Config. #2 is a method of presenting the puzzle which I saw a long time ago. It is fairly useful for a 3D observer. However, because of the lack of shrink with Configuration #2, a Flatlander looking at a 1D projection of this would be at a serious disadvantage for being able to infer the state of the puzzle or follow any twisting. A Flatlander operating in the 2D projection space needs the shrinking in order to be able see 'behind' (or 'around') the stickers. The program also generates 1D projections of the 2D image (which is "the model" from a Flatlander's perspective) in order to allow a user to simulate the behaviour of a Flatlander solving the 3D puzzle. This is analogous to the difficulty of us 3D beings trying to solve the 4D puzzle using a 2D rendering of the 3D model resulting from the 4D to 3D projecton. (It appears to me that solving the 3D puzzle under flatland restrictions is actually more difficult than solving the 4D puzzle based on the 2D rendering. The Flatlander must observe via a 1D projection of the 2D model resulting from the 3D to 2D projection.)
Config. #1 is what I set out to implement initially. It is still
like a box with the lid off with, but face and sticker shrinking have
been added. You can vary either of these parameters continuously with
the slider to observe how the shrink effect works.
If the centers coincide with Config. #1, the stickers of the upper
face seriously occlude our view of other stickers.
Config. #3 was actually a surprise for me. It turns out that, with
the right amounts of face and sticker shrink and an appropriate 3D
viewing distance, you can achieve a configuration similar to #1 but in
which the centers coincide and (almost) all the stickers are visible
looking between the spaces separating the stickers of the up face.
The one sticker you cannot see is that opposite the up-facing face
sticker. However, that can only be the one face sticker which is
complementary to that on top, so there is no real loss of information
due to the fact that the down-facing face sticker is occluded.
Similar to Config. #3 is Config. #7. Again, all stickers are
visible. In this case, the effect has been achieved with no face
shrink, so it is easier to perceive as a group the set of stickers
which belong to the same cubie. In this configuration, the effect is
achieved by giving up coordinate axis alignment for the orientation of
the puzzle in the 3D to 2D viewing transformation. The eyepoint is
only slightly off the the z-axis in this case.
(At one point, it seemed to me that there is a sense in which
Configuration #7 is 'physically' achievable (since sticker shrink is
no problem), while those with face-shrink are not. (Think of glass
cubies.) However, it has now also occurred to me that face-shrink can
be achieved physically as well. You have to put the stickers up on
'stilts'. Imagine a block for which one pair of opposing faces is a
1x1 square. The other dimension can be whatever length (like 1.3).
Now stick one of these blocks on each exposed cubie facelet (matching
up the 1x1 faces) and stick the sticker for that facelet on the
outside end of this extension block. Such a puzzle is still
physically workable (Not that I'd care to.), and it exhibits the
equivalent of face-shrink. (We did not actually shrink the faces, but
we grew the cube instead. The faces are only shrunk relative to the
larger cube which would circumscribe the extended facelets.))
With the possibility of placing the eyepoint anywhere in 3-space
(sufficiently far from cube), we get possibilities like the nearly
isometric view in Config. #4. It is not exactly isometric, as there
is still some perspective transformation to enhance the 3D effect. I
can use all these configurations for the purpose of working the cube,
but this one is really my favorite. I think I can work the cube
faster with the program in this configuration than I can work a real
physical puzzle. With this isometric sort of projection, I find that
the definition of turning direction as perceived in the 2D projection
to be quite natural. I must admit that in the box/lid type
configurations, I find the 3D turning direction to be what I am
expecting (with not even the d-face being much of a surprise).
It was not really in the 'plan', but the program can be configured to present a very normal looking puzzle. The only weird thing about Config. #5 is that when you twist a slice, then, during the animation, you can see between the 'cracks', into the the 'inside' the cube, and through to some in-facing cubies on the far side which are not visible unless an animation is in progress.
Since it is so easy to tumble and rotate one's view of the 3D
puzzle, I find that I can work the puzzle quite effectively using this
no-shrink presentation. Much as when I am working with a physical
puzzle, I find that I am tumbling the picture of the cube to see where
certain cubies are located on the far side and to see how they are
oriented. This does slow me down, so I prefer Config. #4. Again, the
definition of turning direction in terms of the 3D view seems much
more natural to me in this configuration.
The ability to make out-facing stickers semi-transparent sounded
like a good idea; but it did not result in a presentation that it made
it any easier to see what is going on. On the other hand, it does
produce some cool looking pictures; so I left the feature in. Try
tumbling and scrambling.
The Colored Tiles presentation turns out to be surprisingly similar
to Config. #2. This was not by design. The intent was to find a way
of coloring the tile corresponding to a cubie in such a way as to
intuitively suggest the orientation of the stickers on it. This was
the most obvious way of doing it which occurred to me.
The Text Display takes an entirely different approach to representing the orientation of a cubie. It adds identification of each cubie in the form of a name in the top row of characters. The characters of the name are the the characters which name the colors of the stickers on the cubie. The order corresponds to xyz relative to the axis of the direction in which the stickers face when the puzzle is initialized. These names are all unique. Under the character for each sticker color name, is a character which specifies the direction in which the sticker faces. As a further aid, the backgrounds behind the characters are colored with the color associated with the character. When a cubie is correctly placed, the colors in the top and bottom rows of its cell will match. (A scrambled view is shown here for contrast.)
A very significant aspect of the Text Display is that this
completely 'flat' method for representing the state of the puzzle
generalizes to higher dimensions in a straightforward manner. When I
am working with higher dimensional analogues of the puzzle, this
presentation method is actually easier for me to think about than the
methods based on rigorous dimension-reducing projections - mainly
because it makes it easier to think about where the cubies themselves
are and how they are oriented.
Here we have an isometric-type view with lots of face shrink. The
roving eye is placed right in the middle. With this arrangement, you,
acting in the role of a Flatlander, can use panning of the rover's
heading to look closely at the stickers of each face. You can use
sideways sliding motion of the rover so that the parallax effect
allows you to be able to see all the stickers in a group.
There is no particular practical value to this configuration. Both
the orthographic and the rover 1D projections are presented. The
geometry is such that what the rover sees is virtually the same
picture as the orthographic projection. In a sense, this is a sanity
check for the program, because the code which generates the 1D
orthographic projection is very different from the code which
generates the 1D perspective transformation from the roving eye point
of view. There is still enough of a perspective effect that the
pictures are not identical.
Config. #B took me a while to discover. It is an arrangement in which every sticker is visible to the roving eye. In this configuration the animation duration has been increased to 2 seconds to make it easier to follow what's happening in 1-space during the animation. There is so much face shrink and sticker shrink and the scale factor is so small that you cannot even see the individual stickers in the 2D display. (I left the face sticker labels on in this one so you can at least see where they are.) This configuration effectively prevents 'cheating' by a solver trying to emulate the behaviour of a Flatlander, since the 2D display is actually useless in this case anyway. The best way to get an even better feel for the set stickers in the 3x3 group for each face is to initiate a twist of the face and then take over the animation with the mouse. Then you can rock the face back and forth through small angles and get clear impression about how the stickers are situated with respect to one another.