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*, 
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.
PS
ancient footnotes are everywhere:
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] 
(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!
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.
the livingston review 