3 by 3 by 4 Tensors
====================

We now study the case of :math:`3\times 3 \times 4` tensors.

Visualizations over the Real numbers
------------------------------------

Since the maximum rank of :math:`3\times 3 \times 3` tensors is 5, the
maximum rank of :math:`3\times 3 \times 4` tensors is at least 5 and
at most 8.
[C-B-L-C2009]_ (table 1) states (as know previously) that the 
typical rank is 5.  

A Random Example
^^^^^^^^^^^^^^^^^

We choose :math:`T_1`, :math:`T_2`, and :math:`T_3` randomly in
:math:`S_3`. Since these three tensors are in :math:`S_3`, any tensor
on the line connecting any two of them is in :math:`S_6`, and a tensor
in the plane is generically in :math:`S_9`,  which we is large
enough to capture general :math:`3\times 3 \times 4` tensors.


We first produce a visualization of :math:`S_5` on
:math:`[-1,2]\times[-1,2]`, and obtain

.. image:: 3x3x4pr5random.*

[data file `<3x3x4pr5random.dat>`_]
Next, using the same set of random :math:`T_1`, :math:`T_2`, and
:math:`T_3`, we produce a visualization of :math:`S_6`, and obtain

.. image:: 3x3x4pr6random.*

[data file `<3x3x4pr6random.dat>`_]
Initially the upper-right corner was generally lighter, so we added
more points on :math:`[0,2]\times[0,2]`, which explains the greater
density in this region.

Based on these visualizations, we would conjecture that the maximum
rank of :math:`3\times 3 \times 4` tensors is probably 5, and at most 6. 
As noted above, [C-B-L-C2009]_ says it is known (proven) to be 5.