Photo on 4-17-15 at 10.02 AM

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.

Photo on 4-10-15 at 4.37 PM

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.

Photo on 4-10-15 at 12.11 PM

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.
5201764791_840763f68b

Photo on 4-9-15 at 8.43 AM

these little balls of clay will be *very* useful
in explaining “finite geometries”. or would,
if i could actually find anybody to endure my
goings-on about them. on jugera.

Photo on 4-8-15 at 9.43 AM

the other four ways can be found
by holding the page up to a mirror
(rotated 90 degrees from this
orientation).

this is a fairly old drawing.
here is my recent discovery
of the “five-way symmetric” version
of the same situation (the “ten
circles theorem” as i call it).
finding this stuff out… color
displays in projective geometry…
is my probably my proudest accomplishment
in mathematics-as-such. my classroom
work impressed me quite a bit, too,
but those days are over now most likely.
here’s fano’s cube from 2011.

i wish i’d stayed a grill cook.
by now, i’d’ve probably gotten
pretty good at it.

Photo on 3-31-15 at 10.54 AM

but i never found the pages
that the plot’s in.
so all i’ve done is put
a lot of blots in.
you know, my meh, thuds,
watson.

Photo on 3-31-15 at 9.25 AM

cf
Photo on 3-27-15 at 7.33 AM

the second, smaller, sketch is from
desargues’ theorem in color (but let’s call it
desargues’ rainbow from here on
to match fano’s rainbow,
posted the next day).

in the first, newer, bigger sketch,
i’ve used my mystical “projective
geometry” powers to bend all the lines
into circles. so we now (as you can see)
have a red *circle* (at the “omega” point)
along with (arcs of) blue and yellow
*circles* (replacing the red, blue, and
yellow *lines* on the original [textbook]
drawing).

again, as one should expect from the
names-of-colors aspect of all this,
we find certain yellow-and-blue
point-pairs occurring on the arcs
of certain circles… and two such
circles meet in the *green* point
(green is the “blend” of yellow
and blue). and then likewise for
the other “secondary” colors: an
*orange* point at the intersection
of two red-and-yellow circles, and
a “purple” point where two red-and-
-blue circles meet.

desargues’ theorem is then that the
secondary colors are on one of the
“lines” of the system at hand.

[in this case, (an arc of) another
“wide circle” (our system consists of
ten circles; the “narrow circles”
appear as circles in the diagram
and arcs of three of the “wide”
circles are indicated by three-
-point arcs).]

Photo on 3-28-15 at 11.00 AM

the 7 points, as usual:
{m, r, b, g, p, y, o}.

Let P = {mud, red, blue, green,
purple, yellow, orange}.

wipe all traces of ROYGBIV
away for now and consider
the “colors of the rainbow”
as MRBGPYO instead.

the virtue of this renaming-
-and-reordering is that i can
now present the seven “lines”
of fano space using the color
scheme (*without* reference
to geometric or numeric data).
specifically, as
“the blends”, “the blurs”, and
“the ideal”, where
{
{red, blue, purple},
{blue, green, yellow},
{yellow, orange, red}
}
is the set of “blends”,
{
{mud, red, green},
{purple, yellow, mud},
{orange, mud, blue}
}
is the set of “blurs”, and
{green, purple, orange}
is the “ideal”.

the lines then fall in pleasant places.
in two ways that matter to me rather a lot.

we can jam {red, blue, yellow}…
the so-called “primaries” in my scheme…
onto the x-, y-, and z- axes of some
three-dimensional vector space
via
red —> (1,0,0)
yellow —> (0,1,0)
blue —> (0,0,1)
(say… one of course has five
other ways to assign colors to
axes).

*or*, as shown above, we can jam
the colors onto the “corners” of
the well-known fano diagram
representing the two-dimensional
projective space over the field
of two elements.

together, the colors give me
a very convenient way to talk about
certain correspondences between
the situations (7 non-zero
corners of an algebraic “cube”,
on the one hand, and
7 points of fano space
on the other): certain “planes”
of the cube become “lines” of
fano space, for instance…
with, of course, the green
*plane* in 3-space (say) associated
to the green *line* in fano space.

on and on it goes, this thing of ours.

Photo on 3-27-15 at 7.33 AM

the diagram is, essentially, traced from
bruce e.~meserve’s _fundamental_concepts_-
_of_geometry_ (dover reprint of 1983;
originally addison-wesley 1959). but
i added in the colors. with which, one
has as follows.

there’s a red line, a blue line, and a yellow line.
to start with. all sharing a point.
then two “triangles” are constructed:
each of these is to have a red, a blue, and a yellow
“vertex”.
(on the diagram, the “red”, “blue”, and “yellow”
vertices of one of the triangles is actually black;
this helps [maybe… it helps *me*] in determining
which vertices belong to which triangles.)

next we can think of the *edges* of these two triangles
as “blending two primary colors”; for instance we can
think of the red-and-yellow edge of either triangle as
determiing an *orange* line… and go on to construct
an “orange vertex” at the intersection of these lines.

likewise, the yellow-and-blue edges of the two triangles
will determine a *green* point (at the intersection of
two green lines) and the blue-and-red edges will determine
a *purple* point.

desargues’ two-triangle theorem then says that the
orange, green, and purple points will “line up”.

proof: meserve (or any of *many* other sources)

footnote: alas, one has *exceptional* cases
in “euclidean two-space”, i.e., the ordinary
(two-dimensional) *plane* of high-school
geometry. one remedies this by working
instead in a (so-called) *projective* plane
(in such a plane, there are no “parallel” lines;
nevertheless, much of “ordinary” plane geometry
becomes *easier* in projective geometry [example:
this theorem]).

putting color names on things is a common trick in math,
but if anybody else is using blends-at-intersections in
anything resembling this way (elementary projective
geometry, i suppose i mean), i don’t know about it.
(priority claim; if you use this amazingly good idea,
remember where you got it. please.)

combinations with repetition (from a course by one “miguel a.~lerma” at northwestern).

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