Archive for the ‘Projective Spaces’ Category

more (08/15).

longer and worse (10/16).

Photo on 2012-05-23 at 11.19, originally uploaded by vlorbik.

so here’s how to memorize the alphabetic
version of “ten point reverse tic-tac-toe”.
just learn to draw half an icosa and fill
in the letters: A in the middle, BC,
DE, FG around the edge, and HIJ
back the other way. simple.

(the “poles” are then in the rather
disturbing order JIHGEFDCBA.
something fishy here.)

with “pairs of opposite faces” of an icosahedron as “points”, the ten “lines” of the
desargues configuration have the
very pleasant form seen here.

stand by for another post on this.

ten poles, ten polars,
and ten pairs-of-triangles:
ten ways to use one drawing
(from MathEdZine, of course)
to illustrate one theorem.

(desargues’ theorem at w’edia.)

performing calculations using
actual blobs of color rather than
alphabetical symbols *standing*
for colors is time-consuming
(and demanding of special tools)…
but one literally “sees” certain things
*much more readily* than with
symbol manipulation.

hey, i’m a visual learner.

the three directions associated
with “arrows’… 001, 010, and 100
in the binary code… are here
colored yellow, red, and blue
(it didn’t reproduce as well as
one would’ve liked).

these so-called “primary” colors
(for pigments; for colored lights
the rules are different) blend
along the appropriate cross-sections
of the diagram (the easy-to-see
ones with the arrows attached)…
*or* the sides of the triangle:
yellow and red form orange,
yellow and blue form green,
red and blue form purple.

blending *opposite* colors…
the primary/”secondary” pairs
yellow/purple, red/green, blue/orange…
gives a muddy brown (“mahogany”
sez crayola) and again the appropriate
cross-sections of the cube (and lines
of the “fano plane” triangle) associate
the appropriate “blends” with the
pairs-of-pigments combining to form them.

finally, there’s the weird cross-section
(the 3-5-6 arc in the fano diagram)
consisting of all three secondaries.
on the cube diagram, this appears
as a tetrahedron-through-the-origin
rather than a plane-through-the-origin.

colored pencils (with erasers) rock.
we now return to zeerox-friendly B&W.

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:

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.