# Message #1436

From: Eduard Baumann <baumann@mcnet.ch>

Subject: Re: [MC4D] Social dream

Date: Mon, 21 Feb 2011 13:47:00 +0100

Fortunatly the 120-cell HAS unmoving center of the cells.

Ed

—– Original Message —–

From: Andrey

To: 4D_Cubing@yahoogroups.com

Sent: Monday, February 21, 2011 8:11 AM

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:

- 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