Message #549

From: David Vanderschel <>
Subject: Re: [MC4D] Something interesting and strange about permutations
Date: Mon, 11 Aug 2008 17:16:57 -0500

On Sunday, August 10, "Melinda Green" <> wrote:
>MC2D may not be a proper analogy but it is not a
>misnomer because a square is definitely a 2D cube.

I would agree that, in the context of multidimensional
puzzles, a square for MC2D is every bit as much a cube
(2-cube, in this case) as is the tesseract (4-cube) in
the MC4D case. So, in saying "misnomer", with
emphasis on "cube", I think Nelson got a bit carried
away. However, and more importantly, he correctly
observed that, unless you allow reflecting twists, the
‘puzzle’ does not twist at all. (So Nelson’s
"misnomer" could probably be taken appropriately with
respect to "magic".)

The reference to "equivalent 2D puzzle" regarding MC2D
on the Superliminal MC4D page is misleading. In the
description for MC2D, it needs to be made clear that,
to permit it to be at all interesting, mirroring
twists are allowed and that this does violate the
analogy with MC4D and the regular 3D puzzle. Melinda,
if you fix the description, this would also be a good
point to mention that MC3D also allows the mirroring
twists extension in the context of the 3D puzzle. In
either case (and in higher dimensions as well), it
could be argued that reflections can be thought of as
being achieved by embedding the n-dimensional puzzle
in (n+1)-dimensional space, so that reflection can be
achieved by performing a 180 degree rotation for which
the extra spatial axis is in the plane of rotation.

>Regarding rotations, I really don’t think that it is
>helpful to try to think of N-D rotations as involving
>rotation axes. The fact that 3D rotations are easily
>visualized as happening *about* an axis is really
>just a quirk of three dimensions. A better way to
>think of rotations is that they always occur *within*
>a 2D plane.

I don’t think that there is an important distinction
to be made here. The (n-2)-dimensional subspace
orthogonal to the plane of rotation is also called the
"fixed space" for the rotation. In 3D, it is just a
line. In 4D, the fixed space for a rotation is a 2D
subspace. Defining a rotation in terms of its fixed
space or its plane of rotation are essentially
equivalent, since the two subspaces are always related
by orthogonality. When Nelson wrote, "The axis of
rotation of a figure in N-space will always be
composed of a segment of N - 2 dimensions.", his "axis
of rotation" would more appropriately be referred to
as the "fixed space for the rotation" and his
"segment" would more appropriately be referred to as
"subspace" or "hyperplane".

>In other words, while an object moves under the
>influence of any single rotation in any number of
>dimensions, any point of that object will move in a
>circular arc within a single 2D plane. In 3
>dimensions there will be a single rotation axis that
>cuts through the centers of rotation of all those
>parallel planes but in 4 dimensions there can be more
>than one axis that does that, so try to forget about
>axes and just look for the planes of rotation.

I think this is poor advice. It is often very useful
to be able think about a rotation in terms of its
fixed space. Indeed, the reasoning for what to use
for a rotation often involves thinking about the
aspects of state that you do NOT want to change.
I.e., it is a constraint on the fixed space that
may motivate the choice of rotation plane.

>Now as to the legitimacy of classifying MC2D with the
>other puzzles, it all depends upon how we want to
>define these puzzles. We can choose to allow mirror
>operations or not, and we can allow twists involving
>higher dimensions or not. I don’t have a strong
>opinion on the best choice, and I’m perfectly happy
>if there does appear to be a best choice which does
>not allow a valid 2D puzzle.

There does not have to be just one choice; and, in the
presence of multiple possibilities, there is no need
to evaluate any choice as "best". E.g., the 3D puzzle
is interesting whether or not you allow reflecting
twists. Why try to exclude a choice? Regarding
mirroring, only one choice makes sense in the 2D case;
but, for a higher dimension, in addition to the
non-mirroring twists normally considered, one can also
consider a variation which permits reflections. (You
could argue that making this sort of choice is
analogous to whether or not one regards the
orientation of face stickers to be relevant for the 3D
puzzle. By making a different choice, you create a
somewhat different puzzle.)

>For me, the most interesting thing about MC2D as
>implemented is that one can easily sketch the entire
>state graph for the puzzle (8 states!) and thereby
>begin to get an idea of what the topology of other
>similar puzzles might look like.

Where does this "8 states!" come from? Orientation of
a corner 2-cubie depends only on its position.
However, the 4 corner 2-cubies can be permuted in all
24 different ways. In what sense can 3 different
permutations all be regarded as the same state? I had
pointed out the apparent discrepancy in Melinda’s
analysis in somewhat greater detail a couple years
Since Melinda is now repeating the dubious claim, I
wonder if she ever saw my old message replying to
hers. In that old message, I also touched on some of
the other issues which have arisen again in the
current discussion as well as some other issues which
have not rearisen (yet).

David V.