### someday i’ll learn how to scan a document

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…

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