back-of-the-envelope calculations

June 17, 2020 in binary tetr'al, Finite Geometry

here’s A-four-hat three ways.

but really *four* ways; like we agreed upthread,
the trit-string version is inherent in the very
positioning of the table entries, to wit.
consider
-+___++
– -___+-
—the “quadrants” of beginning algebra…
iterate: each of the three “versions” of our group
has each of its entries in one of the (16) “positions”
-+-+___-+++___++-+___++++
-+- -___-++-___++- -___+++-
– – -+___- -++___+- -+___+-++
– – – -___- -+-___+- – -___+-+-
; now just remember that, e.g.
“+- -+” in this context means
(1-i-j+k)/2—a “hurwitz unit”
in —the quaternions
(or, if you prefer… as i do… in
—the *integral*
quaternions). where was i.

the matrix notation is “mod 3”;
the generators-and-relators version
requires one to work with “relators”
like “hi = jh”—(this is, like, the
very *textbook example* of a
“semi-direct product”, if you want
my opinion… anyhow, this is quite
close to the actual way *i* actually
got it if i can be said to have it now]).

finally, the “permutation notation” version
is very much the easiest to work with (and you
should learn right away how to work with these
if you haven’t already; i had to be dragged
slowly and painfully into accepting this stuff
but maybe you’ll be one of the lucky one in
a million): one readily sees which elements
have order six, for example.

anyhow, this is one of the coolest things i ever
put on one sheet of paper or so it seems to me now.

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