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
hypercube.
--------------------------------------------------------------
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.
-Melinda
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 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.
-Melinda
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.
-Alex
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.
-Melinda
--------------------------------------------------------------
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.)
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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
hypercube.
--------------------------------------------------------------
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.
-Melinda
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 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.
-Melinda
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.
-Alex
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.
-Melinda
--------------------------------------------------------------
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
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On Friday, August 17, "Alexander Goldberg"
>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.
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I just had a couple minor comments. First is that while I recognize this should be considered a matter of pe= To me, there is not much difference between moving the lid off to the si= Another possible way to deal with this could be to implement the shift-h= I do find that the MC4D/MC5D sight lines (eyepoint->lookat point) are= On Friday, August 17, "Alexander Goldberg" &= Fair enough, it wouldn't hurt to have the "box with lid off&quo= There are some questions in my mind about the proper way to display= On the inside/outside semantics discussion, yes, it is a dynamic animati= and keep my eye on one of the faces moving down through the center, it a= Roice
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I just had a couple minor comments.
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), I find the "box with lid off" presentation undesirable because
it introduces an artificial manipulation of the projection. I imagine that
statement could be a source of contention depending on what one considers
artificial, but one is effectively taking apart the puzzle to present it in
this way. For a 3D being looking at a real puzzle, you'd never be able to
physically produce a Rubik's cube that worked but had the out-face removed
and off to the side. For this reason, I wouldn't want to see the projection
of the 4D puzzle done as such.
To me, there is not much difference between moving the lid off to the side
vs. doing a display somewhat like Alex presented (which would leave the lid
position true, but alter the projection and/or shrink parameters applied to
that single face so that the stickers did not obscure everything else).
Both solutions introduce unnatural effects into the result. 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).
Another possible way to deal with this could be to implement the
shift-highlighting done in MC5D. If you press shift while hovering over a
cubie, all the stickers on that cubie become visible (whether the face is
set to be visible or not). The stickers are rendered with the current
projection settings regardless of where this places them in the view. This
would allow one to more easily probe the invisible 8th face without having
to move it into view. I can see this being considered an artificial
solution as well and so it is not ideal, but at least the projection would
remain true.
I do find that the MC4D/MC5D sight lines (eyepoint->lookat point) are
constrained to the coordinate axes limiting, and I'd like to extend MC5D on
this front at some point. The reason this hasn't been too big of a thorn in
my side is that for doing solutions, the current setup is probably the most
desirable and what I would end up using anyway. It'd still be nice to have
extra this control over the projection to study the nature of the puzzle
though. In the case of MC5D, it'd be nice if the 4th and 5th dimensions
weren't both centrally projected.
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. To us, limited 3D beings,
that is what the stickers appear to do, even though they are not actually
turning inside out within the larger dimensional space. (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.) Anyway, talking about turning "inside out" seems as
fair as talking about sticker "distortions", both appearances only being due
to projection effects. I still think it was a nice point to make, and gives
a more subtle understanding of what is going on with the projections.
Sorry, my minor comments turned out more lengthy than I intended.
Take care all,
Roice
On 8/19/07, David Vanderschel
>
> On Friday, August 17, "Alexander Goldberg"
> 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.
>
>
>
>
------=_Part_128836_21966388.1187631108441
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable
Content-Disposition: inline
rsonal preference (and my preference is therefore no better than another=
9;s), I find the "box with lid off" presentation undesirable beca=
use it introduces an artificial manipulation of the projection. I ima=
gine that statement could be a source of contention depending on what one c=
onsiders artificial, but one is effectively taking apart the puzzle to pres=
ent it in this way. For a 3D being looking at a real puzzle, you'=
d never be able to physically produce a Rubik's cube that worked but ha=
d the out-face removed and off to the side. For this reason, I wouldn=
't want to see the projection of the 4D puzzle done as such. =20
de vs. doing a display somewhat like Alex presented (which would leave the =
lid position true, but alter the projection and/or shrink parameters applie=
d to that single face so that the stickers did not obscure everything else)=
. Both solutions introduce unnatural effects into the result. T=
o give a last bit of further weight to my position, I'd like to point o=
ut the programming model becomes less simple/elegant if one wants to introd=
uce either of these behaviors (I'm not saying it becomes too =
difficult, just less elegant because of the special casing).
