What is the dimension of chess board ? ( in mathematical model)

Let's think about this question. After all, we believe that a chessboard is 2D, right?
But the fact that the diagonal "distances" on this board are "equal" to the horizontal ones tells us that there are gaps in this board if we imagine its abstract model. If our leg is equal to the hypotenuse, then we skip some segments in our hypotenuse, right?

Going further, we can take the idea from the next video, about coefficients in fractals.

And to assume that the dimension of the chess (in the mathematical model) board is less than two.

What do you think about it ?