out" effect on the image
seen. (The last sentence above will make better sense
after reading my paragraph second below.)

>To us, limited 3D beings, that is what the stickers
>appear to do,

I don’t see this. I am not sure what you are talking
about. I just imagine them going on edge and then
fattening out again. They reappear mirror reflected.
It strikes me as being more nearly analogous to what
the animation in MC3D does if you use the mirroring
prefix on a twist: 2D ‘objects’ collapse into a line
and then reemerge mirrored.

>even though they are not actually turning inside out
>within the larger dimensional space.

But the projected-into-3-space stickers are not
turning inside out in 3-space either. Inside/outside
is not the same as front-side/back-side. Note that
the words "inside" and "outside" _are_ relevant for
describing the position of the eyepoint relative to a
hyperface. A point at the center of the puzzle is on
the inside side of all hyperfaces. But, in that
context, "inside out" is not meaningful. The eyepoint
can make a transition from the inside side to the
outside side of the hyperplane in which a hyperface is
embedded. But nothing turns inside out. What does
happen is that our view of what’s in the hyperplane
undergoes mirror reflection.

>(In addition to the handedness, if one side of a box
>lid was painted one color, and the other side
>another, you would see the color of the lid change as
>your viewpoint moved from the outside to the inside,
>so the issue of which side we are viewing from is not
>necessarily meaningless - for the 120 cell program I
>did, I actually allow one to control coloring of the
>different sides of the cells and depending on the
>projection parameters, one can see the colors change
>with viewpoint changes. It really does appear as if
>the cells are turning inside out.)

I agree that, if stickers can have different colors on
their two sides, that changes things considerably.
However, that is not the situation we are dealing
with.

>Anyway, talking about turning "inside out" seems as
>fair as talking about sticker "distortions", both
>appearances only being due to projection effects.

There is nothing imaginary about the distortion of the
faces due to the perspective projection from 4D to
3D. The ‘images’ of the hyperfaces as projected into
3D are distinctly not cubical (unless the hyperface
happens to be orthogonal to the line of view). It
appears to me that the "inside out" phrase is being
used to refer to a dynamic phenomenon which occurs
during animation. I think you can argue that this
effect can occur whether there is perspective
transformation or not. It arises because the eyepoint
moves from one side to the other of the hyperplane in
which is embedded the thing that it said to turn
"inside out". I still don’t see it.

>Sorry, my minor comments turned out more lengthy than
>I intended.

No penalty for that. Those who are not interested do
not need to read all this discussion. Furthermore, I
hope you will continue the discussion. I suspect that
this is precisely the sort of discussion that Alex was
hoping to provoke.

After I posted my previous message, it occurred to me
that there was something I had not explained very
well. I had written:
>With respect to hyperstickers in the Out face, one’s
>ability to ‘see’ which set of hyperstickers lie on
>the same hypercubie is quite similar to the cognition
>required for the Box with the Lid Off presentation of
>the 3D puzzle. It’s just that, when one of them is
>in the Out face, a pair of hyperstickers stuck on the
>same hypercubie no longer appear to ‘face’ each other
>at close range. Instead, they both seem to be
>displaced in the same direction relative to the
>centers of their respective faces.

First of all, the only stickers in the regular MC4D
presentation which are candidates for being on
hypercubies in the Out slice are those in the 3x3
groups of stickers which plot on the outside (relative
to the whole 3D scene) of each of the 6 faces other
than the In face. They happen to be the stickers
which render the largest because they are those which
are displaced in the Out direction relative to the
particular face and are thus the closest to the
eyepoint, which lies on the In-Out axis in the Out
direction. So, considering those 54 stickers, we can
talk about which of those and the ones in the Out face
are stuck on the same hypercubie. Note that, for a
sticker in 3D-corner position relative to the Out
face, there are 3 other faces which have a corner
sticker among the 54 which appears to be displaced in
the same direction from the center of its face as is
the original one in the Out face. So this identifies
the 4 stickers on a given 4D-corner hypercube.
Similarly, for a sticker in 3D-edge position relative
to the Out face, there are 2 faces which have
similarly displaced stickers among the 54; and, for a
sticker in 3D-face position, there is only one other
face which has a sticker among the 54 which is
displaced in the same direction relative to its face.
This seems long-winded, but I don’t think it takes
much time to learn to ‘see’ it this way. Indeed I
think it is as easy to make the associations this way
as with the display technique Alex suggested.

It occurs to me that what Alex suggested could be
regarded as follows: Present the Out face with
extreme sticker-shrink and a little face-grow. The
center sticker of the Out face is not drawn, as it
would coincide with the center sticker of the In face.
(This is not a problem, as the center sticker in a
face does not move when you twist the corresponding
slice.) In other words, the Out face is projected in
place on top of the other faces, but the shrink is so
extreme for the stickers in the Out face that,
combined with a little face-grow, they do not obscure
the rendering of the other stickers. My point is that
the challenge I made to show the mathematics of the
transformation for the Out face can be satisfied. As
it turns out, it is the _same_ transformation. What
is different is the pair of parameters used for
sticker-shrink and face-shrink, which are consistent
on all faces except the Out face.

With the move-aside approach I suggest, the Out face
can be rendered with the same sticker- and face-shrink
as is used for all the other faces. I think I would
be able to comprehend the translated Out face better
than I could comprehend Alex’s finely dispersed one,
parts of which will likely be obscured by stickers
from the other faces. But, in a strong sense, they
are equivalent. They both address the issue of how
one can present the Out face in a geometrically
interpretable way that does not obscure the other
stickers. You can make it diaphanous on top or
substantial on the side. Note that, even with Alex’s
approach, the Out face is still unique in that it is
being seen from the outside. With either approach,
the rendering of the face that was Out will undergo
mirror reflection when the puzzle is turned so that it
is Out no more.

As it turns out, what Alex suggests is already almost
doable from an analogous point of view in MC3D - the
hitches being that the extreme sticker shrink must be
applied uniformly and no individualized face-grow is
possible. The configurations numbered 3 and 7 on the
ConfigSelect menu could be viewed in this spirit.
Shrink and eyepoint position have been manipulated so
that, even when drawn in correct position on top
(according to the 3D-to-2D perspective transformation),
the stickers of the Up face do not obscure our view of
the others.

Regards,
David V.