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 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.

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).]

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.

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).