ighlighting done in MC5D. If you press shift while hovering over a cu=
bie, all the stickers on that cubie become visible (whether the face is set=
to be visible or not). The stickers are rendered with the current pr=
ojection settings regardless of where this places them in the view. T=
his would allow one to more easily probe the invisible 8th face without hav=
ing to move it into view. I can see this being considered an artifici=
al solution as well and so it is not ideal, but at least the projection wou=
ld remain true.
constrained to the coordinate axes limiting, and I'd like to extend MC=
5D on this front at some point. The reason this hasn't been too b=
ig of a thorn in my side is that for doing solutions, the current setup is =
probably the most desirable and what I would end up using anyway. It&=
#39;d still be nice to have extra this control over the projection to study=
the nature of the puzzle though. In the case of MC5D, it'd be ni=
ce if the 4th and 5th dimensions weren't both centrally projected.
ly happening with the outer face relative to the eyepoint, I don't find=
the description of it turning "inside out" offensive. To u=
s, limited 3D beings, that is what the stickers appear to do, even though t=
hey are not actually turning inside out within the larger dimensional space=
. (In addition to the handedness, if one side of a box lid was painte=
d 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 issu=
e of which side we are viewing from is not necessarily meaningless - for th=
e 120 cell program I did, I actually allow one to control coloring of the d=
ifferent sides of the cells and depending on the projection parameters, one=
can see the colors change with viewpoint changes. It really does app=
ear as if the cells are turning inside out.) Anyway, talking abo=
ut turning "inside out" seems as fair as talking about sticker &q=
uot;distortions", both appearances only being due to projection effect=
s. I still think it was a nice point to make, and gives a more subtle=
understanding of what is going on with the projections.
David Vanderschel <DvdS@austin=
.rr.com> wrote:=20px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">
lt;to:ajgold04%40gmail.com" target=3D"_blank">ajgold04@gmail.com> wrote=
:
>For those of you who have also been curious where
>those 8th &quo=
t;face" facets disappear to when they get
>cycled outward(?), th=
is post is open to your
>thoughts. I hope some discussion of this sub=
ject
>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
8th face which yo=
u 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 depic=
t 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
dis=
tance 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 th=
e 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 outs=
ide which just makes a mess.
This is true. The hypercubies o=
f the Out face project
into the same 3D volume as do all the others and
obscure one's v=
iew of the other hyperfaces which
project smaller.
=
>>We don't want to draw an unrealistic projection so
>&=
gt;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&q=
uot;
presentation of the 3D puzzle. That is the default
configuration for=
presentation that I used in my
implementation of the 3D puzzle, which y=
ou may try
here: is)" href=3D"http://david-v.home.texas.net/MC3D/" target=3D"_blank">
http://david-v.home.texas.net/MC3D/
(The MC3D program can present t=
he 3D puzzle in a
surprisingly large variety of other manners, some of
even more compellin=
g version of "Box with the Lid Off"
configuration is the second one on the ConfigSelect
menu.)s=3D"q">
>I've attached a couple of renderings of a quick 3d<=
br>>model I made showing what the 8th face would look
>like. Is th=
is 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' project=
ion. If you could show the
mathematics of it, I might change my mind; bu=
t I
cannot imagine a realistic projection from 4-space to
3-space that w=
ould 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 whic=
h
are suggestive of their geometric relationship in
4-space. In so doi=
ng, 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
&=
gt;>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
>>transformation=
s on the 3D projections. Given that
>>restriction, the outer face&=
#39;s stickers would not end
>>up outside the others but rather wo=
uld be turned
>>inside-out and would intersect other parts of the
>>vi=
sible puzzle. It would just be a mess.
The intersection prob=
lem can be solved easily by
simply shifting the picture that results fro=
m the Out
face over so that it no longer falls atop the
renderings of the othe=
r faces.
The "inside-out" issue is a non-issue. All the
The issue is wheth=
er what we are seeing of one is the
side which faces the center of the 4D cube or the side
which faces o=
utward. 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 di=
fference 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 li=
es in the
same 3D space into which the 4D scene is projected.
The iss=
ue 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 th=
e viewer is never on _either_ side as
might be defined in 4-space.
