Archive for March, 2015

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.

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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).]

fano’s rainbow

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.)