## Archive for June, 2020

### the same old thing

here’s yet another version of this thing
(which, like all its predecessors, is unreadable
in the version posted here; don’t get me started):
four “representations” of the 24-element group
called variously “the binary tetrahedral group”,
“the hurwitz units”, SL_2(F_3), and A-four-hat
(among other things).

new here are the yin-yangs on the g-coset points
of the generators-and-relators version at upper left.
and nothing else. still, it’s an entire course
in group theory summarized on one page and it cost me
quite a bit of effort figuring out what to put where.
maybe some of the bits-into-graphics pages of my book
were as much trouble as this but i doubt it. anyhow,
i’ve long since given up on ever again getting anybody
to read *that* damn thing so i’m stuck with this until
i find a better obsession. read the ten page news.

### i do not like it on the net. i would not do it on a bet.

i tried posting this 6 months ago
but wordpress ate it. here it is now.

### they say it’s one d-mn thing after another

… but really it’s the *same* d-mn thing,
over and over.

i can now say for sure that some of the drawings
in these old Math Ed Zines scattered all around
the place could be adapted for coloring books.

the B&W drawing underlying both colorations
is from a zine of about ten years ago and shows
graphically (“lectures without words”) that
$P^2(\Bbb{F}_3) = P^2(\Bbb{F}_3)^{*}$. you’d see this code
on the actual display if i could get a decent
shot. a poor workman blames his tools. and i
intend to make the most of it.

### the abyss looks back into you

here’s looking, not at euclid, but rather at this
nameless-by-me-i-think mac book. behind me a few
of the many shelves here at the livingston libe.
madeline’s encyclopedia was here before my arrival
(all those years ago). most of the rest was brought
by me or acquired since. the hugely-cool coffee-table
book about public libraries (lavish production;
photos of libes) is next to the top row
under that, typography & dictionaries…
this is way too slow… the “connect four”
game… fast forwarding…

there’s a whole martin gardner *section* for
hecksake including two issues of th’ _notices_;
under his _last_recreations_ is a bunch of stuff
by the great knuth (that made up $\TeX$—thanks, don!).
the jacket for th’ _annotated_alice_ (the caroll section is
in my sight from here but not in this shot) also
deserves mention. i loved that book even before
i knew who gardner *was*.

and a few of my many dollar-coins. there’s a map
more or less of course—part of a streets-of-columbus
folded-up-for-gloveboxen thingum in this case
(one of many… want one?).

the enormous _handbook_of_physics_ at lower right
was a cursemas gift from public-school days.
too awkward to be good for much actual *reading*
but nonetheless known by me to have an impressive

okay. enough stalling around.

announcing _the_ten_page_ten_yearly_ (2020).
this i vow.
if it comes out at all
(i) it’ll be ten pages
(
digest-size front and back
in glorious [and expensive]
full color; 8 pages inside
in glorious cheap B&W
), and
(ii)
i won’t do it again for ten years.

watch this space.

### i am awesome… somebody buy me a drink

from blank file-folder (and no idea)
to conceived, drafted, penciled, and inked
before finishing my third cup of coffee:
behold: MEdZ # (1+i+j+k)/2!—
th’ G-mod-H issue!! in which we can see
no less than *four* (count ’em) more-or-less
familiar examples of “modding by a subgroup”.

namely,
(-+) {E,O}—even & odd: the ol’ cosmic yin-yang…
(++) time considered as an endless spiral of half-days…
(- -) the “unit circle” & the “periodic functions”…
(+-) time considered as an endless spiral of weeks.

one can easily look up the (- -) diagram expanded
at arbitrary length in textbooks considering “trig”.
and the helices of endless time are too familiar
to say much more about. the “clock face”, though,
is a bottomless well of shorthand examples—there’s
a car trying to run us off the road at three o’clock—
and so might be worth some further development
if it were useful in, say, fixing notations.

