someday i’ll learn how to scan a document

February 15, 2011 in Projective Spaces

until that day, i’ve set up a process
involving “photo booth” (on the mac)
and “flickr” (on the web). optionally,
i can involve the mac’s “iPhoto”.
not sure yet if that’s ever going to
be useful… but it organized the
“photo booth” stuff in maybe a better
way than Photo Booth itself.

i’m *writing* this passage in “flickr”
(but i’ll probably *edit* it in wordpress;
in particular, there’ll be a redundant
line ID’ing this as a flickr shot i think;
if so i’ll kill it).

“the medium is (part of) the message”
as i’m given to saying. as for the drawing
itself? also an exercise in media (of course!):
the very-sloppy-looking double line

(mostly along the bottom and RHS
[RHS abbreviates “right hand side”;
very useful in making terse remarks
(typically in chalk or marker on a board
black- or white-) about equations])

shows pretty clearly (if you didn’t know
already) that marker-on-paper is *not*
something i’m very practiced in at this scale.
even this took a couple of drafts
(i’m out of whiteout or id’ve used
it at the first sign of trouble… so
a careful description of the “medium”
would include the information
“no corrections allowed”).

anyhow. as to the “content”.

the 13 circles making up “the big picture”
represent ,
the Projective Plane constructed on
the Field of Order 3.

there’s a 3-by-3 square of points.
call these points Finite Points.

the other four are called Points At Infinity
(together, these form the Line At Infinity…
so the “double line” i mentioned earlier
was the “line at infinity” all along).

the P-at-I at the top is associated
with the *vertical* direction,
the P-at-I on the left is associated
with the *horizontal* direction,
and the two P’s-at-I in the corners
stand for the two *diagonal* directions.

as an example of the “diagonal” directions,
i’ve drawn the three “lines of slope one”.
each one passes through three Finite Points
*and* through the Infinite Point at lower-left.

of course the lines-of-slope-one are *parallel*
in Finite Space… in our context, this means
that the dotted lines meet *only* at the lower-left
Infinite Point (accounting for the “association”
of this point with the upward-sloping diagonal
to which i referred two paragraphs up):
the slogan “parallel lines never meet”
is replaced in projective spaces
with “parallel lines meet at infinity”.

(sort of. in the most general setting,
*any* line [or none] can be thought of
as “the” line at infinity… so it’d be more
accurate to replace “parallels never meet”
with “there *are no* parallel lines”.)

this is probably the best example of the
the concept of Ideal Points: one creates
new elements to include into a set to make
certain nice things happen. another example:
, the “extended line”.

i’ve gone on (in my exuberance) to draw
*another* little P_2(F_2) inside the lower-left
(infinite) point.

when the drawing is finished, the points of
the Big Picture that correspond (in the
“obvious” way: unreoriented-blowup-and-shift
[a “homothety” if i understand correctly])
to the points of the “shaded” line will be
precisely those points whose Little-Picture
lines-“inside”-of-points *pass through*
the point-at-lower-left itself.

this “little” P_2(F_2) isn’t *necessarily*
found at this spot… i *put* it there.
there are many other ways…
in some other lecture, i’ll want to
look at *how* many…
to set up a points-to-lines correspondence
like the ones i keep on drawing over and over
(and this is true even *after* selecting a
“line at infinity” as i’ve done here).

enough for today; actual work is still
a long busride away…

Advertisement