## Archive for April, 2015

i’ve given the equations of the (seven)

planes-in-ordinary-(x, y, z)-space

that pass through the various

“color triples” of rainbow space—

the blends, the blurs, and the ideal.

(upon putting the “primaries” [*not* one

of our triples] {R, Y, B} onto the vertices

(100, 010, and 001) of a cube

[namely, the “unit cube” I^3=

…

if you wanna get all *technical*…].)

somewhat awkwardly, *one* of our planes

(up in the upper right somewhere) does

*not* pass through (0,0,0)

(or 000 as it’s also called here).

what gives?

mod 2 arithmetic, is what.

the equations in the display “work”

in good old-fashioned —

or E^3(**R**) [euclidean 3-space]—

but to get the 7-point-space

— P^2 (F_2) [projective 2-space

over the 2-element field]… if

you wanna get all *technical*…—

we have to “work mod 2”.

and, believe it or not, 0 = 2

on this model. (so we pick up

(0, 0, 0) as a solution to

“x+y+z = 2”

[which it now becomes more

convenient to write as

“x + y + z = 0 (mod 2)”

]).

and that’s essentially it.

this “dualization” i’ve been

going on about for years, now?

here it is.

the set-of-seven *equations*

(

or their planes in three-space,

or their color-triples,

or their color-triples-plus-000,

or the “lines” of P^2(F_2)…

these are all ways of saying

the same thing…

)

can *also* be given a 7-point

“fano space” structure.

and has been (in some sense),

here on the page, via the [X:Y:Z] notation.

(note that the set-of-seven *points*

already *has* such a structure

[i.e., any two distinct points

determine a unique 3-point line]).

(details suppressed with great effort…

part of the point here is that we

*don’t* need the [“linear algebra”]

formalism [usually learned, oddly

enough, in “calculus” classes (if at

all)]—“dot products” and so on—

to achieve our dualization: we

can just *draw* the doggone thing

and check directly that our structure

“works”.)

thank you and good day.

the “rotation model”

suggested by the lower-left

(addition-mod-3) table really

only “works” for addition

(insofar as it lacks any [obvious]

*multiplicative identity*);

so too for the “eight-hour shift”

version (what would it *mean*

to “multiply” by “adding 8

hours on a 24-hour clock”?).

and working in color presents

quite a few technical difficulties.

so quite the done thing

is to use something typo-

graphically “nice” like

{[0]_3, [1]_3, [2]_3 }…

or even {0, 1, 2}, when we get

right down to actually *calculating*…

when *presenting* this kind of thing.

but the eye-appeal of colorful or

geometrically-suggestive notations

can actually make things, well,

*easier to see*…

so we beat on…

(the “typed” image will be munged by nearly any interface,

i suppose… it looks lousy enough in mine, i know…

i never thought i’d miss DOS boxes *then*…)

^ >==========> ^

| ____________________ |

| ____________________ |

| ____________________ |

| ____________________ |

^ >==========> ^

the interior of our square

can be conveniently co-ordinatized as

I = { (x,y) | 0 < x < 1, 0 < y < 1}.

the *boundary*, , then

consists of four line segments

L = {(0,y) | 0 =< y =< 1} &

R = {(1,y) | 0 =< y =< 1}

("left" & "right"), and

T = {(x,1) | 0 =< x =< 1} &

B = {(x,0) | 0 =< x =< 1}

("top" & "bottom").

(convince yourself of this

if you can. hint: pencil

& paper.)

the Torus, T^2, can then be modeled as

"the union of I with its boundary, mod 1".

but never mind the fancy terminology.

instead we can imagine the situation

by thinking of the inside of the square

as if it were a video screen…

with the condition that when we

"move the mouse"

upwards and get to the Top (say),

we will "vanish off of" the Top

boundary and "re-emerge" at

(the corresponding point—

"having the same x-co-ordinate"—of)

the Bottom (still going upward).

