## 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] {000, 100, 010, 110}.

(excuse me my “joy of ” here;

has what i hope is its obvious

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.