Understanding 3+ spatial dmensions?

[ughaibu]

In the ticktactoe arrangement there is a square of sixteen 4x4 squares, each of these represents a slice of a 4x4x4 cube, so those on the outside represent faces of cubes and for those in the corners there are cubes on the diagonals as well as cubes along the sides, whereas those on the outside but not in the corners are faces of cubes crossing the entire arrangement but are internal slices of the cubes along the edge.

[Qfwfq]

That's how I was reckoning the 4x4x4x4 lattice, but I don't see the cubes along the diagonal that you mention as really being significant. Now that I've figured it through, I see where the eight 3-cubes, which form the boundary of the 4-cube, are hidden. They are 4 + 4 of the columns and 4 + 4 of the rows, not so easy to visualize B).

 

This attachment took a bit more doing, it's a topological breakdown of the 4-cube, which Salvador Dalì used as a crucifiction symbolic of the fourth dimension. For clarity, I gave 3-D rendering, using thicker black lines as visible and thinner grey for hidden ones, and I gave two examples of vertices that are the same actual vertex of the true 4-cube. One is the group of 4 red dots (the 4 lines directly between these and the blue dot are also the same actual line). Likewise holds for the three yellow and one green dot.

[Qfwfq]

Here, four instances of the 4-D tictactoe, each has two of the boundary 3-cubes shaded, one in green and one in blue.

 

errata corrige: In the previous post I should have said, it's the boundary of a 4-cube.