The problem with the physicist's definition is that the larger the N the less the geometrical interpretation makes sense. For 1, 2, and even 3-dimensional tenso

The problem with the physicist's definition is that the larger the N the less the geometrical interpretation makes sense. For 1, 2, and even 3-dimensional tensors there is some connection to geometry, but eventually it loses all meaning. Physicist has to give up and "admit" that an N-dimensional tensor really just is a collection of N-1-dimensional tensors.