Message #232

From: Roice Nelson <>
Subject: More MagicCube5D
Date: Fri, 07 Apr 2006 00:54:43 -0500

Melinda, you are such a great motivator :) Thanks for your positive
response and excellent feedback. The other upbeat responses were great too,
and made me think there is enough interest from you guys to warrant further
work on this. Plus my good friend Charlie joined me in the effort, which
has made this even more fun to work on. I just posted the next incremental
version we’ve made (, and have some inline
responses and additional comments.

*Inline Comments:*

> * The wireframe representation does help one to see the structure though
> it may be harder to select stickers. BTW, the 4D stickers look *very*
> cool! With a wireframe design I’d personally prefer side-by-side
> cross-eyed stereo views otherwise you can only make out the structure
> while rotating it, but I know that is not for everyone.

I don’t think it would be too difficult to do a side-by-side stereo version,
and so I will hope to do that at some point for you. My personal stereo
preference is the LCD goggles (I got mine at I’ve
made Gravitation3D work with those and it’d be cool to do the same with
this. The bummer is that they don’t work with windowed programs, and making
it a usable full-screen mode program might be a bit of work.

> * The 3D drag rotations clearly applies all of one axis rotation before
> the other which makes it different to predict what will happen. I
> suggest that you use the quaternion code that I use in the Java version
> which makes it so that each drag increment always does what you
> naturally expect.

I definitely agree. The current mouse handling code was just pulled from
Gravitation3D. (I will add this suggestion to the to do list, but will give
it lower priority than the other items that need to be done first to make
this a workable puzzle.)

> * Instead of axis drop-downs for 5D rotations I would suggest a
> lower-triangular grid of 15 small rotate buttons.

Charlie had wanted these quick buttons as well, and I agree with you guys.
The drop down combos are a bit cumbersome. Charlie has already started
doing this. I think he is planning to have 10 buttons (xy, xz, xu, xv, yz,
yu, yv, zu, zv, uv) and to make it so left button clicks rotate one way and
right button clicks the other. It would be good to allow rotations using
only sticker-clicks too, but I think also having the buttons for convenience
would be nice.

> * I suggest that toggling of individual faces should operate on their
> rotated logical positions rather than on their colors. IOW, like MC4D
> does with the outside face.

I agreed very much with this idea too, and this is now implemented.

> * I understand that you wrote this mostly as a GUI feasibility study but
> I encourage you to give some serious consideration to working out a UI
> and implementation of a twist. I know that will be a huge task, and that
> once you do that you’ll be all but compelled to turn it into a workable
> puzzle and then compelled to solve it but it would still be way cool to
> see some twists regardless of whether you ever take it further.

I got a big smile reading the sentence about being compelled to do this
stuff. It’s so true and I wonder why that is sometimes.

I also wanted to let Remi know that even though there is no option to save
the projection settings yet, Charlie is also planning to work on that.

*Here are the changes done besides those already described above*:

Charlie optimized the hell out of my nD vector class. The Debug version
sped up by a factor of 12 and the release version by a factor of 4! Things
are much smoother looking now.

I added sticker highlighting, in the form of making the 4-cube lines thicker
when you mouse over stickers. If you hold down shift with a sticker
selected, all the connected 4-stickers of that 5-cubie will highlight. This
is really nice for gaining more intuition about the puzzle and how it is all
connected together (idea totally stolen from MC4D). At the very center of
the puzzle, a bunch of stickers overlap, but there is currently no good way
to highlight specific ones.

Proof-of-concept twisting: If you right or left click any sticker, you’ll
see face twisting. Unfortunately, this is not fully functional right now
because I have not yet developed exactly how to choose the axes for twisting
(more on that below). Right now, I just picked a couple of axes. However,
you do get to see twisting and the mess of a scrambled puzzle :) A
scrambled screenshot is at

Unfortunately, I broke allowing n^5 for n != 3. It should not be too bad to
fix at some point, but I wanted to get this other stuff working first
without worrying about that.

*User Interface for the Twists:*

I’d like to get the group’s input on twisting control. I’ve been thinking
about it off and on this past week and have what I think is a decent plan. But
the best approach is really not clear and feedback would be nice.

