### …and then a hockey game broke out…

my previous post was about

an earlier version of this drawing.

i’ve added most of the rest of the lines.

if the fine print emerges in your version,

you’ll see

*three vertical

*three horizontal

and

*three upward-sloping

lines, plus the “line at infinity”.

i’ve also added (what i’ll now call)

“polar lines”

for a few more of the points

(specifically, the four points of the

line at infinity… recall that the

“finite points” of the diagram

are the 9 still-blank circles

forming a 3-by-3 square in the middle).

for the *polarity* (a certain

pairing-of-points-with-lines)

i’ve begun to define here,

each point-at-infinity is the

*pole* (or “polar point”)

associated with a *vertical* line

(its so-called “polar line”,

typically called just the *polar*

[for the given pole]).

the topmost point (i hereby declare)

is the Point At Infinity

(the “distinguished” point of

the Line At Infinity).

the topmost point considered as a Pole

has *the line at infinity* as its Polar.

so (assuming the drawing eventually

*does* present some particular Polarity)

the point-at-infinity is a *self-conjugate* point

for the polarity we are beginning to consider.

(quoting meserve, “a point that is on its own

polar is called a *self-conguate point*

of the polarity” (p. 137).)

it’s not by accident that i chose the *vertical* lines

for the polars of the other “infinite” points

of the system. the notion that

parallel lines of “ordinary” (3-by-3) space

“meet at infinity” (in projective space)

suggests that *points at infinity*

can be associated with

*slopes of lines*… and indeed that’s

precisely what’s been done here.

the three lines of each “parallel class”

(vertical, horizontal, or upward-sloping in the diagram)

come together at some *particular*

point-at-infinity…

and so i’ve placed the points in what

i hope are suggestive parts of the picture:

the verticals with the point on top;

the horizontals with the point to the side;

the diagonals with the points at the corners.

(“easy”) exercise: fill in the missing three lines.

(“hard”) exercise: finish filling in the poles-to-polars

bubbles. (i do *not* claim that there is only one way

to do this). hint.

February 20, 2011 at 7:22 pm

February 20, 2011 at 7:24 pm

(still one point missing; exercise.)