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

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.