Rabbits Five

Rank 1 Tensor

\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}

Rank 2 Tensor

\begin{bmatrix} \begin{bmatrix} v_{11} \\ v_{12} \\ v_{13} \end{bmatrix} \\ \\ \begin{bmatrix} v_{21} \\ v_{22} \\ v_{23} \end{bmatrix} \\ \\ \begin{bmatrix} v_{31} \\ v_{32} \\ v_{33} \end{bmatrix} \end{bmatrix}

Rank 3 Tensor

\begin{bmatrix} \begin{bmatrix} \begin{bmatrix} v_{111} \\ v_{112} \\ v_{113} \end{bmatrix} \\ \\ \begin{bmatrix} v_{121} \\ v_{122} \\ v_{123} \end{bmatrix} \\ \\ \begin{bmatrix} v_{131} \\ v_{132} \\ v_{133} \end{bmatrix} \end{bmatrix} \\ \\ \begin{bmatrix} \begin{bmatrix} v_{211} \\ v_{212} \\ v_{213} \end{bmatrix} \\ \\ \begin{bmatrix} v_{221} \\ v_{222} \\ v_{223} \end{bmatrix} \\ \\ \begin{bmatrix} v_{231} \\ v_{232} \\ v_{233} \end{bmatrix} \end{bmatrix} \\ \\ \begin{bmatrix} \begin{bmatrix} v_{311} \\ v_{312} \\ v_{313} \end{bmatrix} \\ \\ \begin{bmatrix} v_{321} \\ v_{322} \\ v_{323} \end{bmatrix} \\ \\ \begin{bmatrix} v_{331} \\ v_{332} \\ v_{333} \end{bmatrix} \end{bmatrix} \end{bmatrix}

Do you see how we could keep going to higher and higher ranks, using only up-and-down?

We could have done the samething going only left-and-right.

The choices are arbitrary–whatever math you do matches those choices.

We might devise a system where each index is given one of two labels and whether we have it going up-and-down or left-and-right depends on which label it has.

We might do this with the two labels being Covariant and Contravariant.