### back-of-the-envelope calculations

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
-+___++
– -___+-
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 $\Bbb{H}$—the quaternions
(or, if you prefer… as i do… in
$\Bbb{Z}[h,i,j,k]$—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
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.

1. the RSV denotes not “revised standard version” but “rosetta stone five”.

1. 1 allegedly “classic” | the livingston review

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Vlorbik On Math Ed ('07—'09)
(a good place to start!)