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 Flat=
lander 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 a=
rises
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 depi=
ct 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 whi=
ch currently implements the rendering of the
In face would work with ver=
y little modification for
rendering the Out face. The trickiest part probably
lies in determin=
ing when, during animation, the
hyperplane of a face passes through the =
eyepoint,
necessitating changing which image (shifted or not)
the sti=
ckers 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
>>in=
visible face it gets large and almost completely
>>flat. It is act=
ually turning inside out.=20
This is not a correct explanation. What is actually
happe=
ning is that the hyperplane containing the face
is intersecting the eyep=
oint of the 4D-to-3D
projection. When the eyepoint is _in_ the hyperplan=
e
of the face, the rendering of that face does go flat.
The non-flat r=
enderings 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.
<=
br>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 give=
n 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 t=
his instant that
the program moves the presentation of the sticker from
being plotted relative to one of the virtual centers
to the other. Y=
ou can study this effect in detail by
using the mouse to control the ani=
mation, 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 defa=
ult
configuration, which presentation (without the shifted
Up face) i=
s 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 MC4=
D to
render the Out face in the manner analogous to "Box
with the Li=
d 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 i=
n the Out face, one's ability to
'see' which set of hypersti=
ckers lie on the same
hypercubie is quite similar to the cognition requi=
red
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 ot=
her at
close range. Instead, they both seem to be displaced
in the same dir=
ection 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 vers=
ion 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 tran=
sformation and there is not enough
of that in the Out faces as I have sh=
own 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.
------=_Part_128836_21966388.1187631108441--
From: David Vanderschel <DvdS@Austin.RR.com>
Date: 21 Aug 2007 01:03:05 -0500
Subject: Re: [MC4D] On Rendering the Elusive 8th "Face" of the 4D Puzzle
On Monday, August 20, "Roice Nelson"
>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.
From: "Roice Nelson" <roice@gravitation3d.com>
Date: Tue, 21 Aug 2007 17:19:23 -0500
Subject: Re: [MC4D] On Rendering the Elusive 8th "Face" of the 4D Puzzle
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Fair enough, it wouldn't hurt to have the "box with lid off" option (and the
additional option Alex suggested with altered sticker-shrink and face-grow
for the out face). Those like me who don't prefer it for their own personal
aesthetic reasons simply don't have to turn it on. Good points about the
artificial nature of the shrink settings too. As Einstein said, "Everything
should be made as simple as possible, but not simpler." I guess we just
have different opinions on where to draw the line in this case. Ultimately,
I think it is fair for the implementer to impose their own preference,
especially when the source code is available for modification to anyone who
would like to alter it otherwise.
There are some questions in my mind about the proper way to display a
removed lid. Do you flip it over, like you were opening a flap on a box? or
just cut the lid completely off and slide it over to the side as you
suggest? (in other words, does the representation of that face need to be
mirrored or not?) I can't think of a great reason to truly favor one over
the other, so perhaps it would need to be an option? Also, if you want to
open the lid like a flap, maybe there is a case to be made that you should
go ahead and flatten the entire box, not by a mixture of projection and
unfolding, but entirely by unfolding. The resulting representation would be
a 3D cross, and none of the faces would have the severe distortion we see on
some of the faces in the current projection (since they would all lie in one
3D hyperplane and no 4D->3D projection would even be required). Such a
representation would seem a little more logically pure to me than one that
mixes projection/unfolding. Maybe I've just circuitously given an argument
for sliding the lid over instead of unfolding it :)
On the inside/outside semantics discussion, yes, it is a dynamic animation
effect I was describing. When I watch this:
http://www.mathematik.com/4DCube/4DCubePovray.html
and keep my eye on one of the faces moving down through the center, it
appears (to me) as if the cube turns inside out. This was all I was
saying. To accept this language, you must interpret the scene as if it was
3D only, and look at the face like it had an inside and an outside (the
terms in this case are not being used for "describing the position of the
eyepoint relative to a hyperface"). I recognize this appearance effect is
also due to the simplicity of the face geometry for a 4D cube. If you
placed a more complicated shape like me inside one of the faces, the
animation of the projection wouldn't appear to turn me inside out and show
my guts. It would flatten me, then mirror me, as you've described. And the
possibility of the effect is sensitive to the movement of the viewpoint as
well. So anyway, I'll concede on this too. You've made a good argument
that your language is more precise here...
