# 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:

- 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