…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.

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  1. (still one point missing; exercise.)




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