Message #409

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: [MC4D] On Rendering the Elusive 8th "Face" of the 4D Puzzle
Date: Sun, 19 Aug 2007 23:04:30 -0500

On Friday, August 17, "Alexander Goldberg" <ajgold04@gmail.com> wrote:
>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.

Alex, you are not the first to raise this issue.
Myself and others have raised it in the past and we
got the same arguments against attempts to render the
8th face which you got from Melinda. I do not buy all
those arguments. I say why below.

Alex:
>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
>hypercube.

Melinda:
>>There is currently no way to draw the 8th face.

There may be no way that is currently implemented in
the program, but there is absolutely no difficulty in
rendering it. I refer to the hyperfaces as Left,
Right, Front, Back, Up, Down, In, and Out. By this
naming scheme, it is the Out face that the program is
not drawing. The In face is the small, but
undistorted one shown in the middle. (It is
undistorted because all its points lie at the same
distance from the eyepoint along the viewing
direction - the axis on which the eyepoint lies.) The
Out face, if drawn by the _same_ transformation, would
render in the same manner as does the In face - just
bigger, since it is closer to the eyepoint. (The eyepoint
is on the In-Out axis.)

>>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.

This is true. The hypercubies of the Out face project
into the same 3D volume as do all the others and
obscure one’s view of the other hyperfaces which
project smaller.

>>We don’t want to draw an unrealistic projection so
>>it’s best to just not draw it.

But there is an easy solution: Just shift the
projection of the Out face over so that it does not
fall on top of the others. This amounts to a
presentation analogous to the "Box with the Lid Off"
presentation of the 3D puzzle. That is the default
configuration for presentation that I used in my
implementation of the 3D puzzle, which you may try
here: http://david-v.home.texas.net/MC3D/
(The MC3D program can present the 3D puzzle in a
surprisingly large variety of other manners, some of
which are more useful for solving it. Actually, an
even more compelling version of "Box with the Lid Off"
configuration is the second one on the ConfigSelect
menu.)

>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.

Alex, I agree with Melinda that this is an
‘unrealistic’ projection. If you could show the
mathematics of it, I might change my mind; but I
cannot imagine a realistic projection from 4-space to
3-space that would produce this rendering. As far as
I can tell, you have just plotted the stickers from
the Out face in positions relative to the others which
are suggestive of their geometric relationship in
4-space. In so doing, the face loses its integrity
and its pieces are spread literally all over the
place.

>>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.

The intersection problem can be solved easily by
simply shifting the picture that results from the Out
face over so that it no longer falls atop the
renderings of the other faces.

The "inside-out" issue is a non-issue. All the
hyperfaces (which are 3D constructs) are flat in 4D.
The issue is whether what we are seeing of one is the
side which faces the center of the 4D cube or the side
which faces outward. As it turns out, for each of the
faces which is being drawn, the side seen is that
which faces inward. The Out face, if drawn, would be
as seen from the outside. For a 3D scene embedded in
4-space, the difference in its appearance depending on
which side of the 3D hyperplane (in which the 3D scene
is embedded) it is viewed from is that of mirror
reflection. Aside from this switching of the
handedness, there is no difference in appearance to a
3D observer whose eyepoint necessarily lies in the
same 3D space into which the 4D scene is projected.
The issue of "which side" we are viewing it from in 3D
is meaningless, since our viewpoint lies _in_ the 3D
hyperplane onto which the 4D scene has been projected.
We see _everything_ in a manner which is "on edge"
relative to the 4D scene. The viewer’s point of view
is constrained to be in the same hyperplane as the
projection, so the viewer is never on _either_ side as
might be defined in 4-space.

If what I was saying in the above paragraph is not
clear, you might be able to gain some additional
insight by playing around with the 1D rendering
capabilities in MC3D. With MC3D you can approach the
3D puzzle from a 1D rendering as a Flatlander would
have to. This is analogous to us 3D beings trying to
approach the 4D puzzle based on a 2D rendering such as
we get from MC4D. (MC4D must do a second
projection to 2D to render the 3D scene which arises
from the initial projection from 4-space. We think
in terms of the 3D scene, but its rendering is 2D.)

>>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.

However, rendering the Out-face with a shift in
3-space is easily achieved. I would expect that the
code which currently implements the rendering of the
In face would work with very little modification for
rendering the Out face. The trickiest part probably
lies in determining when, during animation, the
hyperplane of a face passes through the eyepoint,
necessitating changing which image (shifted or not)
the stickers in that face are plotted relative to.
But nothing blows up. (E.g., no division by zero.)

>>Notice also that right before a face becomes the
>>invisible face it gets large and almost completely
>>flat. It is actually turning inside out.

This is not a correct explanation. What is actually
happening is that the hyperplane containing the face
is intersecting the eyepoint of the 4D-to-3D
projection. When the eyepoint is _in_ the hyperplane
of the face, the rendering of that face does go flat.
The non-flat renderings just before and after the
crossing differ by which side in 4D the face is viewed
from and thus bear a mirror image relationship. So it
is _not_ "actually turning inside out" - the 4D side
from which we are viewing it is flipping. Nothing is
turning inside out.

You can see the analogous phenomenon in my 3D program.
When you twist one of the laterally facing slices (not
the Up or Down one), some of the stickers will move
out of the plane of the Up face (the removed lid) and
some will move into it. For a given sticker during
the animation, the switch occurs when the plane of the
sticker passes through the eyepoint of the projection
from 3-space. At that instant, the sticker projects
to a line segment. It is also at this instant that
the program moves the presentation of the sticker from
being plotted relative to one of the virtual centers
to the other. You can study this effect in detail by
using the mouse to control the animation, slowing or
stopping it at the critical instant. (There are other
configurations of the program in which as many as
three of the faces are seen from the outside rather
than the single Up face as in the default
configuration, which presentation (without the shifted
Up face) is analogous to that of MC4D. A variety of
other presentation configuations arise in MC3D
because, unlike MC4D, MC3D does not require the
eyepoint for the first projection to lie on a
coordinate axis.)

I would very much like to see an option in MC4D to
render the Out face in the manner analogous to "Box
with the Lid Off" for the 3D puzzle. It seems to me
that the code required to do this would be rather
trivial compared to some of the impressive
improvements Melinda has added recently. 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. It is easy to learn to see this
relation with the default configuration for the MC3D
program; and it would be easy to to transfer this
learning to a version of MC4D which draws a displaced
rendering of the Out face.

I have attached a couple examples of roughly what it
could look like. One is unscrambled and the other has
a couple of twists from the unscrambled state. I
mentioned that the In face is undistorted, but I was
really talking about the more profound distortion
which arises from the 4D-to-3D transformation. There
should still be a little perspective distortion from
the 3D viewing transformation and there is not enough
of that in the Out faces as I have shown them. (I did
it simply by blowing up the In face after rotating
what had been the Out face into there, correcting
for the mirror imaging.)

Regards,
David V.