Archive for the ‘Projective Spaces’ Category


here’s another picture of a dualization
of the projective plane on the field
of four elements: P^2(\Bbb F_4)^*.

in the first of two tweaks since the last version
published here
, i reversed
the positions of the roots of x^2 + x +1…
i’ve been calling ’em alpha and beta…
reversed their positions, i say,
for the vertical axis (as opposed
to the horizontal:

0a 1a aa ba
0b 1b ab bb
01 11 a1 b1
00 10 a0 b0

as opposed to the earlier

0b 1b ab bb
0a 1a aa ba
01 11 a1 b1
00 10 a0 b0
).
this had the pleasing effect that
*all* the “lines” now had naked-eye
symmetry. the earlier drafts had
some twisty-looking “lines” once
the alphas and betas got involved;
this was in some part due to the
artificial “symmetry breaking” that
i’d indulged in by *putting alpha
first*. the new version had beta
close to the origin just as often
as alpha… which better describes
their relation in F_4.

anyhow, the second, more radical
“tweak” involved swapping (0,0)
and (1,1)… and causing “lines”
actually *looking like lines* to
appear as *broken* lines
(but simultaneously causing the
alpha-points and beta-points
to behave better still).

this looks like a pretty good board
for pee-two-eff-four tic-tac-toe.
(or, ahem, “21-Point Vlorbik”…
he who blowet not his own horn,
that one’s horn shall never be blown).

P_2(\Bbb F_4) Tic Tac Toe
Official Rules (all rights reserved).

“The Board” is the 21-line
array of letters:

ABCDE
AFGHI
AJKLM
ANOPQ
ARSTU
BFJNR
BGKOS
BHLPT
BIMQU
CFKPU
CGJQT
CHMNS
CILOR
DFLQS
DGMPR
DHJOU
DIKNT
EFMOT
EFLNU
EHKQR
EIJPS
.
two players alternate turns.
in the first turn the first player
chooses any letter from A to U
and “colors” all five copies of
that letter on the board;
the second player then chooses
any *other* letter and “colors”
all five copies (in some other
color… X’s and O’s can be made
to do in a pinch if colored pencils
aren’t available).

players continue alternating turns,
each coloring all five copies of
some previously uncolored letter
at each turn.

play ends if one player… the winner…
has colored *all five letters* of any row.
otherwise play continues until all (21)
letters are used.

if one player… the winner… now has more
“four in a row” lines than the other, so be it;
otherwise the game ends in a draw.

i actually played this for the first time
this morning on the bus with madeline:
a four-to-four draw.

in the thus-far-imaginary computer version,
upon selection of a letter, five dots…
all the same relative position in their
respective “crosses”… light up and
“five in a row” becomes “an entire
cross lights up”. this might be worth
learning to write the code for.

better if somebody else did it, though,
i imagine. i just want a “game designer”
credit and a small piece of every one sold.


i’ve finally put the “points at infinity”
where they’ve belonged all along.
this’ll probably be the last draft
as far as “where do the points go”.

the color scheme is still almost
entirely up in the air. all i know
so far is that nobody not already
interested in maths will look
at the damn thing in b&w.

:)


i did this before rearranging the points.
now i suppose i oughta do it again.
anyhow there sits the plane on
the field with *two* elements
in red right inside the plane
on the field with *four* elements.


another microzine 8-pager
(one side of a single sheet
of typing paper, cut & folded).

pages 6 & 7 are this new version
of P^2({\Bbb F}_4)^*. i’ve used
the (more “obvious”) ordering
00 01 0a 0b
10 11 1a 1b
a0 a1 aa ab
b0 b1 ba bb
on the “finite points”;
last time i posted a drawing
of this space
i used the weird
aa ab ba bb
a0 a1 b0 b1
0a 0b 1a 1b
00 01 10 11
.
it seemed like a good idea
at the time. part of the point
is that there *are* different
ways to go about putting in
a co-ordinate system.
but mostly i just wanted
to see how it would look.


new drawing today.

you can check my work…
as of course i already *have*:
start with any “cross”, C, and
verify on five cross-diagrams
(by shading one “circle” on each…
namely the one whose position
corresponds to that of C
in the “big picture”) that
the “Point-at-C” belongs
to each of the “Lines-through-C”.

i’ll edit in some algebra; probably soon.
but the point now is that i *don’t* need
to refer to any but purely *visual*
calculations.

somewhat to my surprise, i’ve decided
in middle age to become much more
of a “visual learner”. in my case, this
amounts to “how can i represent results
from abstract algebra as pictures?”.
it still counts.


not a projective space though
(there are “parallel lines”, for example).

ten “lines” through ten “points”.
you can get this by using
2-element subsets of {0,1,2,3,4} as points.
the lines are then triples
(like {0,1}–{1,2}–{0,2})
such that each point of the triple
is the symmetric difference
(A \Delta B = (A \cap B') \cup (A' \cap B))
of the other two.

the point associated to {0,1}–{1,2}–{0,2}
in the points-to-lines correspondence
i’ve illustrated here is then {3,4}…
the complement of the union of
the three points of the line.

so you can re-create this at will.
you just have to fiddle out a
nice symmetric version of the picture.

anyhow. pick a big circle.
look at the *three* big circles
matching the dark dots inside.
the three “lines” meet at the
original point.

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.


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 P_2({\Bbb F}_3),
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:
\{-\infty\} \cup {\Bbb R} \cup \{\infty\}, 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…

Photo on 2011-02-07 at 14.40

here’s MEdZ #1 in both editions: last year’s “mini”
and this years “digest” sizes. the 2011 edition
features, along with “the hip-pocket vocab”
(a glossary for math-for-humanities), some
remixed drawings from some “micro” zines
(also from last year), along with some (new,
brief) handwritten commentary.

the seven-sections pictures look way better
cut together (so here *that* is).

Photo on 2011-02-07 at 14.44

this last one’s previously unpublished.

Photo on 2011-02-07 at 14.45

anyhow. it exists. there’s a even a (proof) copy in circulation.
mostly it’s just masters, though. i’ll be running off the first
big print run in the next few days i think. numbered & dated.

Photo on 2011-01-31 at 19.49

so here’s my latest version of P_2({\Bbb F}_3),
the two-dimensional projective space constructed on
the field of three elements.

and the story. there are thirteen “windows”.
through each window, one sees a “line”.
each line is associated with four windows;
these in their turn, upon “looking through”,
show the four lines through the original window.

i went through essentially the same explanation
(on an earlier drawing of a different space) here
(and left a bunch of footnotes).

my post, “some finite projective spaces”
consists of a bunch of photos of zines with pictures
of projective spaces in them; some of these could
be considered “drafts” of this version of P_2(F_2).

i’m trying to make stuff simple where possible.
there’s been some progress since *this*:

Photo on 2011-01-31 at 20.08