& likewise for L & R

( (0, y) ~ (1, y)… i.e., for any

particular "y", we consider these

two boundary-points-of-the-square

to represent *one point* of T^2 ):

moving the mouse rightward, one

will, as it were, "go through"

the Right edge of the screen and re-

-emerge on the Left

(still going rightward).

thus far, so familiar, i hope.

using "identification diagrams" of this

kind is one of the coolest tricks

i ever learned in a math department.

now we just replace the "field of Real Numbers"

with "the" field with three elements (F_3).

all of (x,y)-space over this field

consists of *nine* ordered pairs

(and we "calculate mod 3"…

1+1+1 == 0 in this world…

but otherwise "ordinary algebra"

works…

(1, 2) + (1, 2) == (2, 4) == (2, 1),

for example…

the upshot is that we need only the

three elements of {0, 1, 2} to represent

*any integer pair* (x, y) of our

good old-fashioned "euclidean" space

):

(0,2) (1,2) (2,2)

(0,1) (1,1) (2,1)

(0,0) (1,0) (2,0).

i'll call this E^2(F_3)…

"euclidean two-space over the three-element field".

mod three arithmetic produces

"the video game effect" on this

space… one might imagine starting

with (a very flexible copy of)

(0,3) (1,3) (2,3)

(0,2) (1,2) (2,2) (3,2)

(0,1) (1,1) (2,1) (3,1)

(0,0) (1,0) (2,0) (3,0),

and then "wrapping it around"

to form a tube with each (x, 0)

covering up its corresponding (x, 3);

next (here's where "very flexible"

comes into it) "wrap around" again

the other way to produce the

(doughnut shaped) Torus by overlapping

each (3, y) with its corresponding (0, y).

now just replace the *ordered pairs of numbers*

with (unordered) pairs of "direction icons"

& draw the whole thing on the nearest

tape-doughnut to hand; voila. a pleasant

time not grading papers.

the colors-to-corners assignment here

is nonstandard… i.e., inconsistent

with the one i’ve been using as my

“base case” in earlier blog entries.

so for heck sake don’t memorize ’em this way.

the “algebraic” aspect here…

whereby certain “vertex triples” are assigned

“dual vectors” according to the scheme

[1] = [0:0:1] —> {010, 100, 110}

[2] = [0:1:0] —> {001, 100, 101}

…

[7] = [1:1:1] —> {011, 101, 110}…

isn’t affected by my having “renamed the colors”.

some mod two equations are included.

this justifies the orientation-reversal

in [7]. it’s still a pretty lousy draw-

ing, though.

“mod two arithmetic” underlies our whole

algebraic “scheme” (*not* a technical term here):

the duality constructed here will turn out to be

“the *dot product* (mod two)” when our scheme has

been fully carried out.

also i don’t vouch for the stuff in pencil.

i drew this a few years ago and’ve only

checked the parts i inked in today.

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!

here one has labeled the vertices of a cube

with (“euclidean”) 3-space co-ordinates.

the “origin”… we can think of it as

our “point of view”… is at 000.

[the “point” in 3-space usually

denoted (x, y, z) is here written

as “xyz”; we restrict our attention

to the values in {0, 1} for these

variables (since the (8) points

000, 001, 010, 011,

100, 101, 110, 111

then form the vertices of our cube).]

putting the primary colors (Red, Yellow, & Blue)

“on the x, y, & z axes (respectively)”, i.e., putting

Red—>001

Yel—>010

Blu—>100,

we may then conveniently put the *secondaries*

(Green, Orange, & Purple) at the “third vertex” of

the back left wall (“G = Y + B”),

the front left wall (“P = R + B”), and

the floor (“O = R + Y”)

(respectively).

the last vertex is the “ideal” point 111;

all the colors blend here to form Mud.

cool algebra ensues. but not till after dinner.

or you could look it up.

one has already learned the 7-cycle

sunday-monday-tuesday-wednesday-thursday-

-friday-saturday-(sunday-…)

as part of one’s cultural heritage

as an english speaker.

and it’s high time i *used* that fact

in studying “fano’s rainbow”

(aka “the seven-point [projective] space”).

and so to each of the

*positions* in the well-known

“three corners, three midpoints, one center”

representation of fano space,

i have assigned a *day of the week*.

