Mathematical objects have dimensions associated with them. The temperature outside my house is measured as a single number or scalar. It is a one-dimensional quantity. This fact can be observed in how mercury thermometers are built: they are a long tube or line. Thermometers are never built as squares.

The position of house in a city is an example of a two dimensional quantity. It requires two numbers to specify and is therefore two-dimensional. This fact is obvious in that maps of cities are usually printed in a sheet of paper not a very thin strip of paper. The position of the house is expressed as a vector. This vector can be expressed as an ordered series of numbers of the form $(v_1, v_2,\ldots,v_n)$. Another way to represent the vector is just with the single symbol $v_i$. You tell me the value of $i$ and I go down the list and read off the component $v_i$.

Generalizing further, we can have matrices like the precision error covariance matrix I have been going on and on about all these months. This matrix can be represented by the symbol $m_{i,j}$. You now have to tell me two numbers, $i$ ad $j$, for me to read off the correct entry in the matrix.

We can keep playing this game forever. It is possible to invent mathematical quantities of the form $m_{i,j,k}$. Three “indices” need to be specified to read off an entry. You can think of this as a cube of numbers.

Precision error covariance matrices can also be generalized to precision error tensors. Instead of just asking how are the errors between two models correlated, we can ask how are the errors of three models correlated. We can have a cube of cross-correlations between the different model errors!