Archive for the ‘Lectures Without Words’ Category
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.
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.
at least one stranger “liked” last week’s
hell hound on my trail, so here’s another
guitar-neck diagram in the same format.
this one’s in “open G” tuning: DGDGBD
(“dog dug bed”). in key-neutral notation
one has 0-5-12-17-21-24 (“half steps” above
the lowest note). (the thing to memorize
here is probably the gaps-between-strings:
5, 7; 5, 4, 3 [“twelve & twelve”; there are
(of course) 12 half-steps… guitar frets
or piano keys for example… in an “octave”];
in the same notation, the “open D” tuning
from the previous slide would be “7,5;4,3,5”.)
adding “7” (and reducing mod-12 again) gives us
7-0-7-0-4-7; the point of this admittedly rather
weird-looking move is that the “tonic note”
for the open chord (“G” in the dog-dug-bed
notation) is placed at the “zero” for the system.
(“open D” is 0-7-0-4-7-0 in this format; the
tonic note falls on the high and low strings.)
my source informs me that open-G tuning is also
known as “spanish” tuning; a little experience
shows me that it’s even more convenient for
noodling-about-on-three-strings. that is all.
here’s a diagram showing some of the notes for
a guitar tuned in “open D” tuning. i posted it
if f-book not long ago but of course one soon
loses all track of whatever is posted there.
three “inside strings” (the 3, 4, & 5; marked
here with a purple “{” [a “set-bracket” to me;
i’m given to understand that typographers call
it a “brace”])…
three “inside strings”, i say, can be played
as shown here to produce “the same” major chord
(the 0, 4, and 7 from a 12-note scale) in
three different “inversions” (namely, 7-0-4, 0-4-7,
and 4-7-0).
meanwhile, quasi-random bangings of the “outside”
strings—i like to use my thumb for these some-
times—particularly at frets 5, 7, & 12—will
blend in nicely and one needn’t worry much about
“muting” strings-not-played.
yet another sketch from the
“lectures without words” run
of MEdZ. here improved with
colored inks (and spoiled by
flouting the “without words” rule).
“binary arithmetic” is here exploited
to assign *number values* to the corners;
the symbol “xyz” chosen from
000, 001, 010, 011, 100, 101, 110, 111
corresponds on this model to
4x + 2y + z.
(this follows the usual “place-value”
conventions typically used in the
context bases-other-than-ten
[in base ten, the same symbol “xyz”
would denote 100x + 10y + z].)
the “front face” of our cube (for example)
is now {000, 001, 100, 101}.
these number triples share the feature
y=0…
and are the *only* triples with this feature.
now, we can think of “y=0” as meaning
“don’t move in the y direction at all”
(the “y direction” here is “front to back”…
going [as it were] from the 000 point “back”
toward the 010 point is the only way to get
a y=1… that’s why “y=0” gives us the “front face”.
but the point 010 is not itself a “direction”…
so another notation is introduced: [0:1:0].
the diagram shows (or hopes to) that similarly
[1:0:0] “is perpendicular to”
the left-hand face {000, 001, 011, 010} and
[0:0:1] 
(excuse me my “joy of 
meaning.)
anyhow… there’s real work to be done
(getting to campus and back; the hardest
part of the job some days)… that’s *almost*
it for today.
it remains only to remark that
[0:1:1], [1:0:1], [1:1:0], and [1:1:1]
can also be considered as “perpendicular”
to the other four rainbow-space “lines”
(certain cross-sections of the cube
on the 3-D model)… giving us a
full-blown *algebraic* model of
fano-space duality.
