## Archive for March, 2015

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

determining 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.)