To start off (sorry if this is obvious given the recent set of emails), I
want to point out that faces of MagicCube5D twist in the manner of a 4-cube.
To extend the analogy of presented by MC4D, you remove a face, twist it any
way you like in the 4 dimensions of the face, then put it back. But unlike
3-twists, a 4-twist needs 2 axes to pick a unique rotation plane. This is
the main problem I’ve been having in trying to create a good user input

I only finally appreciated this past week one beauty of the MC4D interface,
which is how 4-view-rotations are done by ctrl-clicking. The beauty is that
you only have to select a single sticker to define the 2 axes of the
rotation. The second axis is determined implicitly because the rotation
will always move the clicked face to the "center" face position. I’m not
sure how the MC4D code does it, but a way to define faces is by their face
center offsets in space coordinates. In MC4D, for normalized values of
these, we have:

+-X face    (+-1,0,0,0)

+-Y face    (0,+-1,0,0)

+-Z face    (0,0,+-1,0)

+-U face    (0,0,0,+-1)

So ctrl-clicking always involves a rotation between one of the first 6 faces
and the –U face (and you can’t do a view rotation from the +U face). The
two axes defining the plane of rotation are the 2 non-zero values of the
face offsets, from one selected face and one implicit face. For example,
ctrl-clicking the Y face does a view rotation using the plane of the y and u

View rotations in MC5D can be done exactly the same if we want. In this
case, the face centers are at:

+-X face    (+-1,0,0,0,0)

+-Y face    (0,+-1,0,0,0)

+-Z face    (0,0,+-1,0,0)

+-U face    (0,0,0,+-1,0)

+-V face    (0,0,0,0,+-1)

So we could make ctrl-click always move the selected face into the –V face
for instance. The selected face and the implicit destination determine the
2 needed axes defining the rotation, allowing easy view rotations.

Ok, so what is the big deal then? Note that all stickers of a face share
the same face offset, so for a given 4-face, the sticker components we are
working with during a twist are the zero components of the face (e.g. for
the +X face, it is only the y,z,u,v components of stickers that are changing
when twisting) The problem is that for a 4-twist in MC5D, there is no good
way to set the implicit part of the rotation because the 4 relevant
component axes change from face to face. For the U and V faces, the
projection does end up making certain stickers look like they are in the
"center", which made made me first think we could define a twist by just
making a sticker click move the clicked sticker to the "center". But for
the X,Y, and Z faces, there is no "center" that is obvious to make the
implicit part of the rotation. Hope this is making sense.

So my best thought at this point is that we define 4-twists with 2 clicks. The
first click picks a sticker. The second click picks where you want to put
that sticker. The direction of the twist would just be determined by the
order in which you clicked 2 stickers, so there would be no left/right
clicking. Note that the 2 stickers selected have to be of the same type
(just like in the 3D cube, where you can’t twist an edge piece into a corner
piece). It would be no problem to help with this visually, by only
highlighting stickers that you can twist into after your first sticker is

I further propose for simplicity that we only allow 90 degree rotations (at
least for now). This means we’ll only allow clicking of "2-color" pieces to
determine twists. This is somewhat limiting, but it does help avoid
complexities. I’m being lazy in that I don’t want to tackle rotations in
non-coordinate planes, so the 90 degree limitation is nice for that, but
there are other issues too. For example, some corner pieces ("5-color
pieces") would require multiple twists in different planes to move from one
location on the face to the other. As an example, for the +V face, to move
the sticker (1,1,1,1) to (-1,-1,-1,-1) would require twists to first alter
two of the components, then a second set of twists to alter the other two
components, since a rotation works in a plane and can only alter 2
components at a time (or 3 at most if the rotation axis is not aligned with
coordinate planes). (Aside: This is related to the "orthogonal inertial
rotations" discussion from last year and this example makes it easier to see
how certain sets rotations have become commutative, not dependent on order,
unlike rotations in 3-space). Anyway, extending the possible face twist
rotations would be good future work.

Does this sound like a good plan/suggestion?

(1) 2 separate clicks determine a twist.

(2) limit twists to 90 degree rotations (for now).

Any better ideas for the interface?

Charlie and I did talk some about this path this evening. One improvement
he suggested was to make it easier to click the second sticker. For
example, once the first sticker is highlighted, all possible 2nd stickers
could immediately be highlighted, and maybe the closest one to the mouse at
that point could even be marked differently and immediately selectable.

As a final note on this topic, I thought it might be nice to provide an
array of buttons and a face selection combo as well, so that the same
behavior could be done without clicking if desired. It could actually be
the same array of buttons Charlie is doing for the 5D view rotations. It
would just be that if a face was also selected, the rotations only affected
the one face instead of the whole puzzle, making it a twist.

*The Main To Do List:*

I think there are 2 items left at a minimum to make this a puzzle people
could start solving. First is to properly finish up the twisting. Second
is to allow saving log files. Hopefully, this will be done in the next
couple weeks and the race for the first 3^5 will be on. Don’t know yet if
I’ll be compelled to join that or not ;)

Of course there are plenty of other improvements that can make it nicer.
Charlie and I want to release the source code for this like Melinda, Don,
and Jay did for MC4D. I’d like to clean up my part of it first though :)

Sorry this email turned out to be long, and I hope it wasn’t too laborious
to read.

Take care everybody,