[exercise. hint: binary arithmetic.]
feed me!
a certain collection of three-point subsets of
{mud, red, blue, green, purple, yellow, orange}–
namely, the “ideal”,
{green, purple, orange}
(AKA “the secondaries”),
the “blends”,
{blue, green, yellow}
{red, purple, blue}
{yellow, orange, red},
and the “blurs”
{green, mud, red}
{purple, mud, yellow}
{orange, mud, blue} —
are called “lines” of 7-color space;
likewise the colors themselves are
called “points”.
the points of 7-color space can then
be made to correspond with the points
of “fano’s 7-point space”… which
is the smallest example of a so-called
“projective geometry”… in such a way
that the “lines” of color-space correspond
to “lines” of fano-space.
all this can be easily verified by comparing
the big “colored lines” diagram of fano space
at left with the uppermost “seven-color” space
to its right.
what we have here moreover is a certain
matching of our color-triple “lines”
in color space (the blends,
the blurs, and the ideal)
with the “colored lines” in fano space
shown in the big “triangle”: namely
Mud~~{green, purple, orange}
Red~~{blue, green, yellow}
Yellow~~{red, purple, blue}
Blue~~{yellow, orange, red}
Green~~{green, mud, red}
Purple~~{purple, mud, yellow}
Orange~~{orange, mud, blue}…
and “Mister Bigpie, oh” order
MRBGPYO, color coded.
to the right of the colored letters
i’ve “bent the line around” into
a circle… “mud” now follows “orange”
just as “red” follows “mud” and
so on.
returning our attention to the upper-right,
i’ve “applied the permutation” MRBGPYO
to the color-points of the first (higher
and to the left) triangle as follows:
the mud point goes where the red point was,
the red point goes where the blue point was,
…
the orange point goes where the mud point was;
as you can now easily see, this permutation
has “preserved the lines”. by this i mean
that in the second (lower and to the left)
triangle of this part of our display, each
“three-color set” of rainbow space lands on
a geometric line in good-old-fashioned fano space.
so this is a pretty cool phenomenon.
back at the “circle” diagram, i’ve highlighted
the “ideal” line (the “secondaries”) and i’ve
dotted-in one of the “blends” (namely {r, g, b});
i now claim that the other five “lines” of our
system are the other five triangles of the same
shape
(“choose a ‘first’ point, go forward one
for the ‘second’, then forward two for the
‘third’, and back to the first”; paraphrasing,
“up one, up two, come back”):
the seven such triangles are yet
another representation of our
seven-line “dual” space
(the Capital Letter color names
in the typographic display of
a few paragraphs ago).
got all that? good. because at least part
of the point here was that, finally, at the
lower-right i’ve calculated out (twice; the
small one was *too* small to content me so
i treated it as a first draft and redrew it)
what happens to the lines of the “original”
color-scheme [which, i hasten to add, is
somewhat arbitrary… “primaries at the
corners” etcetera] upon “applying the
MRBGPYO permutation” to its points.
at about this point, thinks become confusing
enough for me to want to start putting
*algebraic* labels everywhere and start
calculating in monochrome pencil “code”.
which i’ll spare you here.
because another part of the point is that
one simply has no *need* of *numerical*
calculations in most of this work (so far):
the “blending-and-blurring” properties known
(in my day) to every kid on the block
do much of the work for us (as it were).
here’s a years-back draft of this talk
from before i knew about this whole
“geometry’s rainbow” phenomenon.
ladies and gentlemen, PSL(2,7) (w’edia).
here’s a hex board
with seven icons on it;
each icon has seven colors;
the seven permutations of
colors-into-hexes (each icon
has seven hexes) can be
considered as the objects
of a cyclic group.
specifically
{
() = \identity
(1234567) = \psi
(1357246)= \psi^2
(1473625)=\psi^3
(1526374)=\psi^4
(1642753)=\psi^5
(1765432)=\psi^6
}.
(of course one has \psi^7=\identity,
etcetera… the group here is
essentially just integers-mod-7:
{ [0], [1], [2], [3], [4], [5], [6]},
with addition defined by
“cancelling away” multiples
of 7 (forget about the fancily
denoted “permutation structure”
displayed in defining \phi
and just look at the exponents).
the “cycle notation” used here
is *much* under-used, in my opinion.
but we’ll really only need it
for future slides.
yesterday’s more verbose version.
the livingston review 