what profiteth a man to flee his fate?

June 16, 2020 in binary tetr'al, Notations

let
B =\begin{pmatrix} \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix} \\ \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix} \end{pmatrix}

now let
B=\begin{pmatrix} II & I\\ III & IV\end{pmatrix}, i.e., let
II = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}, etc., so that
I, II, III, and IV (the “quadrants”) denote
two-by-twos of two-by-twos.

we’ll call the smallest matrices in sight “points”.
e.g.,
h =\begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix};
the quadrants are now two-by-two arrays of “points”.

on the graphical models i’ve been going on about all week
the quadrants are (respectively)
I ++
II -+
III —
IV +-
(as is familiar to every calculus student);
expanding on the same logic gives the “trit-code”
for the individual points. for example,
h = ++++
and so on.
one then simply translates the 16 trit-strings
++++, +++-, ++-+, … —-
into “hurwitz units” like
h = {{1+i+j+k}\over2};
voila.

oh, yeah. i forgot to say. the “points” are
matrices *mod 3* (so that 2 = -1). that is all.

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