Message #1440

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Social dream
Date: Tue, 22 Feb 2011 09:26:53 -0600

I very much like what Andrey specified for the twist definition in future
log files.

The only thing I questioned while reading was whether it is necessary to
store the rotation angles in such a way that they depend on the puzzle. So
rather than storing 12/3=4 for a 120 degree rotation on the simplex, why not
simply store 3? This would make those values more readable across puzzles,
since any order-3 rotation would be saved the same. Maybe I am missing
another need there though.

Roice

On Mon, Feb 21, 2011 at 1:11 AM, Andrey <andreyastrelin@yahoo.com> wrote:

> 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:
> - symmetries of 5-cell (15 axes)
> - symmetries of bitruncated 5-cell (70 axes?)
> - symmetries of 8-cell (40 axes)
> - symmetries of 24-cell (120 axes)
> - symmetries of bitruncated 24-cell (624 axes?)
> - symmetries of 120-cell (1320 axes)
> - symmetries of duoprism [n]x[k] (n+k axes?)
> - symmetries of duoprism [n]x[n] (2*n+n^2 axes?)
>
> 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:
> - direction of main axis (orthogonal to the cutting plane)
> - direction of 3D axis (it’s perpendicular to the main axis)
> - rotation angle
> - layers mask
>
> 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