2 by 2 by 2 Tensors
====================

We now study the case of :math:`2\times2\times2` tensors over both the
real and complex numbers. Our purpose is to give visualizations of
theoretical work of others. 

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

We follow the work of Vin De Silva and
Lek-Heng Lim in [DES-LIM2008]_, although several of these results
appeared earlier in [KRUSKA1989]_.  They show that all
:math:`2\times2\times2` tensors are in :math:`S_3` but both
:math:`S_2` and :math:`S_3\setminus S_2` have positive volume. The
tensors in :math:`S_3\setminus S_2` can be divided into the cases
:math:`D_3` and :math:`G_3`.

Degenerate Separation Rank 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The class :math:`D_3` consists of tensors not in :math:`S_2` that are
a limit of tensors in :math:`S_2`. The existence of tensors in
:math:`D_3` shows that :math:`S_2` is not a closed set. A
representative element of :math:`D_3` is 

.. math::
   D_3 = [1,0]\otimes[1,0]\otimes[1,0]
   + [1,0] \otimes [0,1]\otimes [0,1] 
   + [0,1]\otimes [1,0]\otimes [0,1]
   \,,

which can also be viewed as

.. math::
   D_3 = \left[\begin{array}{cc}1&0\\0&0\end{array}\right]\otimes[1,0]
   + \left[\begin{array}{cc}0&1\\1&0\end{array}\right]\otimes [0,1] 
   \,.

   
Using the plane defined by the three tensors

.. math::
   T_1&= [1,0]\otimes [1,0]\otimes [1,0]\,, \\
   T_2&=[1,0] \otimes [0,1]\otimes [0,1]\,,
   \quad\text{and}\\
   T_3&=[0,1]\otimes [1,0]\otimes [0,1]\,,

the point :math:`D_3` is in their center. Since these three tensors
are in :math:`S_1`, any tensor on the line connecting any two of them
is in :math:`S_2`. These lines help us identify the three tensors in
the visualization, and thus help approximately locate the :math:`D_3`
tensor. 

Plotting :math:`S_2` in this way, we obtain the visualization

.. image:: D3.*

[data file `<2x2x2pr2D3.dat>`_]
Notice that we obtain black points near :math:`D_3`, and this lack of
a void validates the result that there are tensors in :math:`S_2` near
:math:`D_3`. It does not allow us to distinguish points in :math:`D_3`
from points in :math:`S_2`.

.. _G3-section-label:

General Separation Rank 3
^^^^^^^^^^^^^^^^^^^^^^^^^

The class :math:`G_3` consists of tensors not in :math:`S_2` that are
not a limit of tensors in :math:`S_2`. A representative element of
:math:`G_3` is

.. math::
   G_3 = [1,1]\otimes[0,1]\otimes[0,1] 
   + [1,-1]\otimes[1,0]\otimes[1,0] 
   + [0,1]\otimes [1,1]\otimes [1,-1]
   \,,

which can also be viewed as

.. math::
   G_3 = \left[\begin{array}{cc}1&0\\0&1\end{array}\right]\otimes[1,0]
   + \left[\begin{array}{cc}0&1\\-1&0\end{array}\right]\otimes [0,1] 
   \,.

Using the plane defined by the three tensors

.. math::
   T_1&=[1,1]\otimes[0,1]\otimes[0,1]
   \,,\\
   T_2&=[1,-1]\otimes[1,0]\otimes[1,0] 
   \,,\quad\text{and}\\
   T_3&=[0,1]\otimes [1,1]\otimes [1,-1]
   \,,

the point :math:`G_3` is in their center. Since these three tensors
are in :math:`S_1`, any tensor on the line connecting any two of them
is in :math:`S_2`. These lines help us identify the three tensors in
the visualization, and thus help approximately locate the :math:`G_3`
tensor. 

Plotting :math:`S_2` in this way, we obtain the visualization

.. image:: G3.*

[data file `<2x2x2pr2G3.dat>`_]
We obtain black points everywhere outside of the
triangle, and part of the region inside of the triangle. However, near
:math:`G_3` there is a distinct void, indicating that this tensor
cannot be approached as a limit of tensors in :math:`S_2`.

We also note that we used only one-third the points to make this
visualization as compared to the :math:`D_3` visualization, and yet
obtained more black points. This indicates that the ALS routine had
more difficulty on the tensors in the :math:`D_3` plane, perhaps
because they are only limit points of :math:`S_2`.

Random Example
^^^^^^^^^^^^^^^

Randomly choosing  :math:`T_1`,  :math:`T_2`, and :math:`T_3`, we
obtained a visualization

.. image:: 2x2x2pr2random.*

[data file `<2x2x2pr2random.dat>`_]
In this visualization we see both black regions and voids, suggesting
that it is relatively easy to hit both types of behaviors.

Visualizations over the Complex numbers
----------------------------------------

Over the complex numbers all :math:`2\times2\times2` tensors are in
:math:`S_2`. We illustrate this by considering the :math:`G_3` tensor
defined above. Using the complex line defined by the two tensors

.. math::
   T_1&=[1,1]\otimes[0,1]\otimes[0,1]+[1,-1]\otimes[1,0]\otimes[1,0] 
   \,,\quad\text{and}\\
   T_2&=[0,1]\otimes [1,1]\otimes [1,-1]
   \,,

the point :math:`G_3` is the (real) midpoint of the line.  Plotting
:math:`S_2` in this way on :math:`[-1,2]\times[-i,i]`, we obtain the
visualization

.. image:: G3C.*

[data file `<2x2x2pr2G3C.dat>`_]
We obtain black points everywhere, indicating that :math:`G_3` is in
:math:`S_2` over the complex numbers (prehaps as a limit point). The
ALS routine had more trouble finding black points in the left half of the
visualization, which is closer to :math:`T_2`. This is
counterintuitive, since :math:`T_2` is only rank 1 whereas :math:`T_1`
is rank 2.