Roice
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t; option (and the additional option Alex suggested with altered sticker-sh=
rink and face-grow for the out face). Those like me who don't pre=
fer it for their own personal aesthetic reasons simply don't have to tu=
rn it on. Good points about the artificial nature of the shrink setti=
ngs too. As Einstein said, "Everything should be made as simple =
as possible, but not simpler." I guess we just have different op=
inions on where to draw the line in this case. Ultimately, I think it=
is fair for the implementer to impose their own preference, especially whe=
n the source code is available for modification to anyone who would like to=
alter it otherwise. =20
a removed lid. Do you flip it over, like you were opening a flap on =
a box? or just cut the lid completely off and slide it over to the side as =
you suggest? (in other words, does the representation of that face ne=
ed to be mirrored or not?) I can't think of a great reason to tru=
ly favor one over the other, so perhaps it would need to be an option? =
; Also, if you want to open the lid like a flap, maybe there is a case to b=
e made that you should go ahead and flatten the entire box, not by a mixtur=
e of projection and unfolding, but entirely by unfolding. The resulti=
ng representation would be a 3D cross, and none of the faces would have the=
severe distortion we see on some of the faces in the current projection (s=
ince they would all lie in one 3D hyperplane and no 4D->3D projection wo=
uld even be required). Such a representation would seem a little more=
logically pure to me than one that mixes projection/unfolding. Maybe=
I've just circuitously given an argument for sliding the lid over inst=
ead of unfolding it :)
on effect I was describing. When I watch this:
ppears (to me) as if the cube turns inside out. This w=
as all I was saying. To accept this language, you must interpret the =
scene as if it was 3D only, and look at the face like it had an inside and =
an outside (the terms in this case are not being used for "describing =
the position of the eyepoint relative to a hyperface"). I recogn=
ize this appearance effect is also due to the simplicity of the face g=
eometry for a 4D cube. If you placed a more complicated shape like me=
inside one of the faces, the animation of the projection wouldn't appe=
ar to turn me inside out and show my guts. It would flatten me, =
then mirror me, as you've described. And the possibility of the e=
ffect is sensitive to the movement of the viewpoint as well. So anywa=
y, I'll concede on this too. You've made a good argument that=
your language is more precise here...
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From: Melinda Green <melinda@superliminal.com>
Date: Wed, 22 Aug 2007 00:20:50 -0700
Subject: Re: [MC4D] On Rendering the Elusive 8th "Face" of the 4D Puzzle
Roice Nelson wrote:
>
> Fair enough, it wouldn't hurt to have the "box with lid off" option
> (and the additional option Alex suggested with altered sticker-shrink
> and face-grow for the out face). Those like me who don't prefer it
> for their own personal aesthetic reasons simply don't have to turn it
> on. Good points about the artificial nature of the shrink settings
> too. As Einstein said, "Everything should be made as simple as
> possible, but not simpler." I guess we just have different opinions
> on where to draw the line in this case. Ultimately, I think it is
> fair for the implementer to impose their own preference, especially
> when the source code is available for modification to anyone who would
> like to alter it otherwise.
>
Very well said, Roice. Einstein was talking about the art of modeling
and in this I agree completely. I'm also trying to balance my own sense
of modeling elegance with the desires of the users. In the end of course
I'll implement what I feel motivated to do, but as you point out, so can
anyone else. I'm not against implementing methods of showing the
invisible faces. (Note that we're not just talking about a single face
but possibly several.) I would just be much more interested in other
designs. My favorite idea would be to model a 3D mirror in 4-space and
to render the projected view of the back of the puzzle where it would
correctly project inside the 3D box that would be the edges of the
mirror cube. This would be analogous to solving a 3D cube while standing
in front of a mirror. There would only be the one 4D->3D projection and
that would preserve the elegance of the model for me.
Still another view that I'd be OK with would be to display two different
projections of the puzzle in separate views. The only difference is that
in each view the 4D eye point would view the puzzle from opposite
directions. It should be possible to perform all the normal operations
in either view.
-Melinda