Message #1454

From: Andrew Gould <agould@uwm.edu>
Subject: RE: [MC4D] Social dream
Date: Fri, 25 Feb 2011 17:09:29 -0600

I’m seeing 46 rotational planes for the tesseract (6 planes for 90-degree
rotations, 24 for 180-degree, 16 for 120-degree), now I’m trying to
translate into Andrey’s 40 axes–I got confused.

I’m keeping my so-called "2D twists" in mind. They seem to use the same
axes/rotational planes that are already used.

http://groups.yahoo.com/group/4D_Cubing/photos/album/1774759718/pic/list

The deal in 5D is that you can twist 4D slices, 3D slices, or 2D
slices…but again I’m seeing that they all get twisted about
axes/rotational planes that are already used for twisting 4D slices.

–

Andy

From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On Behalf
Of Andrey
Sent: Monday, February 21, 2011 1:12
To: 4D_Cubing@yahoogroups.com
Subject: Re: [MC4D] Social dream

Hi all,
About piece finding and percentage of solving, there is a problem in the
most puzzles: if puzzle has no unmoving center of the cell, you cannot say
what is the proper color for this cell. So there is no way to say "what is
the place for this piece", "where this piece should go" and "how many
pieces/stickers are in their places". Program can "guess" colors for faces
(by majority of colors), but is may lead to strange situatons: you solve
something like simplex, get one twisted corner piece, and can’t solve it
without reassigning colors to cells. But the program keeps telling you that
you are going from the correct solution, and number of solved pieces is
decreasing (until you get enough stickers in their new cells)
So I used alternative approach to piece-finding in MC7D. It works, but it’s
much less intuitive than "click in piece and see where it goes".

As for the log files, my first idea is the following.
Let we have some puzzle based on uniform polytope. Now we don’t consider
shape-transformers, so form of puzzle remains the same after each twist.
Cutting hyperplanes are ortogonal to some axes (going through the center of
the puzzle). These axes may contain centers of cells, 2D faces, middles of
edges and vertices of the puzzle, but it doesn’t matter. What is important,
the set of axes for the puzzle is a subset of symmetry/rotation axes of the
puzzle body. There are not many symmetry sets in 4D:

We can enumerate axes for every case in some agreed order.

For every axis we have number and positions of cutting planes. They define
number and connections of stickers, but almost don’t influence rotation
descriptions. We need to define mask of layers for the twist, and for that
we must know only one thing - what is the maximal number M of layers there
is from the center to the surface (for all axes) - not including central
layer. For example, for 3^4 tesseract it will be 1, and for 3^4 with
diagonal cuttings x±y±z±w={-2,0,2} (for x,y,z,w={-1,1}) it will be 2 (there
are 3 layers orthogonal to (0,0,0,1) and 4 layers orthogonal to (1,1,1,1) ).
If such number M is defined, we enumerate layers by each axis so that
central layer (if it exists) has number M. If thare is no central layer, M
is skipped. For the example above layers parallel to cells will have numbers
1,2,3 and layers in 0D direction (orthogonal to (1,1,1,1)) will be 0,1,3,4.
This way we define the mask of the twist.

To define the twist we need two more things - direction of 3D axis of twist
and twisting angle. My guess is that 3D axis is always one of symmetry axes
that is perpendicular to the layer axis, so it is enumerated in our set. The
problem will be with the angle.

Some puzzles (such as 3x3x4x4, or alternated puzzles like snub 24-cell) may
have restricted set of twists enabled by the axes set. So we can’t just
write "turn on smallest possible angle clockwise", we need to define angle
explicitly. I suggest to select some number D for every axes set, that will
be common divisor for all twist orders (not necessarily the least) - 12 for
simplex, hypercube or 24-cell, 60 for 120-cell, [n,k,2] for duoprism,
include it in the set description and write angles assuming D=360 deg.
So every twist will be described by 4 numbers:

For example, 120-deg rotation of 3rd layer of 4^4 may have the form
A1:A2:4:8, where A1=(1,0,0,0) and A2=(0,1,-1,1) (4=12/3, 8=2^3)

I don’t want to include complete stickers mask in the description (stickers
order and description may be different for different implementations), and
don’t see good way to define the starting situation. Of course, we can find
the description for every sticker It may include 4 cutting planes that give
its vertex, and a mask that gives position of the sticker relative to these
planes. But definition of the puzzle state is such terms will be terrible. I
afraid that we will be able to keep only possible positions defined by the
twists sequence.

All the other things - body shape, coloring, position of cutting planes,
mask/description of stickers, not included in puzzle (and the shape of the
remaining stickers), bandaging and so on - it will be in the log header and
its interpretation (= puzzle description) is another story.

Andrey