Originally posted by boaby_phet
their is no mathematical backup to "my theory" as its not a mathematical question.
The maths is the part that im getting at, as the maths is all based on our own human concepts, and not what the concepts may actualy be .. if you get
me .
Ah, but dimensions ARE math. They ARE measurement. And they exist just fine as pure numbers without any physical correlation.
You know how to play tic-tac-toe, right? The game is played on a 3x3 grid and the player who gets three in a row wins.
Let's take that board and describe it as coordinates... 0-2 on the x axis, and 0-2 on the y axis.
So a win, 3 in a row, along a row might be (0,0) (1,0) (2,0) or (0,1) (1,1) (2,1) or (0,2) (1,2) (2,2), right? We're incrementing the X value while
keeping the Y value constant. These are the three horizontal ways to win at tic-tac-toe.
The three vertical wins would be (0,0) (0,1) (0,2) or (1,0) (1,1) (1,2) or (2,0) (2,1) (2,2). Here keeping the X value fixed while incrementing the Y
value.
And the two diagonal wins are (0,0) (1,1) (2,2) or (0,2) (1,1) (2,0) either incrementing both X and Y or incrementing one while decrementing the
other.
OK. Fine. Good. Obvious, right?
Can you do that with three dimensional tic-tac-toe? Let's use a 4x4x4 cube, otherwise, same rules apply.
So a one win might be (0,0,0) (1,0,0) (2,0,0) (3,0,0) another win might be (0,0,0) (0,1,0) (0,2,0) (0,3,0) or a third way would be (0,0,0) (0,0,1)
(0,0,2) (0,0,3), and that's not even considering a diagonal like (0,0,0) (1,1,1) (2,2,2) (3,3,3) (or any of the diagonals along a face)
We've easily extrapolated the rules of 3-D tic-tac-toe based off our knowledge of the 2-D game. The rules for "what's in a row?" still apply,
though in more directions. Diagonals can get a bit more confusing, but we can deduce what they should be. And since we live in a 3-D reality, we can
easily picture a cube and how these things relate.
Now... take that and push it into 4 dimensions. You don't need to be able to picture a hypercube to figure out that a 5x5x5x5 game board will have
winning moves on any one dimension that's variable while the rest remain constant, or any diagonal that's moving in 2, 3 or all 4 variables.
Does a 4th dimension exist? Who knows? But I can mathematically describe some of its attributes and play a game of 4-D tic-tac-toe with you simply by
expanding the rules of what we DO know in one more dimension.