Glotmorf on 29 Dec 2003 05:52:55 -0000


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Re: [spoon-discuss] Junk


On 29 Dec 2003 at 0:13, Zarpint Jeremy Cook wrote:

> 
> Sorry, but my three-dimensional brain is not seeing this. Questions:
> 
> Aren't you just depicting some of the boards rotated? It's not the
> topology that your twisting changes, but the way you represent the boards
> on the paper. Isn't this just really a normal 4x4x4 cube?

One can depict a 4x4x4 cube this way, but the intent was a 
toroid...or, if you prefer, a cube that wraps its bottom side 
around to be above its top side.

Imagine you've got a standard 4x4x4 cube, made of four stacked 
4x4 boards.  Each of these boards has a corner connected to an 
up-and-down post.  Now rotate the boards around the post so 
that none is above the other, and they're in a spiral.  Now 
imagine that the next board in the spiral after the bottom one 
is actually the top one.  Now look at it from the top.

> Then you're adding more lines connecting the points, really. I can't
> visualize this at all. How many neighbors does each point have?
> And how is this hyper-anything? It's 64 points in a 3-D grid.

In the basic 4x4x4 torus, a non-edge point on any 4x4 board 
has eight neighbors on its own board, plus nine neighbors on 
the board above it and nine more on the board below it, for a 
total of 26 neighbors.  This is true for any non-edge point in 
any 4x4 board; in a conventional 4x4x4 cube, non-edge points 
in the top or bottom boards would only have 17 neighbors.  
Similarly, each edge point regardless of board has 17 
neighbors, and each corner point has 11 neighbors.

Now, intersect that one flattened board with two other 
perpendicular boards, and allow rotation along the two new 
axes, and the non-edge points each get 36 additional 
neighbors, the edge points get 24 more, and the corner points 
get 16 more.

> I also question "four in a row with three moves."
> Say we have a 3x3 torus (2-D). If I play at (0,0), (0,1) and (0,2) does
> this really count as "aleph-null in a row"? It's the same issue here...
> I am tempted to say each point of "n points in a row" has to have a different
> coordinate set than the others, or they're not n points.

True.  It's a matter of semantics.  What I meant is that, 
starting from one point, it's possible to take three steps in 
the direction of theoretically connected points and wind up in 
the starting position.  Which means if the goal is to get 
"four in a row", you can't do it that way.

						Glotmorf

> On Mon, 29 Dec 2003, Glotmorf wrote:
> 
> > I tried this once with a 4x4 game, except I made it hyper-
> > toroidal...
> >
> > . . . . | . . . .
> > . . . . | . . . .
> > . . . . | . . . .
> > . . . . | . . . .
> > _ _ _ _   _ _ _ _
> > . . . . | . . . .
> > . . . . | . . . .
> > . . . . | . . . .
> > . . . . | . . . .
> >
> > The idea is that the boards are rotationally above one
> > another, with wraparound.  The upper left corner of the upper
> > left board is above the upper right corner of the upper right
> > board, which is above the lower right corner of the lower
> > right board, which is above the lower left corner of the lower
> > left board...which is above the upper left corner of the upper
> > left board.
> >
> > Then I got carried away.  Imagine that the above board is
> > intersected through its axes by two identical and
> > perpendicular boards...that there exists an X plane, a Y plane
> > and a Z plane.  This makes for a 4x4 board in three toruses at
> > the same time, rotating through a different axis.
> >
> > Problem: Rotate through enough axes, and it's possible to make
> > four-in-a-row in three moves...
> >
> > 						Glotmorf
> 
> -- 
> Zarpint            "All thy toiling only breeds new dreams, new dreams;
> Jeremy Cook         there is no truth saving in thine own heart."
> mcfoufou@xxxxxxxxx       --W.B. Yeats, The Song of the Happy Shepherd
> grep -r kibo /     "Movements are the problem, not the answer to problems."
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