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”);
& 
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—
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
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) 
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!
the livingston review 