## Archive for June, 2020

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 tried posting this 6 months ago

but wordpress ate it. here it is now.

… 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

. 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.

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

of those _britannica_s already mentioned…

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 —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*.

other side of my head. “children’s book” about sacagawea

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

survey-of-advanced-math right up front.

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.

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.

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 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. ☰☱☲☳☴☵☶☷

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 ;

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— 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’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.