## Archive for the ‘binary tetr’al’ Category

so here’s the thing i’ve been in the process

of memorizing ever since i worked it out.

i *almost* know it now. this interface sucks.

bye now.

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.

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.

let

now let

, i.e., let

, 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.,

;

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

;

voila.

oh, yeah. i forgot to say. the “points” are

matrices *mod 3* (so that 2 = -1). that is all.

the monochrome dots-and-lines thingum up top is copied

from a by-hand copy of an online graphics-giz. jpg or

pdf or i dunno what. fuggedaboudit. point is,

the two-tone diagram underneath has the *same* 16 points…

one might almost imagine that one point-set had literally

been *traced* from another… but now there are two

distorto-cubes: a blue & an orange. i’ve filled in

the “trit-string” representation of a projection of

the “h-coset” (the blue “cube”); the orange is the

negative of this one so you can easily fill in

the trits if you should so desire. or could if

you could see the damn things on the blue. anyhow,

point is, i had to *fill in* those co-ordinates

(

h := ++++, g:= h^2 = -+++, … hi, gi… gk…-h=—-:

the sixteen points of the “tesseract” obtained

by shifting the “fundamental quaternion units”

[eight of ’em] to the “right” and “left”

[the “past” & the “future” on another convenient

model].

)

and “connect the dots” literally to see how the

doggone monochrome was a picture of a tesseract at all.

that’s the way i roll. staring at the diagram…

however pleasant.. just wasn’t gonna get it.

and i am reminded of a remark of mine in some blog

or another long ago: the student isn’t learning

if the pencil isn’t moving (the eraser counts).

/*

the (so-called) fundamental quaternion units

can be represented as

with the scalars of the matrices considered

as elements of —i.e., 2=-1 etc.

*/

now let

—the “h” is for hurwitz; let g = h^2

(remember everybody is mod-3; i’d better

get that in somewhere if i’m actually

gonna bring *calculations* into play…)

then \pm (that’s plus-or-minus for you

TeX newbies)

1, i, j, k,

h, hi, hj, hk,

g, gi, gj, gk

is HU—or A-four-hat—or —

or what have you. the binary tetrahedral group.

here’s the other image of a-hat from w’edia.

this one looks quite a bit more like the diagrams

i’ve been drawing in pursuit of a geometric “feeling”

for—what i’ve been calling—the “binary tetrahedral”

group; what i’ll be calling “a-hat” (—this rhymes

with “A-flat”, not “the cat”) henceforth. this is

short for a-four-hat. the idea here (as i found

on some webpage) is that our group

is a “covering” of A_4—th’ “tetrahedral group”

aka the “alternating group on four objects”.

(

the four objects in question, in the unlikely

event that anyone is following so far, “are” the

vertices of the tetrahedron in question…

the group

A_4 = { (),(01)(23),(02)(13),(03)(12),

(012),(021),(013),(031),(023),(032), (123), (132 }

(typing is fun and easy)…

then is made to represent the three 180-degree

“flips” and the eight 120-degree “rotations”

that permute the vertices of a tetr’on [while

preserving its “orientation”—if we allow for

reversals we get the cube-group S_4].

the vertices i call up, down, equal & opp.

in the “covering”, they become the color-pairs

mud-neuter yellow-purple, blue-orange, & red-green.

get it?

)

here on the front of one big envelope are

four versions of the “binary tetrahedral group”:

HU-code,, permos, generators, & matrices.

taken together they form “the BT rosetta stone”.

i posted a bigger version a few days ago.

there’s a semester’s worth of algebra in that display.

i know because it took me much longer than that

to figure out how to do it.

here’s HU—the hurwitz units, aka

BT the binary tetrahedral and also

the 2-D special linear group

for the field of order three… not to mention

the “permutation” representation…—

graphically with pro production-values.

i haven’t groked the layout yet.

image credit: w’edia.