now, the seven points of “rainbow space”…

mud, red, blue, green, purple, yellow, orange (, mud, red, …)

fall neatly… also a part of our cultural heritage… into

“three primaries, three secondaries, and one ideal”

(“mud” is of course the ideal).

so on sunday i put MRBGPYO into the big diagram

by (“arbitrarily”) “coloring the vertices”:

Red—> Lower Left

Blue–> Lower Right

Yellow–> Top.

the other “colors of the rainbow” can now be

filled in (in exactly one way) according to

the rule that

“rainbow lines fall on geometric lines”

(recall that the rainbow lines are

the “blends” RBP, BGY, and YOR,

the “blurs” RMG, BMO, and YMP,

and the ideal GPO.)

turning our attention to monday, i’ve permuted

the colored-triangles “forward one” in the order

m, r, b, g, p, y, o:

“mud goes where red was, red goes where blue was…

… orange goes where mud was”.

then i copied the shot from the zine…

wherein a certain lines-to-points correspondence

is laid out graphically…

into the fano-diagram called “sunday”.

the rest is chasing permutations around.

the question… in all its vagueness…

is “what happens to the geometric lines”

when we “move the colors” around the diagram

by “applying the MRBGPYO permutation”, as it

were, each day of the week.

and i’ve found some stuff out that i didn’t know.

and i’m calling it a day for now.

two triangles are displayed here.

each has a Red, a Yellow, and a Blue

vertex.

joining the red vertices

to form a red *line*…

and forming blue and yellow

lines likewise…

the triangles are arranged in such a way

that the three lines so described

meet at a single point.

(two triangles chosen “randomly” from

triangles-in-the-plane *won’t* have

this property.

of course it’s easy to *construct*

such triangles, though:

just start with the lines

and choose the points from there

[as i’ve done here].)

now we’ve got our two red-yellow-blue

triangles in place (they are said to

be “in perspective with respect to”

the point-of-view point in the lower-

left (X marks the spot).

the next move is to “extend the sides”

of the triangles:

we can imagine that each Blue-Red edge

(for example), forms a Purple line.

the two purple lines now meet

at (what we will call) a purple *point*.

likewise, form Orange and Green points

by intersecting (respectively)

the orange and green “line pairs”

(formed by extending the red-yellow

and the blue-yellow edges [respectively])

of our Two Triangles.

desargues’ theorem: these new “secondary”

points (O, G, P) all fall on a line.

w’edia.

(when cases involving *parallel lines*

are taken out of the discussion.

for example,

ordinary 2-d “euclidean” space

can be enhanced with “ideal” points

in such a way as to make this

*always* true [“parallel lines

meet at infinity” becomes

*formally true* in our (enhanced)

“projective space]

).

now for bus to the office to turn in papers

and pick up other papers. life is good.

marking a “zero” point

at one corner of a cube,

one might then consider

“directions” to “go”, as

it were, “from” that point.

“up”, say, “back”, and “over”…

these are the most obvious ones

in some sense. specifically, in

the diagram, they correspond (in

order) to the “blue”, “yellow”,

and “red” vertices of our

cube-projected-into-the-paper-towel;

each of these is (and only these

are) connected by an “edge” of our

cube to the “zero point”.

the other four “directions” are then

“colored in” using the familiar

paint-mixing rules: the secondary

colors are placed in such a way

that the *faces* of the cube show

the color “blends”… then finally

the “opposite corner to zero” is

filled in with the Muddy color that

you get by dumping in *all three*

primaries (“up”, “over”, *and* “back”).

it is now a peculiar fact that the

various “blends” and “blurs” of

our paint-mixing model, together

with the “ideal” color-triple

{green, purple, orange}

…(aka “the secondaries”;

this triple shares its odd-entity-

-out nature with the color i

have called “mud”)…

corresponding on one hand to

certain *cross-sections* of our cube…

these blurs, bends, and ideal…

can be made *also* to correspond

to the *lines* of the 7-point space

shown (several times) in the rest

of our display.

more on this anon. as i imagine.

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.