## Archive for the ‘Graphics’ Category

starting with the mathy stuff

on what used to be a door

(

at the corner of what used to

be a street: in “the livingston

library”, or, more precisely,

as i think of it, at the living-

ston “branch of the UUCE” libe,

where for “UUCE”, read th’

UU church in reynoldsburg

[ohio; do i have to tell you

*everything*?]… g-d willing

i might make it back to the

*main* branch while it still

stands…):

a bunch of taped-up drawings (and

reproductions of same) by me.

the entire RHS (right-hand side,

natch) is given to various versions

of “desargues’ theorem in color”;

there’s another of these at upper-

-left (& in between, “vlorbik’s

seven-color theorem” [in one of its

many versions]).

then the three big (eight-&-a-half-by-

-eleven) 16-point thingums; these are

versions of the “hurwitz tesseract”

(as i hereby dub it); the two 8-point

“cosets” of the normal 8-group in the

unit integral quaternions. a-four-hat,

as i like to call it. anyway.

and, illegible here more or less of course,

the back cover of an em-ed-zed (M Ed Z —

“math ed zine” to the acronym-averse).

featured here are *more* covers of MEdZ

(what else?): specifically, this very one

(#1—the “hip pocket vocab”, 2010; reissued

in digest size with new [-ly reprinted]

graphics & hand-lettered comments [same year,

i think]); K_n -slash- K_4 (a “remix” of

two “microzines” [eight pages on one side

of an 8 1/2 by 11 each] into one such);

P_2(Z_2) & P_2(F_n) (“projective planes”);

& , , & (“number sets”).

but enough about me. some of my *other* stuff.

r.~crumb’s _art_&_beauty_ (cover shot of mrs.~

~crumb). two “books” with comic-book-swearing

for “titles”; also clowes’ _modern_cartoonist_.

that poster of magazine covers from _starlog_.

you could look at that thing alone for quite

a while. in the right company. (alas.)

what you can’t see covering up part of this

poster (lower left) is the _comic_book_artist_

cover by i-know-not-who showing “magnus

robot fighter” clobbering an evil droid. i did

a song about fighting robots and have a soft

spot for this character.

two pics of fitzgerald & a bunch of his books; you

might also be able to see burroughs. in the brick

under the pez-head of winnie-der-pooh is a period

shot of dad early 60s. you can tell the images

of auden are *there* in my version but not who

they are images of; likewise langston hughes.

this is quite a satisfying way of making the time

go by as long as you don’t wish for somebody else

actually to *read* it…

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.

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.

the seven black triangles are

the blends

Mud Yellow Purple

Mud Red Green

Mud Blue Orange

the blurs

Yellow Blue Green

Yellow Red Orange

Blue Red Purple

and

the ideal

Purple Orange Green.

the “theorem” in question is then that

when the “colors” MRBGPYO are arranged

symmetrically (in this order) around a circle

(the “vertices”of a “heptagon”, if you wanna

go all technical), these Color Triples will

each form a 1-2-4 triangle.

but wait a minute, there, vlorb. what the devil

is a 1-2-4 triangle. well, as shown on the “ideal”

triple (center bottom), the angles formed by these

triangles have the ratios 1:2:4. stay after class

if you wanna hear about the law of sines.

note here that a 1-4-2 triangle is another beast altogether.

handedness counts. (but only to ten… sorry about that.)

anyhow, then you can do group theory. fano plane.

th’ simple group of order 168. stuff like that.

all well known before i came and tried to take

the credit for the coloring-book approach.

with, so far anyway, no priority disputes.

okay then.

behold: the six-color i ching.

as i remarked elsewhere, *any* diagram version

of “the sixty-four things”… the 6-bit “strings”

{000000, 000001, 000010, 000011, … 111111}

being one of the best known… can be considered

as a diagram of the sixty-four hexagrams famous

since the dawn of historical time.

i drew the black-&-white 10 years ago and change.

colorized by my hand today, 6/12/20 vlorbik his mark.

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.

i know it shouldn’t bother me that

for the whole rest of the connected world

it’s trivially easy to capture images

& move ’em around on the net. but by golly

it does anyway. you could look it up.

anyhow, on the image at home you can see

all twenty-four group elements in all four

“panels”… as i called ’em upthread… and so

verify that the four “versions” of the

“binary tetrahedral group” presented

here are pairwise isomorphic. (hence,

duh [six pairs], six iso-isms.)

there are probably mistakes.

i’ll pay whoever spots one before me.

in books by knuth.

here are 2 (of 4) panels from the newly-created

*binary tetrahedral rosetta stone*:

the code… what the hell…

{+000,-000,0+00,0-00,00+0,00-0,000+,000-}…

represents {1,-1,i,-i,j,-j,k,-k} (the “familiar”

unit quaternions; one has i^2 = j^2 = k^2 = ijk = -1

[per w.r.~hamilton; the margin is too small]).

then things get (slightly) messy… for

{—-,—+,–+-,–++,-+–,-+-+,-++-,-+++,

+—,+–+,+-+-,+-++,++–,++-+,+++-,++++}

we have weird sixth-roots of one; for example

++++ denotes h = (1+i+j+k)/2. (the “h” is for hurwitz).

mutatis mutandis for the rest, eg,

-+-+ = (-1+i-j+k)/2;

this is corresponds to “(mgy)(nrp)”

in the permutations-notation version,

as one can see from the photo.

together, the 24 trit-strings ({-,0,+} are

the *trits* in question…) represent the

*hurwitz units* U(Z(h,i,j,k)): the 24

invertible elements of the set of sums

(and differences) of i, j, k, & h.

the other two panels are the “matrix” version…

SL_2(F_3) so called…

and the “semi-direct product” version

where the “hi = jh” relation is made explicit

in the code. six graphical isomorphisms all told.

this would be the best lecture i ever gave

if i ever gave it. here it is for the internet.

the color-scheme is inspired by one-or-the-other of

*a hyperbolic plane coloring & the simple group
of order 168* (dana mackenzie;

*monthly*of 10/95)

or

*why is* PSL(2,7) GL(3,2)?(ezra brown & nicholas loehr;

*monthly* of 10/09)…

okay, it was the mackenzie. but i want you

to look ’em both up. the brown-loehr i’ve

known longer and studied more. anyhow, enough

about the actual math. more about me.

the bits *not* in color show “the desargues

configuration”… the triangle-lookin things

somehow are supposed to depict the version

where the “lines” (sets-of-three “points”)

of the configuration are made to coincide

with triples-of-faces on an icosahedron.

it’s one of the coolest things i know.

there are versions somewhere colorized.

self portrait with comix stuff & art

supplies. happy days, everybody!