why no one-coordinate diagonal plane

[sonata_green]

A plane has two degrees of freedom. Selecting a 2D plane from a 3D space requires choosing what to hold constant. We can pin a single coordinate, giving an orthogonal plane; we can pin the sum or difference of two coordinates, giving a wide staircase; or we can pin the sum/difference of all three coordinates (±x±y±z), giving a qbert staircase. So the answer is that there is a one-coordinate "diagonal", but not a zero-coordinate plane: the sum of zero variables is already constant, and imposing the requirement that a constant must be constant doesn't give us an additional restriction, meaning it fails to reduce the degrees of freedom.