# Two Dimensional Tape of Turing Machine

I was looking at this: http://www.cs.odu.edu/~toida/nerzic/390teched/tm/othertms.html

At the very beginning, they show a 2d tape and (I assume) it's 1d equivalent. However, I cannot figure out they got from the 2d case to the 1d case. I was hoping someone could elaborate.

The illustrative way to think about this is in diagonals: the cell $(1,1)$ is mapped to cell $1$ in the 1D tape. Then the second diagonal, namely cells $(1,2)$ and $(2,1)$ are mapped to cells $2,3$ respectively. Continuing in the same manner, you can map the entire 2D tape to the 1D tape. Of course, any movement of the head needs to be translated to the correct position in the 1D tape, so a single movement in the 2D tape may actually take several operations to simulate in the 1D tape.