but (-+) one is prepared to go on about at any length.
$\Bbb{Z}_2 = \{0, 1\}$ is one of the most useful finite sets there is,
after all. for example, the 64 hexagrams of the i-ching
amount at some level to $\Bbb{Z}_2^6$ displayed pleasingly.
in the zine by me about “the 64 things” the hexagrams are
replaced with “subgraphs of K_4” (where of course K_4 is
the “complete graph on four points”; a diagram having
six edges [giving the six “lines” of the ching in that
version]; the 64 things are then subsets-a-six-set
[say {y, b, r, p, o, g} for the “full color” version]).
for example. i hope to continue in this vein later.
hello out there. ☰☱☲☳☴☵☶☷

### … but we’re only going one way

there’s four directions on the map.
i’ve called ’em Up, Down, Equal, and Op
{U, D, E, O} more affectionately (or when
actually *writing things down*).
never mind why for now; these are just their
names. call ’em table & beermug if you like.

anyhow, the title of this display is “barycentrics”.
it owes this name to the great a.~f.~möbius
(he of the immensely famous non-orientable surface
[and the merely very-famous transformations of $\Bbb{C}$;
also the not-quite-so-well-known (but still
essential!) inversion formula]); that guy…
and his concept of barycentric co-ordinates.

the drawing underlying all this mess was done
freehand by me a few years ago. the idea was
to be sure all 1+2+4+8 points of the tetrahedron
in question—$P^3(\Bbb{F}_2)$
if you must know—
were distinguishable one-from-another. you can
easily look up similar drawings in textbooks and
so on.

anyhow, here the face-centers (of the tetra) are labelled
Yellow, Blue, Red, and Mud (or {Y, B, R, M}—
you know the drill—); the vertices opposite
these points are the “secondaries”
Purple, Orange, Green, and Neuter.

the “four directions” (U, D, E, & O) then correspond
to the (opposite) color-pairs Y-P, B-O, R-G, & M-N.
i hope this is all completely obvious from the drawing.
because it’s very useful for the math.

here it is in code.

$ii = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}$
$i = \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix}$

$iii = \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix}$
$iv = \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix}$

$II= \begin{pmatrix} (mgy) (nrp) & (ybr) (pog) & \\ (mro) (ngb) & (mpb) (nyo) \end{pmatrix}$
$I= \begin{pmatrix} (bo) (mpgnyr) & (mn) (ygbpro) & \\ (yp) (mbrnog)& (rg) (mopnby) \end{pmatrix}$
$III = \begin{pmatrix} (mbp) (nor) & (mor) (nbg) & \\ (yrb) (pgo) & (myg) (npr) \end{pmatrix}$
$IV = \begin{pmatrix} (rg)(mybnpo) & (yp)(mgonrb) & \\ (mn)(yorpbg) & (bo) (mryngp) \end{pmatrix}$

$jj= \begin{pmatrix} hk & g & \\ gk & hi \end{pmatrix}$
$j= \begin{pmatrix} -gi & h & \\ -hj & -gj \end{pmatrix}$
$jjj = \begin{pmatrix} gj & hj & \\ -h & gi \end{pmatrix}$
$jw = \begin{pmatrix} -hi & -gh & \\ -g & -hk \end{pmatrix}$

### if only this power could be used for *good*

$ii = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}$
$i = \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix}$

$iii = \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix}$
$iv = \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix}$

$II= \begin{pmatrix} (mgy) (nrp) & (ybr) (pog) & \\ (mro) (ngb) & (mpb) (nyo) \end{pmatrix}$
$I= \begin{pmatrix} (bo) (mpgnyr) & (mn) (ygbpro) & \\ (yp) (mbrnog)& (rg) (mopnby) \end{pmatrix}$
$III = \begin{pmatrix} (mbp) (nor) & (mor) (nbg) & \\ (yrb) (pgo) & (myg) (npr) \end{pmatrix}$
$IV = \begin{pmatrix} (rg)(mybnpo) & (yp)(mgonrb) & \\ (mn)(yorpbg) & (bo) (mryngp) \end{pmatrix}$

### 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.

• ## (Partial) Contents Page

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